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passes through the point 32. Vertices: foci: 4, 0, 4, 10 foci: 3, 1, 3, 1 0, 5 5, 4 33. Vertices: 0, 4, 0, 0; 34. Vertices: passes through the point 1, 2, 1, 2; passes through the point 5, 1 0, 5 asymptotes: y 4 x 35. Vertices: 36. Vertices: asymptotes: 37. Vertices: 38. Vertices: 1, 2, 3, 2; y x, 3, 0, 3, 6; y 6 x, 0, 2, 6, 2; y 2 3, 0, 3, 4; y 2 asymptotes: 3 x, y 4 2 3x asymptotes: 3 x, y 4 2 3x y x 39. Art A sculpture has a hyperbolic cross section (see figure). y (−2, 13) 16 (2, 13) (−1, 0) −3 −2 8 4 −4 −8 (1, 0) x 2 3 4 (−2, − 13) −16 (2, −13) (a) Write an equation that models the curved sides of the sculpture. (b) Each unit in the coordinate plane represents 1 foot. Find the width of the sculpture at a height of 5 feet. 40. Sound Location You and a friend live 4 miles apart (on the same “east-west” street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.) 3300, 0 3300, 1100, 41. Sound Location Three listening stations located at 3300, 0, monitor an and explosion. The last two stations detect the explosion 1 second and 4 seconds after the first, respectively. Determine the coordinates of the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 100 feet per second.) Model It 42. LORAN Long distance radio navigation for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on |
a hyperbola having the transmitting stations as foci. Assume that two stations, 300 miles apart, are positioned on the rectangular coordinate system at points with coordinates and 150, 0, and that a ship is traveling on a hyperbolic path with coordinates 150, 0 (see figure). x, 75 y 100 50 Station 2 −150 −50 50 −50 Not drawn to scale Station 1 x 150 Bay x (a) Find the -coordinate of the position of the ship if the time difference between the pulses from the transmitting stations is 1000 microseconds (0.001 second). (b) Determine the distance between the ship and station 1 when the ship reaches the shore. (c) The ship wants to enter a bay located between the two stations. The bay is 30 miles from station 1. What should the time difference be between the pulses? (d) The ship is 60 miles offshore when the time difference in part (c) is obtained. What is the position of the ship? 333202_1004.qxd 12/8/05 9:03 AM Page 762 762 Chapter 10 Topics in Analytic Geometry 43. Hyperbolic Mirror A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at a focus will be reflected to the other focus. The focus of a hyperbolic mirror (see figure) has coordinates Find the vertex of the mirror if the mount at the top edge of the mirror has coordinates 24, 24. 24, 0. y (24, 24) x (24, 0) − ( 24, 0) 44. Running Path Let represent a water fountain located in a city park. Each day you run through the park along a path given by 0, 0 x 2 y 2 200x 52,500 0 where and are measured in meters. x y (a) What type of conic is your path? Explain your reasoning. (b) Write the equation of the path in standard form. Sketch a graph of the equation. (c) After you run, you walk to the water fountain. If you how far must you walk 100, 150, stop running at for a drink of water? In Exercises 45– 60, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60 |
. x 2 y 2 6x 4y 9 0 x 2 4y 2 6x 16y 21 0 4x 2 y 2 4x 3 0 y 2 6y 4x 21 0 y 2 4x 2 4x 2y 4 0 x 2 y 2 4x 6y 3 0 x 2 4x 8y 2 0 4x 2 y 2 8x 3 0 4x 2 3y 2 8x 24y 51 0 4y 2 2x 2 4y 8x 15 0 25x 2 10x 200y 119 0 4y 2 4x 2 24x 35 0 4x 2 16y 2 4x 32y 1 0 2y 2 2x 2y 1 0 100x 2 100y 2 100x 400y 409 0 4x 2 y 2 4x 2y 1 0 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 61 and 62, determine whether 61. In the standard form of the equation of a hyperbola, the the larger the eccentricity of the to a, b larger the ratio of hyperbola. 62. In the standard form of the equation of a hyperbola, the trivial solution of two intersecting lines occurs when b 0. 63. Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form. 64. Writing Explain how the central rectangle of a hyperbola can be used to sketch its asymptotes. 65. Think About It Change the equation of the hyperbola so that its graph is the bottom half of the hyperbola. 9x 2 54x 4y 2 8y 41 0 66. Exploration A circle and a parabola can have 0, 1, 2, 3, or 4 points of intersection. Sketch the circle given by x 2 y 2 4. Discuss how this circle could intersect a parabola with an equation of the form Then C find the values of for each of the five cases described below. Use a graphing utility to verify your results. y x 2 C. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection Skills Review In Exercises 67–72, factor the polynomial completely. 67. 68. 69. 70. 71. 72. x3 16x x2 14x 49 2x3 24x |
2 72x 6x3 11x2 10x 16x3 54 4 x 4x 2 x3 In Exercises 73–76, sketch a graph of the function. Include two full periods. 73. 74. y 2 cos x 1 y sin x y tan 2x 75. 76. y 1 2 sec x 333202_1005.qxd 12/8/05 9:04 AM Page 763 10.5 Rotation of Conics Section 10.5 Rotation of Conics 763 What you should learn • Rotate the coordinate axes -term in to eliminate the equations of conics. • Use the discriminant to xy classify conics. Why you should learn it As illustrated in Exercises 7–18 on page 769, rotation of the coordinate axes can help you identify the graph of a general second-degree equation. Rotation In the preceding section, you learned that the equation of a conic with axes parallel to one of the coordinate axes has a standard form that can be written in the general form Ax 2 Cy 2 Dx Ey F 0. Horizontal or vertical axis In this section, you will study the equations of conics whose axes are rotated so x that they are not parallel to either the -axis or the -axis. The general equation for such conics contains an -term. xy y Ax 2 Bxy Cy 2 Dx Ey F 0 Equation in xy -plane xy -term, you can use a procedure called rotation of axes. The To eliminate this objective is to rotate the - and -axes until they are parallel to the axes of the conic. The rotated axes are denoted as the -axis, as shown in Figure 10.42. -axis and the FIGURE 10.42 After the rotation, the equation of the conic in the new form xy -plane will have the Ax2 Cy2 Dx Ey F 0. Equation in xy -plane -term, you can obtain a standard form by Because this equation has no completing the square. The following theorem identifies how much to rotate the axes to eliminate the -term and also the equations for determining the new coefficients xy A, C, D, E, and F. xy Rotation of Axes to Eliminate an xy-Term The general second-degree equation Ey F 0 can be rewritten as Ax 2 Bxy Cy 2 Dx Ax2 Cy2 Dx Ey F 0 by rotating the coordinate axes through an angle |
where, cot 2 A C B. The coefficients of the new equation are obtained by making the and y x sin y cos. substitutions x x cos y sin 333202_1005.qxd 12/8/05 9:04 AM Page 764 764 Chapter 10 Topics in Analytic Geometry Remember that the substitutions x x cos y sin and y x sin y cos were developed to eliminate the xy -term in the rotated system. You can use this as a check on your work. In other words, if your final equation contains an xy -term, you know that you made a mistake. (x ′)2 2 ( 2( − (y ′)2 2 ( 2( = 1 y x ′ x 1 2 xy − 1 = 0 y ′ −2 −1 2 1 −1 -system: Vertices: xy In In xy-system: FIGURE 10.43 2, 0, 2, 0 1, 1, 1, 1 Example 1 Rotation of Axes for a Hyperbola Write the equation xy 1 0 in standard form. Solution Because A 0, B 1, and C 0, you have cot 2 A C B 0 2 2 4 which implies that 4 y sin y 1 2 4 x x cos x 1 2 x y 2 and 4 y cos y 1 2 4 y x sin x 1 2 x y. 2 -system is obtained by substituting these expressions in The equation in the the equation x y 2 xy xy 1 0. x y 2 1 0 1 0 x2 y2 2 x 2 22 y 2 2 2 1 Write in standard form. xy In the ± 2, 0, xy -system, this is a hyperbola centered at the origin with vertices at as shown in Figure 10.43. To find the coordinates of the vertices in the ± 2, 0 in the equations -system, substitute the coordinates x x y 2 and y x y. 2 1, 1 This substitution yields the vertices also that the asymptotes of the hyperbola have equations correspond to the original - and -axes. and x y in the 1, 1 xy y ±x, -system. Note which Now try Exercise 7. 333202_1005.qxd 12/8/05 9:04 AM Page 765 Section 10.5 Rotation of Conics 765 Example 2 Rotation of Axes for an |
Ellipse Sketch the graph of 7x 2 63xy 13y 2 16 0. Solution Because A 7, B 63, and C 13, you have cot 2 A C B 7 13 63 1 3 which implies that making the substitutions 6. The equation in the xy -system is obtained by 6 x x cos x3 y sin y 1 6 2 2 3x y 2 and 6 6 y cos y3 2 y x sin x1 2 x 3y 2 y ′ y 2 (x ′)2 22 + (y ′)2 12 = 1 x ′ in the original equation. So, you have 7x2 63 xy 13y2 16 0 2 73x y 2 13x 3y 2 16 0 2 633x y 2 x 3y 2 −2 −1 1 2 x which simplifies to −1 −2 7x2 − 6 3xy + 13y2 − 16 = 0 Vertices: xy In In xy-system: -system: ±2, 0, 0, ±1 3, 1, 3, 1,, 4x 2 16y2 16 0 4x 2 16y 2 16 x 2 4 x 2 22 y2 1 y2 12 1 1. Write in standard form. This is the equation of an ellipse centered at the origin with vertices the -system, as shown in Figure 10.44. xy ±2, 0 in FIGURE 10.44 Now try Exercise 13. 333202_1005.qxd 12/8/05 9:04 AM Page 766 766 Chapter 10 Topics in Analytic Geometry Example 3 Rotation of Axes for a Parabola Sketch the graph of x 2 4xy 4y 2 55y 1 0. Solution Because A 1, B 4, cot 2 A C B and 1 4 4 C 4, you have 3 4. Using this information, draw a right triangle as shown in Figure 10.45. From the figure, you can see that you can use the half-angle formulas in the forms To find the values of cos 2 3 5. cos, sin and sin 1 cos 2 2 and cos 1 cos 2. 2 So, sin 1 cos 2 2 1 3 5 2 cos 1 cos. Consequently, you use the substitutions x x cos y sin y 1 x 2 5 5 2x y 5 y x sin y cos y 2 x 1 5 5 x 2y 5. Substituting these expressions |
in the original equation, you have 2x y 2 5 42x y 5 x 2y 5 x 2 4xy 4y 2 55y 1 0 55x 2y 1 0 4x 2y 2 5 5 which simplifies as follows. 5y2 5x 10y 1 0 5y2 2y 5x 1 5y 1 2 5x 4 y 1 2 1x 4 5 The graph of this equation is a parabola with vertex to the -axis in the shown in Figure 10.46. -system, and because xy x Now try Exercise 17. Group terms. Write in completed square form. Write in standard form. 5, 1. 4 sin 15, Its axis is parallel 26.6, as 5 4 2θ 3 FIGURE 10.45 x2 − 4xy + 4y2 + 5 5y + 1 = 0 y y ′ x ′ θ ≈ 26.6° x 2 1 −1 −2 (y′ + 1)2 = (−1) x′ − 4 5 ( ) Vertex: xy In -system: In xy-system: FIGURE 10.46 5, 1 4 13 55, 6 55 333202_1005.qxd 12/8/05 9:04 AM Page 767 Section 10.5 Rotation of Conics 767 Invariants Under Rotation In the rotation of axes theorem listed at the beginning of this section, note that the constant term is the same in both equations, Such quantities are invariant under rotation. The next theorem lists some other rotation invariants. F F. Rotation Invariants The rotation of the coordinate axes through an angle equation Ax 2 Bxy Cy 2 Dx Ey F 0 Ax 2 Cy 2 Dx Ey F 0 that transforms the into the form has the following rotation invariants. 1. F F A C A C 2. 3. B 2 4AC B 2 4AC term in the xyIf there is an equation of a conic, you should realize then that the conic is rotated. Before rotating the axes, you should use the discriminant to classify the conic. You can use the results of this theorem to classify the graph of a second-term in much the same way you do for a the invari- -term. Note that because B 0, xy xy degree equation with an second-degree equation without an ant reduces to B 2 4AC B 2 4AC 4AC. Discriminant |
This quantity is called the discriminant of the equation Ax 2 Bxy Cy 2 Dx Ey F 0. Now, from the classification procedure given in Section 10.4, you know that the sign of determines the type of graph for the equation AC Ax 2 Cy 2 Dx Ey F 0. Consequently, the sign of original equation, as given in the following classification. B 2 4AC will determine the type of graph for the Classification of Conics by the Discriminant The graph of the equation is, except in degenerate cases, determined by its discriminant as follows. Ax 2 Bxy Cy 2 Dx Ey F 0 1. Ellipse or circle: 2. Parabola: 3. Hyperbola: B 2 4AC < 0 B 2 4AC 0 B 2 4AC > 0 For example, in the general equation 3x2 7xy 5y2 6x 7y 15 0 you have A 3, B 7, and B2 4AC 72 435 49 60 11. So the discriminant is C 5. Because 11 < 0, the graph of the equation is an ellipse or a circle. 333202_1005.qxd 12/8/05 9:04 AM Page 768 768 Chapter 10 Topics in Analytic Geometry Example 4 Rotation and Graphing Utilities For each equation, classify the graph of the equation, use the Quadratic Formula and then use a graphing utility to graph the equation. to solve for y, a. c. 2x2 3xy 2y2 2x 0 3x2 8xy 4y2 7 0 b. x2 6xy 9y2 2y 1 0 Solution a. Because B2 4AC 9 16 < 0, the graph is a circle or an ellipse. Solve for as follows. y 2x2 3xy 2y2 2x 0 2y2 3xy 2x2 2x 0 Write original equation. ay2 by c 0 Quadratic form 3x ± 3x2 422x2 2x 22 y 3 Graph both of the equations to obtain the ellipse shown in Figure 10.47. y 3x ± x16 7x 4 −1 5 3x x16 7x 4 3x x16 7x 4 y1 y2 Top half of ellipse Bottom half of ellipse −1 FIGURE 10.47 4 0 0 FIGURE 10.48 −15 FIGURE 10.49 10 −10 b. Because B2 |
4AC 36 36 0, x2 6xy 9y2 2y 1 0 9y2 6x 2y x2 1 0 the graph is a parabola. Write original equation. Quadratic form ay2 by c 0 y 6x 2 ± 6x 22 49x2 1 29 Graphing both of the equations to obtain the parabola shown in Figure 10.48. c. Because B2 4AC 64 48 > 0, the graph is a hyperbola. 3x2 8xy 4y2 7 0 4y2 8xy 3x2 7 0 Write original equation. Quadratic form ay2 by c 0 y 8x ± 8x2 443x2 7 24 The graphs of these two equations yield the hyperbola shown in Figure 10.49. Now try Exercise 33. 6 15 W RITING ABOUT MATHEMATICS Classifying a Graph as a Hyperbola In Section 2.6, it was mentioned that the graph of is a hyperbola. Use the techniques in this section to verify this, and justify each step. Compare your results with those of another student. f x 1x 333202_1005.qxd 12/8/05 9:04 AM Page 769 10.5 Exercises Section 10.5 Rotation of Conics 769 VOCABULARY CHECK: Fill in the blanks. 1. The procedure used to eliminate the xy- term in a general second-degree equation is called ________ of ________. 2. After rotating the coordinate axes through an angle, the general second-degree equation in the new xy- plane will have the form ________. 3. Quantities that are equal in both the original equation of a conic and the equation of the rotated conic are ________ ________ ________. B 2 4AC 4. The quantity is called the ________ of the equation Ax 2 Bxy Cy 2 Dx Ey F 0. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. xy degrees from the In Exercises 1–6, the -coordinate system has been -coordinate system. The rotated xy -coordinate system are coordinates of a point in the xy given. Find the coordinates of the point in the rotated coordinate system. 90, 0, 3 30, 1, 3 45, 2, 1 45, 3, |
3 60, 3, 1 30, 2, 4 3. 1. 6. 5. 4. 2. 22. 23. 24. 25. 26. 40x 2 36xy 25y 2 52 32x 2 48xy 8y 2 50 24x2 18xy 12y2 34 4x 2 12xy 9y 2 413 12x 613 8y 91 6x2 4xy 8y2 55 10x 75 5y 80 In Exercises 27–32, match the graph with its equation. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y y′ (b) y y ′ In Exercises 7–18, rotate the axes to eliminate the -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. xy 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. xy 1 0 xy 2 0 x 2 2xy y 2 1 0 xy x 2y 3 0 xy 2y 4x 0 2x 2 3xy 2y 2 10 0 5x 2 6xy 5y 2 12 0 13x 2 63 xy 7y 2 16 0 3x 2 23 xy y 2 2x 23 y 0 16x 2 24xy 9y 2 60x 80y 100 0 9x 2 24xy 16y 2 90x 130y 0 9x 2 24xy 16y 2 80x 60y 0 −2 −3 y′ y 3 (c) −3 −2 3 2 x′ x 3 −3 (d) y y′ x′ x 3 x x′ x′ x 1 3 4 x′ −3 −4 y 4 2 In Exercises 19–26, use a graphing utility to graph the conic. Determine the angle through which the axes are rotated. Explain how you used the graphing utility to obtain the graph. (e) y y′ x′ (f) y′ 19. 20. 21. x 2 2xy y 2 20 x 2 4xy 2y 2 6 17x 2 32xy 7y 2 75 −4 −2 −2 −4 x −4 −2 2 4 x −2 −4 333202_1005.qxd 12/8/05 11:07 AM Page 770 770 Chapter 10 Topics in Analytic Geometry |
27. 28. 29. 30. 31. 32. xy 2 0 x 2 2xy y 2 0 2x 2 3xy 2y 2 3 0 x 2 xy 3y 2 5 0 3x 2 2xy y 2 10 0 x 2 4xy 4y 2 10x 30 0 In Exercises 33– 40, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation. y, 33. 34. 35. 36. 37. 38. 39. 40. 16x 2 8xy y 2 10x 5y 0 x 2 4xy 2y 2 6 0 12x 2 6xy 7y 2 45 0 2x 2 4xy 5y 2 3x 4y 20 0 x 2 6xy 5y 2 4x 22 0 36x 2 60xy 25y 2 9y 0 x 2 4xy 4y 2 5x y 3 0 x 2 xy 4y 2 x y 4 0 In Exercises 41– 44, sketch (if possible) the graph of the degenerate conic. 41. 42. 43. 44. y 2 9x 2 0 x 2 y 2 2x 6y 10 0 x 2 2xy y 2 1 0 x 2 10xy y 2 0 53. 54. 55. 56. 57. 58. x 2 y2 4 0 3x y 2 0 4x 2 9y2 36y 0 x 2 9y 27 0 x 2 2y2 4x 6y 5 0 x y 4 0 x 2 2y2 4x 6y 5 0 x 2 4x y 4 0 xy x 2y 3 0 x 2 4y2 9 0 5x 2 2xy 5y 2 12 0 x y 1 0 Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 59 and 60, determine whether 59. The graph of the equation x2 xy ky2 6x 10 0 where k is any constant less than 1 4, is a hyperbola. 60. After a rotation of axes is used to eliminate the xy -term from an equation of the form Ax2 Bxy Cy2 Dx Ey F 0 the coefficients of the respectively. x2 - and y2 -terms remain A and C, In Exercises 45–58, find any points of intersection of the graphs algebraically and then |
verify using a graphing utility. 61. Show that the equation x 2 y 2 r 2 45. 46. 47. 48. 49. 50. 51. 52. x 2 y2 4x 6y 4 0 x 2 y 2 4x 6y 12 0 x 2 y 2 8x 20y 7 0 x 2 9y2 8x 4y 7 0 4x 2 y 2 16x 24y 16 0 4x 2 y2 40x 24y 208 0 x 2 4y2 20x 64y 172 0 16x 2 4y 2 320x 64y 1600 0 x 2 y 2 12x 16y 64 0 x 2 y 2 12x 16y 64 0 x 2 4y 2 2x 8y 1 0 x 2 2x 4y 1 0 16x 2 y 2 24y 80 0 16x 2 25y 2 400 0 16x 2 y 2 16y 128 0 y 2 48x 16y 32 0 is invariant under rotation of axes. 62. Find the lengths of the major and minor axes of the ellipse graphed in Exercise 14. Skills Review In Exercises 63–70, graph the function. 63. 65. 67. 69. f x x 3 gx 4 x2 ht t 23 3 f t t 5 1 64. 66. 68. 70. f x x 4 1 gx 3x 2 ht 1 t 43 2 f t 2t 3 In Exercises 71–74, find the area of the triangle. 71. 72. C 110, a 8, b 12 B 70, a 25, c 16 a 11, b 18, c 10 73. 74. a 23, b 35, c 27 333202_1006.qxd 12/8/05 9:05 AM Page 771 10.6 Parametric Equations Section 10.6 Parametric Equations 771 What you should learn • Evaluate sets of parametric equations for given values of the parameter. • Sketch curves that are represented by sets of parametric equations. • Rewrite sets of parametric equations as single rectangular equations by eliminating the parameter. • Find sets of parametric equations for graphs. Why you should learn it Parametric equations are useful for modeling the path of an object. For instance, in Exercise 59 on page 777, you will use a set of parametric equations to model the path of a baseball. Jed Jacobsohn/Getty Images Plane Curves x y. Up to this point you have been representing |
a graph by a single equation involving the two variables and In this section, you will study situations in which it is useful to introduce a third variable to represent a curve in the plane. To see the usefulness of this procedure, consider the path followed by an object that is propelled into the air at an angle of If the initial velocity of the object is 48 feet per second, it can be shown that the object follows the parabolic path y x2 72 Rectangular equation 45. x as shown in Figure 10.50. However, this equation does not tell the whole story. Although it does tell you where the object has been, it doesn’t tell you when the object was at a given point on the path. To determine this time, you can introduce a third variable, called a parameter. It is possible to write both and y to obtain the parametric equations x, y x t t as functions of x 242t y 16t 2 242t. Parametric equation for x Parametric equation for y From this set of equations you can determine that at time the point Similarly, at time 242, 242 16, and so on, as shown in Figure 10.50. 0, 0. t 1, t 0, the object is at the object is at the point Rectangular equation: − x2 72 + = y x Parametric equations: x t = 24 2 2 − = 16 + 24 2 y t t y 18 9 t = 3 2 4 (36, 18) (0, 0) t = 3 2 2 (72, 0) t = 0 9 18 27 36 45 54 63 72 81 x Curvilinear Motion: Two Variables for Position, One Variable for Time FIGURE 10.50 For this particular motion problem, t, and the resulting path is a plane curve. (Recall that a continuous function is one whose graph can be traced without lifting the pencil from the paper.) are continuous functions of and y x Definition of Plane Curve f If and are continuous functions of on an interval pairs The equations C. t g ft, gt x ft is a plane curve y gt and I, the set of ordered are parametric equations for C, and t is the parameter. 333202_1006.qxd 12/8/05 9:05 AM Page 772 772 Chapter 10 Topics in Analytic Geometry Sketching a Plane Curve When sketching a curve represented by a pair of parametric equations, you still plot points in the is determined from a |
value chosen for the parameter Plotting the resulting points in the order of increasing values of traces the curve in a specific direction. This is called the orientation of the curve. -plane. Each set of coordinates x, y xy t. t Example 1 Sketching a Curve Sketch the curve given by the parametric equations x t2 4 and y t 2, 2 ≤ t ≤ 3. Solution Using values of shown in the table. t in the interval, the parametric equations yield the points x, y y y 6 4 2 −2 −4 6 4 2 −2 −1 FIGURE 10.51 FIGURE 10.52 x = t2 − 2 −2 ≤ t ≤ 3 x = 4t2 − 1 − 12 0 12 1 32 C By plotting these points in the order of increasing shown in Figure 10.51. Note that the arrows on the curve indicate its orientation t to 3. So, if a particle were moving on this curve, it would as increases from start at and then move along the curve to the point you obtain the curve 0, 1 5, 3 2. t, 2 Now try Exercises 1(a) and (b). Note that the graph shown in Figure 10.51 does not define as a function of This points out one benefit of parametric equations—they can be used to x. represent graphs that are more general than graphs of functions. y It often happens that two different sets of parametric equations have the same graph. For example, the set of parametric equations x 4t2 4 and y t, 1 ≤ t ≤ 3 2 t has the same graph as the set given in Example 1. However, by comparing the values of in Figures 10.51 and 10.52, you see that this second graph is traced out more rapidly (considering as time) than the first graph. So, in applications, different parametric representations can be used to represent various speeds at which objects travel along a given path. t 333202_1006.qxd 12/8/05 9:05 AM Page 773 Section 10.6 Parametric Equations 773 Eliminating the Parameter Example 1 uses simple point plotting to sketch the curve. This tedious process can sometimes be simplified by finding a rectangular equation (in and ) that has the same graph. This process is called eliminating the parameter. y x Parametric equations x t2 4 y t2 Solve for t in one equation. t 2y Substitute in other equation. Rectangular |
equation x 2y2 4 x 4y2 4 Now you can recognize that the equation a horizontal axis and vertex 4, 0. x 4y2 4 represents a parabola with When converting equations from parametric to rectangular form, you may need to alter the domain of the rectangular equation so that its graph matches the graph of the parametric equations. Such a situation is demonstrated in Example 2. Example 2 Eliminating the Parameter Sketch the curve represented by the equations x 1 t 1 and y t t 1 by eliminating the parameter and adjusting the domain of the resulting rectangular equation. in the equation for produces x x2 1 t 1 Solution Solving for t x 1 t 1 which implies that t 1 x2 x2. Now, substituting in the equation for 1 x2 x2 1 x2 x2 1 y, you obtain the rectangular equation 1 x2 x2 1 x2 x2 1 x2. x2 x2 1 From this rectangular equation, you can recognize that the curve is a parabola that opens downward and has its vertex at Also, this rectangular equation but from the parametric equation for you can see is defined for all values of that the curve is defined only when This implies that you should restrict the domain of t > 1. to positive values, as shown in Figure 10.53. 0, 1. x, x x Exploration Most graphing utilities have a parametric mode. If yours does, enter the parametric equations from Example 2. Over what values should you let vary to obtain the graph shown in Figure 10.53? t Parametric equations1 −2 −3 − t = 0.75 −2 − FIGURE 10.53 Now try Exercise 1(c). 333202_1006.qxd 12/8/05 9:05 AM Page 774 774 Chapter 10 Topics in Analytic Geometry It is not necessary for the parameter in a set of parametric equations to represent time. The next example uses an angle as the parameter. To eliminate the parameter in equations involving trigonometric functions, try using the identities sin2 cos2 1 sec2 tan2 1 as shown in Example 3. Example 3 Eliminating an Angle Parameter Sketch the curve represented by x 3 cos and y 4 sin, 0 ≤ ≤ 2 by eliminating the parameter. Solution Begin by solving for cos and sin in the equations. 3 2 1 −1 −2 −3 θ = π −4 −2 − cos x 3 and sin y 4 Solve for |
cos and sin. Use the identity sin2 cos2 1 to form an equation involving only and x y cos y = 4 sin θ θ cos2 sin2 1 y x 4 3 1 2 2 x2 9 y2 16 1 Pythagorean identity Substitute x 3 for cos and for sin. y 4 Rectangular equation 0, 0, From this rectangular equation, you can see that the graph is an ellipse centered as at shown in Figure 10.54. Note that the elliptic curve is traced out counterclockwise as varies from 0 to and minor axis of length with vertices 0, 4 2b 6, 0, 4 2. and FIGURE 10.54 Now try Exercise 13. In Examples 2 and 3, it is important to realize that eliminating the parameter is primarily an aid to curve sketching. If the parametric equations represent the path of a moving object, the graph alone is not sufficient to describe the object’s motion. You still need the parametric equations to tell you the position, direction, and speed at a given time. Finding Parametric Equations for a Graph You have been studying techniques for sketching the graph represented by a set of parametric equations. Now consider the reverse problem—that is, how can you find a set of parametric equations for a given graph or a given physical description? From the discussion following Example 1, you know that such a representation is not unique. That is, the equations 1 ≤ t ≤ 3 2 x 4t2 4 y t, and produced the same graph as the equations x t2 4 and y t 2, 2 ≤ t ≤ 3. This is further demonstrated in Example 4. 333202_1006.qxd 12/8/05 9:05 AM Page 775 x = 1 − t y = 2t − 2 t = 0 x 2 −1 −2 −3 t = −1 t = 3 FIGURE 10.55 PD In Example 5, represents the arc of the circle between.D points and P Te c h n o l o g y Use a graphing utility in parametric mode to obtain a graph similar to Figure 10.56 by graphing the following equations. X1T Y1T T sin T 1 cos T Section 10.6 Parametric Equations 775 Example 4 Finding Parametric Equations for a Graph Find a set of parametric equations to represent the graph of the following parameters. t 1 x t x b. a. y 1 x 2, |
using Solution a. Letting x t t x, you obtain the parametric equations and t 1 x, y 1 x 2 1 t 2. you obtain the parametric equations b. Letting x 1 t and y 1 x2 1 1 t2 2t t 2. In Figure 10.55, note how the resulting curve is oriented by the increasing values of For part (a), the curve would have the opposite orientation. t. Now try Exercise 37. Example 5 Parametric Equations for a Cycloid Describe the cycloid traced out by a point on the circumference of a circle of radius as the circle rolls along a straight line in a plane. P a Solution As the parameter, let be the measure of the circle’s rotation, and let the point P x, y 0, is at a maximum point is back on the -axis at and when 2a, 0. So, you have begin at the origin. When APC 180. is at the origin; when P 2,, x P P a, 2a; From Figure 10.56, you can see that sin sin180 sinAPC AC a BD a cos cos180 cosAPC AP a BD a sin. and a. OD PD AP a cos which implies that x along the -axis, you know that DC a, you have Because the circle rolls BA Furthermore, because x OD BD a a sin So, the parametric equations are and x a sin y BA AP a a cos. y a1 cos. and Cycloid: x = a( − sin ), y = a(1 − cos ) θ θ (3 a, 2a) π θ y 2a a P = (x, y) π ( a, 2a2 a, 0) π 3 a π (4 a, 0) x FIGURE 10.56 Now try Exercise 63. 333202_1006.qxd 12/8/05 11:08 AM Page 776 776 Chapter 10 Topics in Analytic Geometry 10.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. If and are continuous functions of on an interval t x f t ________ ________ The equations and C. g f I, y gt the set of ordered pairs are ________ equations for f t, gt C, is a and t is the ________. 2. The ________ of a curve is the direction in which the curve |
is traced out for increasing values of the parameter. 3. The process of converting a set of parametric equations to a corresponding rectangular equation is called ________ the ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1. Consider the parametric equations x t x (a) Create a table of - and -values using and t 0, y 3 t. 1, 2, 3, y and 4. (b) Plot the points x, y graph of the parametric equations. generated in part (a), and sketch a (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ? 2. Consider the parametric equations x 4 cos2 and y 2 sin. (a) Create a table of 4, 4, (b) Plot the points x - and -values using 2. and x, y generated in part (a), and sketch a graph of the parametric equations. 2, 0, y (c) Find the rectangular equation by eliminating the parameter. Sketch its graph. How do the graphs differ? In Exercises 3–22, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. 3. 5. 7. 9. 11. 13. x 3t 3 y 2t 1 x 1 4t 2t 1 y t 2 x 3 cos y 3 sin 4. 6. 8. 10. 12. 14. x 3 2t y 2 3t cos y 3 sin 15. 17. 19. 21. x 4 sin 2 y 2 cos 2 x 4 2 cos y 1 sin x et y e3t x t3 y 3 ln t 16. 18. 20. 22. x cos y 2 sin 2 x 4 2 cos y 2 3 sin x e2t y et x ln 2t y 2t 2 In Exercises 23 and 24, determine how the plane curves differ from each other. 23. (a) (c) 24. (a) (c) x t y 2t 1 x et y 2et 1 x t y t 2 1 x sin t y sin2 t 1 (b) (d) (b) (d) x cos y 2 cos 1 x |
et y 2et 1 x t 2 y t 4 1 x et y e2t 1 In Exercises 25 –28, eliminate the parameter and obtain the standard form of the rectangular equation. x2, y2 x1, y1 25. Line through, tx2 x1 x x1 x h r cos, 26. Circle: x h a cos, 27. Ellipse: and y y1 : t y2 y1 y k r sin y k b sin 28. Hyperbola: x h a sec, y k b tan 29. Line: passes through In Exercises 29–36, use the results of Exercises 25–28 to find a set of parametric equations for the line or conic. 0, 0 2, 3 radius: 4 30. Line: passes through 3, 2; 6, 3 6, 3 31. Circle: center: and and 333202_1006.qxd 12/8/05 9:05 AM Page 777 32. Circle: center: 3, 2; radius: 5 33. Ellipse: vertices: 34. Ellipse: vertices: ±4, 0; foci: 4, 7, 4, 3; ±3, 0 foci: 35. Hyperbola: vertices: 36. Hyperbola: vertices: (4, 5, 4, 1 ±4, 0; ±2, 0; foci: foci: ±5, 0 ±4, 0 Section 10.6 Parametric Equations 777 53. Lissajous curve: x 2 cos, y sin 2 54. Evolute of ellipse: 55. Involute of circle: 56. Serpentine curve: y 6 sin3 x 4 cos3, cos sin x 1 2 sin cos y 1 2 x 1 2 cot, y 4 sin cos In Exercises 37– 44, find a set of parametric equations for t 2 x. the rectangular equation using (a) and (b) t x 37. 39. 41. 43. y 3x 2 y x 2 y x2 1 y 1 x 38. 40. 42. 44. x 3y 2 y x3 y 2 x y 1 2x v0 Projectile Motion A projectile is launched at a height of h feet above the ground at an angle of with the horizontal. The initial velocity is feet per second and the path of the projectile is modeled by the parametric equations x v0 |
cos t y h v0 sin t 16t 2. In Exercises 57 and 58, use a graphing utility to graph the paths of a projectile launched from ground level at each value of and For each case, use the graph to approximate the maximum height and the range of the projectile. and v0. In Exercises 45–52, use a graphing utility to graph the curve represented by the parametric equations. x 4 sin, x sin, y 41 cos 45. Cycloid: 46. Cycloid: 47. Prolate cycloid: 48. Prolate cycloid: y 1 cos x 3 2 sin, x 2 4 sin, y 1 3 2 cos y 2 4 cos 49. Hypocycloid: x 3 cos3, y 3 sin3 50. Curtate cycloid: 51. Witch of Agnesi: 52. Folium of Descartes: y 8 4 cos x 8 4 sin, x 2 cot, x 3t 1 t 3, y 2 sin2 y 3t 2 1 t 3 In Exercises 53–56, match the parametric equations with the correct graph and describe the domain and range. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 2 1 −2 −1 −1 x 1 2 y 5 (b) (d) y 2 1 1 −1 −1 −2 y 4 x 5 −5 −5 −4 2 −4 x x 57. (a) (b) (c) (d) 58. (a) (b) (c) (d) 60, 60, 45, 45, 15, 15, 30, 30, v0 v0 v0 v0 v0 v0 v0 v0 88 132 88 132 60 100 60 100 feet per second feet per second feet per second feet per second feet per second feet per second feet per second feet per second Model It 59. Sports The center field fence in Yankee Stadium is 7 feet high and 408 feet from home plate. A baseball is hit at a point 3 feet above the ground. It leaves the bat at an angle of degrees with the horizontal at a speed of 100 miles per hour (see figure). 7 ft θ 3 ft 408 ft Not drawn to scale (a) Write a set of parametric equations that model the path of the baseball. (b) Use a graphing utility to graph the path of the baseball when 15. Is |
the hit a home run? (c) Use a graphing utility to graph the path of the baseball when 23. Is the hit a home run? (d) Find the minimum angle required for the hit to be a home run. 333202_1006.qxd 12/8/05 9:05 AM Page 778 778 Chapter 10 Topics in Analytic Geometry 60. Sports An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an 10 angle of with the horizontal and at an initial speed of 240 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utility to graph the path of the arrow and approximate its maximum height. (d) Find the total time the arrow is in the air. 61. Projectile Motion Eliminate the parameter t from the and y h v0 sin t 16t2 parametric equations x v0 cos t for the motion of a projectile to show that the rectangular equation is y 16 sec 2 x 2 tan x h. 2 v0 62. Path of a Projectile The path of a projectile is given by the rectangular equation y 7 x 0.02x 2. (a) Use the result of Exercise 61 to find the parametric equations of the path. h, v0,. and Find (b) Use a graphing utility to graph the rectangular equation for the path of the projectile. Confirm your answer in part (a) by sketching the curve represented by the parametric equations. (c) Use a graphing utility to approximate the maximum height of the projectile and its range. 63. Curtate Cycloid A wheel of radius units rolls along a straight line without slipping. The curve traced by a point P is called a curtate cycloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. that is units from the center b < a b a π ( a, a + b) y 2a P b θ a (0, a − b) π a π 2 a x 64. Epicycloid A circle of radius one unit rolls around the outside of a circle of radius two units without slipping. The curve traced by a point on the circumference of the |
smaller circle is called an epicycloid (see figure). Use the angle shown in the figure to find a set of parametric equations for the curve. y 4 3 1 θ 1 (x, y) 3 4 x Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 65 and 66, determine whether 65. The two y t 2 1 rectangular equation. sets of parametric and y 9t 2 1 x 3t, equations x t, have the same 66. The graph of the parametric equations x t 2 and y t 2 is the line y x. 67. Writing Write a short paragraph explaining why parametric equations are useful. 68. Writing Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve? Skills Review In Exercises 69–72, solve the system of equations. 69. 71. 11 13 5x 7y 3x y 3a 2b c 2a b 3c a 3b 9c 8 3 16 70. 72. 9 14 3x 5y 4x 2y 5u 7v 9w 4 u 2v 3w 7 8u 2v w 20 In Exercises 73–76, find the reference angle in standard position. and, and sketch 73. 75. 105 2 3 230 74. 76. 5 6 333202_1007.qxd 12/8/05 9:06 AM Page 779 10.7 Polar Coordinates Section 10.7 Polar Coordinates 779 What you should learn • Plot points on the polar coordinate system. • Convert points from rectangular to polar form and vice versa. • Convert equations from rectangular to polar form and vice versa. Why you should learn it Polar coordinates offer a different mathematical perspective on graphing. For instance, in Exercises 1–8 on page 783, you are asked to find multiple representations of polar coordinates. Introduction on the rectangular coordinate system, where and So far, you have been representing graphs of equations as collections of points x, y y represent the directed In this section, you will distances from the coordinate axes to the point study a different system called the polar coordinate system. x x, y. To form the polar coordinate system in the plane, fix a point called the an initial ray called the polar axis, as P in the plane can be assigned polar O, O pole (or origin), and construct from shown in Figure 10.57. Then each point |
coordinates r as follows. PO to r, 1. 2. directed distance from directed angle, counterclockwise from polar axis to segment OP P r= (, )θ r = directed distance θ O = directed angle Polar axis FIGURE 10.57 Example 1 Plotting Points on the Polar Coordinate System a. The point r, 2, 3 lies two units from the pole on the terminal side of the angle 3, as shown in Figure 10.58. b. The point side of the angle r, 3, 6 6, r, 3, 116 lies three units from the pole on the terminal as shown in Figure 10.59. coincides with the point 3, 6, as shown c. The point in Figure 10.60. =θ π 3 2,( θ π 6 3, −( π ) 6 π 3 2 2 3 =θ 0 π 11 6 3,( π 11 6 ) π 3 2 FIGURE 10.58 FIGURE 10.59 FIGURE 10.60 Now try Exercise 1. 333202_1007.qxd 12/8/05 9:06 AM Page 780 780 Chapter 10 Topics in Analytic Geometry Exploration Most graphing calculators have a polar graphing mode. If yours does, graph the equation r 3. (Use a setting in which 6 ≤ x ≤ 6 You should obtain a circle of radius 3. and 4 ≤ y ≤ 4.) a. Use the trace feature to cursor around the circle. Can you locate the point 3, 54? b. Can you find other polar representations of the point 3, 54? how you did it. If so, explain π 2 π 3 2 π 3, −( π ) 3 4 = −, = −3, − 3, − 4 FIGURE 10.61 ( y r θ Pole (Origin) x FIGURE 10.62 3, =... 4 θ (r, ) (x, y) y x Polar axis (x-axis) x, y In rectangular coordinates, each point has a unique representation. and represent the same point, as illustrated in Example 1. Another way r. represent This is not true for polar coordinates. For instance, the coordinates r, 2 to obtain multiple representations of a point is to use negative values for r Because the same point. In general, the point is a directed distance, the coordinates r, r, r, and r, can be represented as r, |
r, ± 2n 1 r, r, ± 2n or is any integer. Moreover, the pole is represented by 0,, where is any n where angle. Example 2 Multiple Representations of Points Plot the point point, using 3, 34 2 < < 2. and find three additional polar representations of this Solution The point is shown in Figure 10.61. Three other representations are as follows. 3, 3 4 3, 3 4 3, 3 4 2 3, 5 4 Add 2. to 3, 7 4 3, 4 Now try Exercise 3. Replace r by –r; subtract from. Replace r by –r; add. to Coordinate Conversion x x, y Moreover, for To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive -axis and the pole with the origin, as shown it follows that in Figure 10.62. Because r 2 x 2 y 2. the definitions of the trigonometric functions imply that tan y x lies on a circle of radius cos x r sin y r r > 0, and r,,,. If r < 0, you can show that the same relationships hold. Coordinate Conversion The polar coordinates as follows. r, are related to the rectangular coordinates x, y Polar-to-Rectangular x r cos y r sin Rectangular-to-Polar tan y x r 2 x 2 y 2 333202_1007.qxd 12/8/05 9:06 AM Page 781 2, 0) 1 −1 −2 FIGURE 10.63 π ) 6 ) 3 2 2 Section 10.7 Polar Coordinates 781 Example 3 Polar-to-Rectangular Conversion Convert each point to rectangular coordinates. 3, 2, b. a. 6 x Solution a. For the point r, 2,, you have the following. x r cos 2 cos 2 y r sin 2 sin 0 x, y 2, 0. (See Figure 10.63.) you have the following. b. For the point The rectangular coordinates are, r, 3, 6 33 2 31 2 x 3 cos y 3 sin 6 6 3 2 3 2 The rectangular coordinates are x, y 3 2., 3 2 Now try Exercise 13. Example 4 Rectangular-to-Polar Conversion Convert each point to polar coordinates. a. 1, 1 b. 0, 2 Solution a. For the second-quadrant point x, y 1, 1, you have 1 2 0 1 tan y x |
3. 4 Because lies in the same quadrant as 1 2 1 2 r x 2 y 2 use positive x, y, 2 r. π 2 2 (x, y) = (−1, 1) θ (r, ) = 2, ( −2 −1 FIGURE 10.64 1 π 4 )3 −1 π 2 (x, y) = (0, 2) 1 −1 −2 −1 FIGURE 10.65 θ (r, ) = 2, ( π 2 ) So, one set of polar coordinates is 10.64. r, 2, 34, as shown in Figure b. Because the point x, y 0, 2 lies on the positive -axis, choose y 1 2 0 2 and r 2. This implies that one set of polar coordinates is in Figure 10.65. r, 2, 2, as shown Now try Exercise 19. 333202_1007.qxd 12/8/05 9:06 AM Page 782 782 Chapter 10 Topics in Analytic Geometry Equation Conversion By comparing Examples 3 and 4, you can see that point conversion from the polar to the rectangular system is straightforward, whereas point conversion from the rectangular to the polar system is more involved. For equations, the opposite is x true. To convert a rectangular equation to polar form, you simply replace by r cos can be by written in polar form as follows. For instance, the rectangular equation r sin. y x 2 and y y x 2 r sin r cos 2 r sec tan Rectangular equation Polar equation Simplest form On the other hand, converting a polar equation to rectangular form requires considerable ingenuity. Example 5 demonstrates several polar-to-rectangular conversions that enable you to sketch the graphs of some polar equations. Example 5 Converting Polar Equations to Rectangular Form Describe the graph of each polar equation and find the corresponding rectangular equation. a. r 2 b. 3 c. r sec Solution a. The graph of the polar equation r 2 consists of all points that are two units from the pole. In other words, this graph is a circle centered at the origin with a radius of 2, as shown in Figure 10.66. You can confirm this by converting to rectangular form, using the relationship r 2 x 2 y 2. r 2 r 2 22 x 2 y 2 22 Polar equation Rectangular equation b. The graph of the polar equation 3 consists of all points on the line that with the positive polar axis, as shown |
in Figure 10.67. 3 makes an angle of To convert to rectangular form, make use of the relationship tan yx. 3 Polar equation tan 3 y 3x Rectangular equation c. The graph of the polar equation r sec is not evident by simple inspection, so convert to rectangular form by using the relationship r sec Polar equation r cos 1 r cos x. x 1 Rectangular equation Now you see that the graph is a vertical line, as shown in Figure 10.68. π FIGURE 10.66 π FIGURE 10.67 FIGURE 10.68 Now try Exercise 65. 333202_1007.qxd 12/8/05 11:09 AM Page 783 Section 10.7 Polar Coordinates 783 10.7 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The origin of the polar coordinate system is called the ________. 2. For the point r,, r is the ________ ________ from and is the ________ ________ counterclockwise from the polar axis to the line segment PO to OP. 3. To plot the point r,, 4. The polar coordinates use the ________ coordinate system. r, are related to the rectangular coordinates x, y as follows: x ________ y ________ tan ________ r2 ________ PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–8, plot the point given in polar coordinates and find two additional polar representations of the point, using 2 < < 2. 4, 3 0, 7 6 2, 2.36 22, 4.71 1. 3. 5. 7. 1, 3 4 5 16, 2 3, 1.57 5, 2.36 2. 4. 6. 8. In Exercises 9–16, a point in polar coordinates is given. Convert the point to rectangular coordinates. 9. 3, 2 π 2 10. 3, ( )3 π 2 11. 1, 5 4 π 2 12. 0, π 2 0 2 4 ( (r, ) = −1, θ )5 π 4 0 2 4 (r, ) = (0, − ) π θ 2, 3 4 2.5, 1.1 13. 15. 2, 7 6 8.25, 3.5 14. 16. 17. 19. In Ex |
ercises 17–26, a point in rectangular coordinates is given. Convert the point to polar coordinates. 3, 3 0, 5 3, 1 3, 1 5, 12 1, 1 6, 0 3, 4 3, 3 6, 9 22. 23. 18. 26. 25. 21. 24. 20. In Exercises 27–32, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. 3, 2 3, 2 5 2, 4 5, 2 3, 2, 32 7 4, 3 28. 32. 31. 27. 30. 29. 3 2 In Exercises 33– 48, convert the rectangular equation to polar form. Assume a > 0. 33. 35. 37. 39. 41. 43. 45. 47. x2 y2 9 y 4 x 10 3x y 2 0 xy 16 y2 8x 16 0 x2 y2 a2 x2 y2 2ax 0 40. 36. 38. 34. x2 y2 16 y x x 4a 3x 5y 2 0 2xy 1 x2 y22 9x2 y2 x2 y2 9a2 46. 48. x2 y2 2ay 0 42. 44. 333202_1007.qxd 12/8/05 9:06 AM Page 784 784 Chapter 10 Topics in Analytic Geometry In Exercises 49–64,convert the polar equation to rectangular form. 49. 51. 53. 55. 57. 59. r 4 sin 2 3 r 4 r 4 csc r2 cos r 2 sin 3 61. r 63. r 2 1 sin 6 2 3 sin 50. 52. 54. 56. 58. 60. r 2 cos 5 3 r 10 r 3 sec r 2 sin 2 r 3 cos 2 62. r 64. r 1 1 cos 6 2 cos 3 sin In Exercises 65–70, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. r 6 65. 66. r 8 3 4 r 2 csc 68. 70. 67. 6 r 3 sec 69. Synthesis In Exercises 71 and 72, determine whether True or False? the statement is true or false. Justify your answer. r, 1 for some integer represent the same point on the polar coordinate system. 2n r, 2 2 71. If then and 1 n, 72. If r1 |
r2, r2, then point on the polar coordinate system. r1, and represent the same 73. Convert the polar equation to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle. r 2h cos k sin 74. Convert the polar equation r cos 3 sin to rectangular form and identify the graph. 75. Think About It r1, 1 and is 2 r1 2 r2 r2, (a) Show that the distance between the points. 2 2 2r1r2 cos (b) Describe the positions of the points relative to each other for Simplify the Distance Formula for this case. Is the simplification what you expected? Explain. 2. 1 1 (c) Simplify the Distance Formula for 1 2 90. Is the simplification what you expected? Explain. (d) Choose two points on the polar coordinate system and find the distance between them. Then choose different polar representations of the same two points and apply the Distance Formula again. Discuss the result. 76. Exploration (a) Set the window format of your graphing utility on rectangular coordinates and locate the cursor at any position off the coordinate axes. Move the cursor horizontally and observe any changes in the displayed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor vertically. (b) Set the window format of your graphing utility on polar coordinates and locate the cursor at any position off the coordinate axes. Move the cursor horizontally and observe any changes in the displayed coordinates of the points. Explain the changes in the coordinates. Now repeat the process moving the cursor vertically. (c) Explain why the results of parts (a) and (b) are not the same. Skills Review In Exercises 77–80, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) log6 x2z 3y ln xx 42 77. 79. log4 2x y ln 5x2x2 1 78. 80. In Exercises 81–84, condense the expression to the logarithm of a single quantity. log7 x log7 3y 1 2 ln 6 ln y lnx 3 ln x lnx 2 log5 a 8 log5 x 1 81. 84. 83. 82. In Exerc |
ises 85–90, use Cramer’s Rule to solve the system of equations. 85. 87. 89. 7y y 5x 3x 3a x 2a a 2x 5x 2b b 3b y 3y 4y 11 3 c 0 3c 0 9c 8 2z z 2z 1 2 4 86. 88. 90. 10 5 9w 3w w 2x3 3x 5y 4x 2y 5u 7v 2v 2v 2x1 x2 2x2 x2 u 8u 2x1 2x1 6x3 15 7 0 4 5 2 91. In Exercises 91–94, use a determinant to determine whether the points are collinear. 4, 3, 6, 7, 2, 1 2, 4, 0, 1, 4, 5 6, 4, 1, 3, 1.5, 2.5 92. 93. 94. 2.3, 5, 0.5, 0, 1.5, 3 333202_1008.qxd 12/8/05 9:08 AM Page 785 10.8 Graphs of Polar Equations Section 10.8 Graphs of Polar Equations 785 What you should learn • Graph polar equations by point plotting. • Use symmetry to sketch graphs of polar equations. • Use zeros and maximum r-values to sketch graphs of polar equations. • Recognize special polar graphs. Why you should learn it Equations of several common figures are simpler in polar form than in rectangular form. For instance, Exercise 6 on page 791 shows the graph of a circle and its polar equation. Introduction In previous chapters, you spent a lot of time learning how to sketch graphs on rectangular coordinate systems. You began with the basic point-plotting method, which was then enhanced by sketching aids such as symmetry, intercepts, asymptotes, periods, and shifts. This section approaches curve sketching on the polar coordinate system similarly, beginning with a demonstration of point plotting. Example 1 Graphing a Polar Equation by Point Plotting Sketch the graph of the polar equation r 4 sin. Solution The sine function is periodic, so you can get a full range of -values by consid0 ≤ ≤ 2, ering values of r as shown in the following table. in the interval r 0 0 6 2 3 23 11 6 2 23 2 0 2 4 2 0 If you plot these points as shown in Figure 10 |
.69, it appears that the graph is a circle of radius 2 whose center is at the point x, y 0, 2. π 2 Circle: r = 4 sin FIGURE 10.69 Now try Exercise 21. You can confirm the graph in Figure 10.69 by converting the polar equation to rectangular form and then sketching the graph of the rectangular equation. You can also use a graphing utility set to polar mode and graph the polar equation or set the graphing utility to parametric mode and graph a parametric representation. 333202_1008.qxd 12/8/05 9:08 AM Page 786 786 Chapter 10 Topics in Analytic Geometry Symmetry the graph is traced out twice. In Figure 10.69, note that as Moreover, note that the graph is symmetric with respect to the line Had you known about this symmetry and retracing ahead of time, you could have used fewer points. increases from 0 to 2. Symmetry with respect to the line 2 is one of three important types 2 of symmetry to consider in polar curve sketching. (See Figure 10.70.) π 2 − θ (−r, − ) θ θ π )− (r, π π θ (r, ) θ 0 π π 3 2 Symmetry with Respect to the Line 2 FIGURE 10.70 π 2 π 3 2 θ (rr, ) θ 0 θ (r, − ) π )− θ (−r, θπ(r, )+ (−r, θ ) π 3 2 Symmetry with Respect to the Polar Axis Symmetry with Respect to the Pole This is Note in Example 2 that cos cos. because the cosine function is even. Recall from Section 4.2 that the cosine function is even and the sine function is odd. That is, sin sin. Tests for Symmetry in Polar Coordinates The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation. 1. The line 2: 2. The polar axis: 3. The pole: Replace Replace Replace r, r, r, by by by r, r, or r, r,. or r,. or r,. Example 2 Using Symmetry to Sketch a Polar Graph π 2 r = 3 + 2 cos θ Use symmetry to sketch the graph of r 3 2 cos |
. r, Solution Replacing So, you can conclude that the curve is symmetric with respect to the polar axis. Plotting the points in the table and using polar axis symmetry, you obtain the graph shown in Figure 10.71. This graph is called a limaçon. r 3 2 cos 3 2 cos. produces r, by FIGURE 10.71 Now try Exercise 27. 333202_1008.qxd 12/8/05 9:08 AM Page 787 Spiral of Archimedes Section 10.8 Graphs of Polar Equations 787 π 4 The three tests for symmetry in polar coordinates listed on page 786 are sufficient to guarantee symmetry, but they are not necessary. For instance, Figure 10.72 shows the graph of to be symmetric with respect to the line 2, and yet the tests on page 786 fail to indicate symmetry because neither of the following replacements yields an equivalent equation Original Equation r 2 r 2 Replacement r, r, by by r, r, New Equation r 2 r 3 The equations discussed in Examples 1 and 2 are of the form r 4 sin f sin r 3 2 cos gcos. and and The graph of the first equation is symmetric with respect to the line the graph of the second equation is symmetric with respect to the polar axis. This observation can be generalized to yield the following tests. 2, FIGURE 10.72 Quick Tests for Symmetry in Polar Coordinates 1. The graph of r f sin 2. The graph of r gcos is symmetric with respect to the line 2 is symmetric with respect to the polar axis.. Zeros and Maximum r-Values r Two additional aids to graphing of polar equations involve knowing the -values For is maximum and knowing the for which instance, in Example 1, the maximum value of and 2, this occurs when when 0. -values for which r r 4 sin is for as shown in Figure 10.69. Moreover, r 0. r 4, r 0 Example 3 Sketching a Polar Graph Sketch the graph of r 1 2 cos. Solution From the equation r 1 2 cos, you can obtain the following Symmetry: Maximum value of r : Zero of : r when With respect to the polar axis r 3 r 0 3 -values in the interval when 11 6 π5 3 π 3 2 θ π 4 3 Limaçon: r = 1 − 2 cos The table shows several corresponding points, you can sketch the graph shown |
in Figure 10.73. By plotting the 0,. 0 6 1 0.73.73 3 Note how the negative -values determine the inner loop of the graph in Figure 10.73. This graph, like the one in Figure 10.71, is a limaçon. r FIGURE 10.73 Now try Exercise 29. 333202_1008.qxd 12/8/05 9:08 AM Page 788 788 Chapter 10 Topics in Analytic Geometry Some curves reach their zeros and maximum -values at more than one r point, as shown in Example 4. Example 4 Sketching a Polar Graph Sketch the graph of r 2 cos 3. Solution Symmetry: Maximum value of r : Zeros of : r 3 0,, 2, 3 or With respect to the polar axis r 2 when 0, 3, 23, r 0 when 6, 2, 56 3 2, 32, 52 r 0 2 12 2 6 0 4 3 5 12 2 2 2 or 2 0 By plotting these points and using the specified symmetry, zeros, and maximum values, you can obtain the graph shown in Figure 10.74. This graph is called a rose curve, and each of the loops on the graph is called a petal of the rose curve. Note how the entire curve is generated as increases from 0 to. Exploration Notice that the rose curve in Example 4 has three petals. How many petals do the rose r 2 cos 4 curves given by r 2 sin 3 and have? Determine the numbers of petals for the curves given by r 2 sin n, r 2 cos n and n where is a positive integer. Te Use a graphing utility in polar mode to verify the graph of r 2 cos 3 shown in Figure 10.74. 0 ≤ ≤ 2 3 FIGURE 10.74 ≤ ≤ Now try Exercise 33 333202_1008.qxd 12/8/05 9:08 AM Page 789 Section 10.8 Graphs of Polar Equations 789 Special Polar Graphs Several important types of graphs have equations that are simpler in polar form than in rectangular form. For example, the circle r 4 sin in Example 1 has the more complicated rectangular equation x 2 y 2 2 4. Several other types of graphs that have simple polar equations are shown below. Limaçons r a ± b cos r a ± b sin a > 0, b > 0 Rose Curves n n petals if 2n petals if n ≥ 2 is odd |
, n is even Limaçon with inner loop Cardioid (heart-shaped Dimpled limaçon Convex limaçon cos n Rose curve r a cos n Rose curve r a sin n Rose curve r a sin n Rose curve Circles and Lemniscates cos Circle r a sin Circle r 2 a 2 sin 2 Lemniscate r 2 a 2 cos 2 Lemniscate 333202_1008.qxd 12/8/05 9:08 AM Page 790 790 Chapter 10 Topics in Analytic Geometry π 2 ( −3 Example 5 Sketching a Rose Curve Sketch the graph of r 3 cos 2. π (3, ) π (3, 0) 0 3 1 2 Solution Type of curve: Symmetry: r = 3 cos 2 θ π 3 2 ( −3, π 2 ) FIGURE 10.75 π 2 ( 33, π 4 ) r2 = 9 sin 2 θ π 3 2 FIGURE 10.76 Rose curve with 2n 4 petals With respect to polar axis, the line and the pole r 3 r 0 0, 2,, 32 4, 34 when when 2, Maximum value of r : Zeros of r: Using this information together with the additional points shown in the following table, you obtain the graph shown in Figure 10.75 Now try Exercise 35. Example 6 Sketching a Lemniscate Sketch the graph of r 2 9 sin 2. Solution Type of curve: Symmetry: Maximum value of r : Lemniscate With respect to the pole r 3 when 4 Zeros of : r r 0 when 0, 2 this equation has no solution points. So, you restrict the values of If sin 2 < 0, to those for which sin 2 ≥ 0. 0 ≤ ≤ 2 or ≤ ≤ 3 2 Moreover, using symmetry, you need to consider only the first of these two intervals. By finding a few additional points (see table below), you can obtain the graph shown in Figure 10.76. r ±3sin 2 0 0 12 ±3 2 4 ±3 5 12 ±3 2 2 0 Now try Exercise 39. 333202_1008.qxd 12/8/05 9:08 AM Page 791 10.8 Exercises Section 10.8 Graphs of Polar Equations 791 2. The graph of VOCABULARY CHECK: Fill in the blanks. r f sin |
1. The graph of r gcos r 2 cos r 2 cos r 2 4 sin 2 r 1 sin 3. The equation 4. The equation 5. The equation 6. The equation represents a ________. represents a ________. represents a ________. is symmetric with respect to the line ________. is symmetric with respect to the ________ ________. represents a ________ ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 6, identify the type of polar graph. 1. π 2 r = 3 cos 2θ 2 sin θ 3. r = 3(1 − θ 2 cos ) π 2 4. π 2 r = 16 cos 2θ 2 0 2 0 5 5. π 2 r = 6 sin 2θ 6. π 2 r = 3 cosθ 0 21 65 21 4 5 0 In Exercises 7–12, test for symmetry with respect to /2, 7. the polar axis, and the pole. r 16 cos 3 8. r r 5 4 cos 2 1 sin r 2 16 cos 2 9. 11. r 3 2 cos r 2 36 sin 2 10. 12. In Exercises 13–16, find the maximum value of zeros of r. r and any 13. 15. r 101 sin r 4 cos 3 14. 16. r 6 12 cos r 3 sin 2 In Exercises 17–40, sketch the graph of the polar equation using symmetry, zeros, maximum -values, and any other additional points. r 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. r 5 r 6 r 3 sin r 31 cos r 41 sin r 3 6 sin r 1 2 sin r 3 4 cos r 5 sin 2 r 2 sec 3 sin 2 cos r 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. r 2 r 3 4 r 4 cos r 41 sin r 21 cos r 4 3 sin r 1 2 cos r 4 3 cos r 3 cos 2 r 5 csc 38. r 6 2 sin 3 cos 39. r 2 9 cos 2 40. r 2 4 sin In Exercises 41– 46, use a graphing utility to graph the polar equation. Describe your viewing window. 41. 43. 45. r |
8 cos r 32 sin r 8 sin cos 2 42. 44. 46. r cos 2 r 2 cos3 2 r 2 csc 5 In Exercises 47–52, use a graphing utility to graph the polar equation. Find an interval for for which the graph is traced only once. 47. r 3 4 cos 48. r 5 4 cos 333202_1008.qxd 12/8/05 9:08 AM Page 792 792 Chapter 10 Topics in Analytic Geometry 49. r 2 cos3 2 51. r 2 9 sin 2 50. 52. r 3 sin5 2 r 2 1 In Exercises 53–56, use a graphing utility to graph the polar equation and show that the indicated line is an asymptote of the graph. Name of Graph 53. Conchoid 54. Conchoid 55. Hyperbolic spiral 56. Strophoid Synthesis Polar Equation r 2 sec r 2 csc r 3 r 2 cos 2 sec Asymptote x 1 y 1 y 3 x 2 True or False? In Exercises 57 and 58, determine whether the statement is true or false. Justify your answer. 57. In the polar coordinate system, if a graph that has symmetry with respect to the polar axis were folded on the line 0, the portion of the graph above the polar axis would coincide with the portion of the graph below the polar axis. 58. In the polar coordinate system, if a graph that has symmetry with respect to the pole were folded on the line 34, the portion of the graph on one side of the fold would coincide with the portion of the graph on the other side of the fold. 59. Exploration Sketch the graph of over each interval. Describe the part of the graph obtained in each case. r 6 cos (a) 0 ≤ ≤ (c) 2 ≤ ≤ 2 (b) (d 60. Graphical Reasoning Use a graphing utility to graph the 0, for (a) polar equation 4, Use the graphs to describe (b) the effect of the angle Write the equation as a function of r 61 cos 2.. for part (c). and (c) r f is rotated about the pole through an Show that the equation of the rotated graph is sin 61. The graph of. angle r f. 62. Consider the graph of r f sin. (a) Show that if the graph is rotated counterclockwise 2 radians about the pole, the equation of |
the rotated graph is r f cos. (b) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated graph is r f sin. (c) Show that if the graph is rotated counterclockwise radians about the pole, the equation of the rotated 32 graph is r f cos. In Exercises 63– 66, use the results of Exercises 61 and 62. 63. Write an equation for the limaçon r 2 sin after it has been rotated through the given angle. 3 2 r 2 sin 2 64. Write an equation for the rose curve (b) (d) (c) (a) 2 4 after it has been rotated through the given angle. (a) 6 (b) 2 (c) 2 3 (d) 65. Sketch the graph of each equation. (a) r 1 sin r 1 sin (b) 4 66. Sketch the graph of each equation. (a) r 3 sec r 3 sec (c) 3 r 3 sec (b) r 3 sec (d) 4 2 67. Exploration Use a graphing utility to graph and identify r 2 k sin for k 0, 1, 2, and 3. 68. Exploration Consider the equation r 3 sin k. (a) Use a graphing utility to graph the equation for Find the interval for over which the graph is k 1.5. traced only once. (b) Use a graphing utility to graph the equation for Find the interval for over which the graph is k 2.5. traced only once. (c) Is it possible to find an interval for over which the k? graph is traced only once for any rational number Explain. Skills Review In Exercises 69–72, find the zeros (if any) of the rational function. 69. 71. f x x2 9 x 1 f x 5 3 x 2 70. 72. f x 6 4 x2 4 f x x 3 27 x2 4 In Exercises 73 and 74, find the standard form of the equation of the ellipse with the given characteristics. Then sketch the ellipse. 73. Vertices: 4, 2, 2, 2; minor axis of length 4 74. Foci: 3, 2, 3, 4; major axis of length 8 333202_1009.qxd 12/8/05 9:09 AM Page 793 10.9 Polar Equations of Conics Section 10. |
9 Polar Equations of Conics 793 What you should learn • Define conics in terms of eccentricity. • Write and graph equations of conics in polar form. • Use equations of conics in polar form to model real-life problems. Why you should learn it The orbits of planets and satellites can be modeled with polar equations. For instance, in Exercise 58 on page 798, a polar equation is used to model the orbit of a satellite. Alternative Definition of Conic In Sections 10.3 and 10.4, you learned that the rectangular equations of ellipses and hyperbolas take simple forms when the origin lies at their centers. As it happens, there are many important applications of conics in which it is more convenient to use one of the foci as the origin. In this section, you will learn that polar equations of conics take simple forms if one of the foci lies at the pole. To begin, consider the following alternative definition of conic that uses the concept of eccentricity. Alternative Definition of Conic The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a conic. The constant ratio is the eccentricity of the conic and is denoted by Moreover, the conic is an ellipse if and a hyperbola if (See Figure 10.77.) a parabola if e 1, e < 1, e > 1. e. In Figure 10.77, note that for each type of conic, the focus is at the pole. π 2 π 2 Directrix π 2 Directrix Q P P Q 0 F = (0, 0) 0 F = (0, 0) 0 < e < 1 Ellipse: PF PQ FIGURE 10.77 < 1 e 1 Parabola: PF PQ 1 Directrix Q P 0 F = (0, 0) P′ Q′ e > 1 Hyperbola PF PF PQ PQ > 1 Digital Image © 1996 Corbis; Original image courtesy of NASA/Corbis Polar Equations of Conics The benefit of locating a focus of a conic at the pole is that the equation of the conic takes on a simpler form. For a proof of the polar equations of conics, see Proofs in Mathematics on page 808. Polar Equations of Conics The graph of a polar equation of the |
form 1. r ep 1 ± e cos or 2. r ep 1 ± e sin p is a conic, where the focus (pole) and the directrix. e > 0 is the eccentricity and is the distance between 333202_1009.qxd 12/8/05 9:09 AM Page 794 794 Chapter 10 Topics in Analytic Geometry Equations of the form r ep 1 ± e cos gcos Vertical directrix correspond to conics with a vertical directrix and symmetry with respect to the polar axis. Equations of the form r ep 1 ± e sin gsin Horizontal directrix 2. correspond to conics with a horizontal directrix and symmetry with respect to the line Moreover, the converse is also true—that is, any conic with a focus at the pole and having a horizontal or vertical directrix can be represented by one of the given equations. Example 1 Identifying a Conic from Its Equation Identify the type of conic represented by the equation r Algebraic Solution To identify the type of conic, rewrite the equation in the form r ep1 ± e cos. r 15 3 2 cos 5 1 23 cos e 2 3 < 1, Because is an ellipse. Write original equation. Divide numerator and denominator by 3. you can conclude that the graph 15 3 2 cos. Graphical Solution 0 You can start sketching the graph by plotting points from. the Because the equation is of the form to r graph of is symmetric with respect to the polar axis. So, you can complete the sketch, as shown in Figure 10.78. From this, you can conclude that the graph is an ellipse. r gcos, π 2 r = 15 2 cos θ − 3 π(3, ) (15, 0) 0 3 6 9 12 18 21 Now try Exercise 11. FIGURE 10.78 For the ellipse in Figure 10.78, the major axis is horizontal and the vertices 15, 0 lie at To find the b2 a 2 c 2 length of the to conclude that major axis, you can use the equations 2a 18. and So, the length of the axis is e ca and minor 3,. b2 a 2 c 2 a2 ea2 a21 e 2. e 2 3, Because b 45 35. analysis for hyperbolas yields Ellipse b2 921 2 2 45, you have So, the length of the minor |
axis is 3 which 2b 65. implies that A similar b2 c 2 a 2 ea2 a2 a2e 2 1. Hyperbola 333202_1009.qxd 12/8/05 9:09 AM Page 795 Section 10.9 Polar Equations of Conics 795 Example 2 Sketching a Conic from Its Polar Equation Identify the conic r 32 3 5 sin and sketch its graph. Solution Dividing the numerator and denominator by 3, you have r 323 1 53 sin e 5. 3 > 1, Because lies on the line and Because the length of the transverse axis is 12, you can see that b, the graph is a hyperbola. The transverse axis of the hyperbola 16, 32. a 6. To find and the vertices occur at 2, 4, 2 write 4 8 0 b 2 a 2e 2 1 625 3 2 1 64. b 8. So, hyperbola are Finally, you can use and a b to determine that the asymptotes of the y 10 ± 3 4 x. The graph is shown in Figure 10.79. π 2 ( −16, )3 π 2 π ( ) 4, 2 r = 32 3 + 5 sin θ FIGURE 10.79 Te c h n o l o g y Use a graphing utility set in polar mode to verify the four orientations shown at the right. Remember that e must be positive, but p can be positive or negative. π 2 Directrix: y = 3 (0, 0 + sin θ FIGURE 10.80 Now try Exercise 19. In the next example, you are asked to find a polar equation of a specified conic. To do this, let be the distance between the pole and the directrix. p 1. Horizontal directrix above the pole: 2. Horizontal directrix below the pole: 3. Vertical directrix to the right of the pole: 4. Vertical directrix to the left of the pole: r r r r ep 1 e sin ep 1 e sin ep 1 e cos ep 1 e cos Example 3 Finding the Polar Equation of a Conic Find the polar equation of the parabola whose focus is the pole and whose directrix is the line y 3. Solution From Figure 10.80, you can see that the directrix is horizontal and above the pole, so you can choose an equation of the form r |
ep 1 e sin. Moreover, because the eccentricity of a parabola is between the pole and the directrix is p 3, you have the equation e 1 and the distance r 3 1 sin. Now try Exercise 33. 333202_1009.qxd 12/8/05 9:09 AM Page 796 796 Chapter 10 Topics in Analytic Geometry Applications Kepler’s Laws (listed below), named after the German astronomer Johannes Kepler (1571–1630), can be used to describe the orbits of the planets about the sun. 1. Each planet moves in an elliptical orbit with the sun at one focus. 2. A ray from the sun to the planet sweeps out equal areas of the ellipse in equal times. 3. The square of the period (the time it takes for a planet to orbit the sun) is proportional to the cube of the mean distance between the planet and the sun. Although Kepler simply stated these laws on the basis of observation, they were later validated by Isaac Newton (1642–1727). In fact, Newton was able to show that each law can be deduced from a set of universal laws of motion and gravitation that govern the movement of all heavenly bodies, including comets and satellites. This is illustrated in the next example, which involves the comet named after the English mathematician and physicist Edmund Halley (1656–1742). astronomical If you use Earth as a reference with a period of 1 year and a distance of 1 astronomical unit (an is defined as the mean distance between Earth and the sun, or about 93 million miles), the proportionality constant in Kepler’s third law is 1. For example, because Mars has a mean distance to the sun of So, the period of Mars is astronomical units, its period d 1.524 P 1.88 is given by d 3 P2. years. unit P π 2 Sun π 0 Earth Halley’s comet Example 4 Halley’s Comet Halley’s comet has an elliptical orbit with an eccentricity of The length of the major axis of the orbit is approximately 35.88 astronomical units. Find a polar equation for the orbit. How close does Halley’s comet come to the sun? e 0.967. Solution Using a vertical axis, as shown in Figure 10.81, choose an equation of the form r ep1 e sin. 2 Because the vertices of the ellipse occur when 32, and |
you can determine the length of the major axis to be the sum of r the -values of the vertices. That is, 2a 0.967p 1 0.967 0.967p 1 0.967 29.79p 35.88. ep 0.9671.204 1.164. p 1.204 So, and equation, you have 1.164 1 0.967 sin r Using this value of ep in the r where (the focus), substitute is measured in astronomical units. To find the closest point to the sun in this equation to obtain 2 π 3 2 r 1.164 1 0.967 sin2 0.59 astronomical unit 55,000,000 miles. FIGURE 10.81 Now try Exercise 57. 333202_1009.qxd 12/8/05 9:09 AM Page 797 Section 10.9 Polar Equations of Conics 797 10.9 Exercises VOCABULARY CHECK: In Exercises 1–3, fill in the blanks. 1. The locus of a point in the plane that moves so that its distance from a fixed point (focus) is in a constant ratio to its distance from a fixed line (directrix) is a ________. 2. The constant ratio is the ________ of the conic and is denoted by ________. 3. An equation of the form r ep 1 e cos 4. Match the conic with its eccentricity. has a ________ directrix to the ________ of the pole. (a) e < 1 (i) parabola (b) e 1 (ii) hyperbola (c) e > 1 (iii) ellipse PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, write the polar equation of the conic for e 1, e 1.5. Identify the conic for each equation. Verify your answers with a graphing utility. e 0.5, and 1. r 3. r 4e 1 e cos 4e 1 e sin 2. r 4. r 4e 1 e cos 4e 1 e sin In Exercises 5–10, match the polar equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) |
(c) (eb) (d) π 2 π 2 (f. r 7. r 9. r 2 1 cos 3 1 2 sin 4 2 cos 6. r 8. r 10. r 3 2 cos 2 1 sin 4 1 3 sin In Exercises 11–24, identify the conic and sketch its graph. 11. r 13. r 15. r 17. r 19. r 21. r 23. r 2 1 cos 5 1 sin 2 2 cos 6 2 sin 3 2 4 sin 3 2 6 cos 4 2 cos 12. r 14. r 16. r 18. r 20. r 3 1 sin 6 1 cos 3 3 sin 9 3 2 cos 5 1 2 cos 22. r 24. r 3 2 6 sin 2 2 3 sin In Exercises 25–28, use a graphing utility to graph the polar equation. Identify the graph. 25. r 1 1 sin 27. r 3 4 2 cos 26. r 28. r 5 2 4 sin 4 1 2 cos 333202_1009.qxd 12/8/05 9:09 AM Page 798 798 Chapter 10 Topics in Analytic Geometry In Exercises 29–32, use a graphing utility to graph the rotated conic. 29. r 30. r 31. r 2 1 cos 4 3 3 sin 3 6 2 sin 6 (See Exercise 11.) (See Exercise 16.) (See Exercise 17.) 32. r 5 1 2 cos 23 (See Exercise 20.) In Exercises 33–48, find a polar equation of the conic with its focus at the pole. Directrix Conic Eccentricity 33. Parabola 34. Parabola 35. Ellipse 36. Ellipse 37. Hyperbola 38. Hyperbola Conic 39. Parabola 40. Parabola 41. Parabola 42. Parabola 43. Ellipse 44. Ellipse 45. Ellipse 46. Hyperbola 47. Hyperbola 48. Hyperbola Vertex or Vertices 1, 2 6, 0 5, 10, 2 2, 0, 10, 2, 2, 4, 32 20, 0, 4, 2, 0, 8, 0 1, 32, 9, 32 4, 2, 1, 2 49. Planetary Motion The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies |
on the polar axis, and the length 2a (see figure). Show that the polar of the major axis is r a1 e21 e cos equation of the orbit is where e is the eccentricity. π 2 Planet r θ Sun a 50. Planetary Motion Use the result of Exercise 49 to show ) from the and the maximum dis- that the minimum distance ( sun to the planet is aphelion tance ( r a1 e ) is r a1 e. perihelion distance distance Planetary Motion In Exercises 51–56, use the results of Exercises 49 and 50 to find the polar equation of the planet’s orbit and the perihelion and aphelion distances. 51. Earth 52. Saturn 53. Venus 54. Mercury 55. Mars 56. Jupiter e 0.0167 a 95.956 106 miles, a 1.427 109 kilometers, a 108.209 106 kilometers, a 35.98 106 miles, a 141.63 106 miles, a 778.41 106 kilometers, e 0.2056 e 0.0934 e 0.0542 e 0.0068 e 0.0484 57. Astronomy The comet Encke has an elliptical orbit with The length of the major axis an eccentricity of of the orbit is approximately 4.42 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun? e 0.847. Model It 58. Satellite Tracking A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by the satellite will have the minimum velocity necessary to escape Earth’s gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). 2, π 2 Circular orbit 4100 miles Parabolic path 0 Not drawn to scale (a) Find a polar equation of the parabolic path of the satellite (assume the radius of Earth is 4000 miles). (b) Use a graphing utility to graph the equation you found in part (a). 0 and the satellite when 30. (c) Find the distance between the surface of the Earth (d) Find the distance between the surface of Earth and the satellite when 60. 333202_1009.qxd 12/8/05 9:09 AM Page 799 Synthesis True or False? statement is true or false. Justify your answer. In Exercises |
59–61, determine whether the 59. For a given value of 2, the graph of e > 1 over the interval 0 to r ex 1 e cos is the same as the graph of r ex 1 e cos. 60. The graph of r 4 3 3 sin has a horizontal directrix above the pole. 61. The conic represented by the following equation is an ellipse. r 2 16 9 4 cos 4 62. Writing In your own words, define the term eccentricity and explain how it can be used to classify conics. 63. Show that the polar equation of the ellipse x 2 a 2 y 2 b2 1 is r 2 b2 1 e 2 cos2. 64. Show that the polar equation of the hyperbola x 2 a 2 y 2 b2 1 is r 2 b2 1 e 2 cos2. In Exercises 65–70, use the results of Exercises 63 and 64 to write the polar form of the equation of the conic. 65. 67. x 2 169 x 2 9 y 2 144 1 y 2 16 1 69. Hyperbola One focus: 70. Ellipse Vertices: One focus: Vertices: 1 66. 68. 1 y 2 16 y 2 4 x 2 25 x 2 36 5, 2 4, 2, 4, 2 4, 0 5, 0, 5, 71. Exploration Consider the polar equation r 4 1 0.4 cos. Section 10.9 Polar Equations of Conics 799 (a) Identify the conic without graphing the equation. (b) Without graphing the following polar equations, describe how each differs from the given polar equation. r1 4 1 0.4 cos, r2 4 1 0.4 sin (c) Use a graphing utility to verify your results in part (b). 72. Exploration The equation r ep 1 ± e sin is the equation of an ellipse with What happens to the lengths of both the major axis and the minor axis when changes? the value of Use an example to explain your reasoning. remains fixed and the value of e < 1. p e Skills Review In Exercises 73–78, solve the trigonometric equation. 73. 75. 43 tan 3 1 12 sin2 9 77. 2 cot x 5 cos 2 74. 76. 6 cos x 2 1 9 csc2 x 10 2 78. 2 sec 2 csc 4 In Exercises 79–82, find |
the exact value of the trigonometric are in Quadrant IV and function given that sin u 3 5 cosu v cosu v u and cos v 1/2. sinu v sinu v and 79. 80. 82. 81. v In Exercises 83 and 84, find the exact values of cos 2u, using the double-angle formulas. tan 2u and sin 2u, 83. sin u 4 5, 2 < u < 84. tan u 3, 3 2 < u < 2 In Exercises 85–88, find a formula for sequence. an for the arithmetic 85. 87. a1 a3 0, d 1 4 27, a8 72 86. 88. a1 a1 13, d 3 5, a4 9.5 In Exercises 89–92, evaluate the expression. Do not use a calculator. 89. 91. 12C9 10P3 18C16 90. 92. 29 P2 333202_100R.qxd 12/8/05 9:11 AM Page 800 800 Chapter 10 Topics in Analytic Geometry 10 Chapter Summary What did you learn? Section 10.1 Find the inclination of a line (p. 728). Find the angle between two lines (p. 729). Find the distance between a point and a line (p. 730). Section 10.2 Recognize a conic as the intersection of a plane and a double-napped cone (p. 735). Write equations of parabolas in standard form and graph parabolas (p. 736). Use the reflective property of parabolas to solve real-life problems (p. 738). Section 10.3 Write equations of ellipses in standard form and graph ellipses (p. 744). Use properties of ellipses to model and solve real-life problems (p. 748). Find the eccentricities of ellipses (p. 748). Section 10.4 Write equations of hyperbolas in standard form (p. 753). Find asymptotes of and graph hyperbolas (p. 755). Use properties of hyperbolas to solve real-life problems (p. 758). Classify conics from their general equations (p. 759). Section 10.5 Rotate the coordinate axes to eliminate the xy-term in equations of conics (p. 763). Use the discriminant to classify conics (p. |
767). Section 10.6 Evaluate sets of parametric equations for given values of the parameter (p. 771). Sketch curves that are represented by sets of parametric equations (p. 772). and rewrite the equations as single rectangular equations (p. 773). Find sets of parametric equations for graphs (p. 774). Section 10.7 Plot points on the polar coordinate system (p. 779). Convert points from rectangular to polar form and vice versa (p. 780). Convert equations from rectangular to polar form and vice versa (p. 782). Section 10.8 Graph polar equations by point plotting (p. 785). Use symmetry (p. 786), zeros, and maximum r-values (p. 787) to sketch graphs of polar equations. Recognize special polar graphs (p. 789). Section 10.9 Define conics in terms of eccentricity and write and graph equations of conics in polar form (p. 793). Use equations of conics in polar form to model real-life problems (p. 796). Review Exercises 1–4 5–8 9, 10 11, 12 13–16 17–20 21–24 25, 26 27–30 31–34 35–38 39, 40 41–44 45–48 49–52 53, 54 55–60 61–64 65–68 69–76 77–88 89–98 89–98 99–102 103–110 111, 112 333202_100R.qxd 12/8/05 9:11 AM Page 801 10 Review Exercises 10.1 In Exercises 1–4, find the inclination degrees) of the line with the given characteristics. (in radians and 1. Passes through the points 2. Passes through the points 1, 2 3, 4 and 2, 5 2, 7 and 3. Equation: 4. Equation: y 2x 4 6x 7y 5 0 In Exercises 5–8, find the angle between the lines. 5. 4x y 2 5x y 1 62 −1 −1 (in radians and degrees) 5x 3y 3 2x 3y 1 y 3 2 1 −1 θ x 1 2 7. 2x 7y 8 0.4x y 0 8. 0.02x 0.07y 0.18 0.09x 0.04y 0.17 In Exercises 9 and 10 |
, find the distance between the point and the line. Point 1, 2 0, 4 9. 10. Line x y 3 0 x 2y 2 0 10.2 In Exercises 11 and 12, state what type of conic is formed by the intersection of the plane and the double-napped cone. 11. 12. Review Exercises 801 In Exercises 17 and 18, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. x 17. 18. x2 2y, x2 2y, 2, 2 4, 8 19. Architecture A parabolic archway is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters (see figure). How wide is the archway at ground level? y y −, 10) ( 4 (0, 12) (4, 10) x 1.5 cm x FIGURE FOR 19 FIGURE FOR 20 20. Flashlight The light bulb in a flashlight is at the focus of its parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation of a cross section of the flashlight’s reflector with its focus on the positive -axis and its vertex at the origin. x In Exercises 21–24, find the standard form of the 10.3 equation of the ellipse with the given characteristics. Then graph the ellipse. 21. Vertices: 22. Vertices: 23. Vertices: 2, 0, 2, 2 3, 0, 7, 0; 2, 0, 2, 4; 0, 1, 4, 1; foci: 0, 0, 4, 0 foci: 2, 1, 2, 3 endpoints of the minor axis: 24. Vertices: axis: 6, 5, 2, 5 4, 1, 4, 11; endpoints of the minor 25. Architecture A semielliptical archway is to be formed over the entrance to an estate. The arch is to be set on pillars that are 10 feet apart and is to have a height (atop the pillars) of 4 feet. Where should the foci be placed in order to sketch the arch? In Exercises 13–16, find the standard form of the equation of the parabola with the given characteristics. Then graph the parabola. 26. |
Wading Pool You are building a wading pool that is in the shape of an ellipse. Your plans give an equation for the elliptical shape of the pool measured in feet as 13. Vertex: Focus: 15. Vertex: 0, 0 4, 0 0, 2 14. Vertex: Focus: 16. Vertex: 2, 0 0, 0 2, 2 Directrix: x 3 Directrix: y 0 x2 324 y2 196 1. Find the longest distance across the pool, the shortest distance, and the distance between the foci. 333202_100R.qxd 12/8/05 9:11 AM Page 802 802 Chapter 10 Topics in Analytic Geometry In Exercises 27–30, find the center, vertices, foci, and eccentricity of the ellipse. x 22 81 x 52 1 y 12 100 y 32 36 1 1 16x2 9y2 32x 72y 16 0 4x2 25y2 16x 150y 141 0 27. 28. 29. 30. In Exercises 31–34, find the standard form of the 10.4 equation of the hyperbola with the given characteristics. 31. Vertices: 32. Vertices: 0, ±1; foci: 2, 2, 2, 2; 0, ±3 foci: 33. Foci: 34. Foci: 0, 0, 8, 0; 3, ±2; 4, 2, 4, 2 y ±2x 4 asymptotes: asymptotes: y ±2x 3 In Exercises 35 –38, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes as an aid. y 52 4 1 35. x 32 16 y 12 4 x2 1 9x2 16y2 18x 32y 151 0 4x2 25y2 8x 150y 121 0 36. 37. 38. 39. LORAN Radio transmitting station A located 200 miles east of transmitting station B. A ship is in an area to the north and 40 miles west of station A. Synchronized radio pulses transmitted at 186,000 miles per second by the two stations are received 0.0005 second sooner from station A than from station B. How far north is the ship? is 40. Locating an Explosion Two of your friends live 4 miles |
apart and on the same “east-west” street, and you live halfway between them. You are having a three-way phone conversation when you hear an explosion. Six seconds later, your friend to the east hears the explosion, and your friend to the west hears it 8 seconds after you do. Find equations of two hyperbolas that would locate the explosion. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.) In Exercises 41–44, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 41. 42. 43. 44. 5x2 2y2 10x 4y 17 0 4y2 5x 3y 7 0 3x 2 2y 2 12x 12y 29 0 4x 2 4y 2 4x 8y 11 0 10.5 In Exercises 45 –48, rotate the axes to eliminate the xy -term in the equation. Then write the equation in standard form. Sketch the graph of the resulting equation, showing both sets of axes. 45. 46. 47. 48. xy 4 0 x2 10xy y2 1 0 5x2 2xy 5y2 12 0 4x2 8xy 4y2 72 x 92 y 0 In Exercises 49–52, (a) use the discriminant to classify the graph, (b) use the Quadratic Formula to solve for and (c) use a graphing utility to graph the equation. y, 49. 50. 51. 52. 16x2 24xy 9y2 30x 40y 0 13x2 8xy 7y2 45 0 x2 y2 2xy 22 x 22 y 2 0 x2 10xy y2 1 0 In Exercises 53 and 54, complete the table for each and 10.6 set of parametric equations. Plot the points sketch a graph of the parametric equations. x, y 53. x 3t 2 and y 7 4t 54. x 1 5 t and In Exercises 55–60, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation, if necessary. (c) Verify your result with a graphing utility. 55. 57. 59. x 2t y |
4t x t2 y t x 6 cos y 6 sin 56. 58. 60. x 1 4t y 2 3t x t 4 y t2 x 3 3 cos y 2 5 sin 333202_100R.qxd 12/8/05 9:11 AM Page 803 61. Find a parametric representation of the circle with center 5, 4 and radius 6. 62. Find a parametric representation of the ellipse with center major axis horizontal and eight units in length, 3, 4, and minor axis six units in length. 63. Find a parametric representation of the hyperbola with vertices 0, ±4 and foci 0, ±5. P 64. Involute of a Circle The involute of a circle is described of a string that is held taut as it is by the endpoint unwound from a spool (see figure). The spool does not rotate. Show that a parametric representation of the involute of a circle is x r cos sin y r sin cos. y P θr x 65. 2 < < 2. In Exercises 65–68, plot the point given in polar 10.7 coordinates and find two additional polar representations of the point, using 2, 4 5, 3 7, 4.19 3, 2.62 66. 67. 68. In Exercises 69–72, a point in polar coordinates is given. Convert the point to rectangular coordinates. 5 4 2, 0, 1, 3 3, 3 4 69. 70. 71. 72. 2 Review Exercises 803 In Exercises 77–82, convert the rectangular equation to polar form. 77. 79. 81. x 2 y 2 49 x2 y2 6y 0 xy 5 78. 80. 82. x 2 y 2 20 x 2 y 2 4x 0 xy 2 In Exercises 83–88, convert the polar equation to rectangular form. 83. 85. 87. r 5 r 3 cos r2 sin 84. 86. 88. r 12 r 8 sin r 2 cos 2 In Exercises 89–98, determine the symmetry of 10.8 the r. maximum value of, and any zeros of Then sketch the graph of the polar equation (plot additional points if necessary). r r, 89. 91. 93. 95. 97. r 4 r 4 sin 2 r 21 cos r 2 6 sin r 3 cos 2 90. 92. 94. 96. 98 |
. r 11 r cos 5 r 3 4 cos r 5 5 cos r cos 2 In Exercises 99 –102, identify the type of polar graph and use a graphing utility to graph the equation. 99. 100. 101. 102. r 32 cos r 31 2 cos r 4 cos 3 r 2 9 cos 2 In Exercises 103–106, identify the conic and sketch 10.9 its graph. 103. r 104. r 105. r 106. r 1 1 2 sin 2 1 sin 4 5 3 cos 16 4 5 cos In Exercises 73–76, a point in rectangular coordinates is given. Convert the point to polar coordinates. In Exercises 107–110, find a polar equation of the conic with its focus at the pole. 73. 74. 75. 76. 0, 2 5, 5 4, 6 3, 4 107. Parabola 108. Parabola 109. Ellipse 110. Hyperbola Vertex: Vertex: 2, 2, 2 5, 0, 1, Vertices: Vertices: 1, 0, 7, 0 333202_100R.qxd 12/8/05 9:12 AM Page 804 804 Chapter 10 Topics in Analytic Geometry 111. Explorer 18 On November 26, 1963, the United States launched Explorer 18. Its low and high points above the surface of Earth were 119 miles and 122,800 miles, respectively (see figure). The center of Earth was at one focus of the orbit. Find the polar equation of the orbit and find the distance between the surface of Earth (assume Earth has a radius of 4000 miles) and the satellite when 3. π 2 Explorer 18 r π 3 Earth 0 a 112. Asteroid An asteroid takes a parabolic path with Earth as its focus. It is about 6,000,000 miles from Earth at its closest approach. Write the polar equation of the path of the asteroid with its vertex at Find the distance between the asteroid and Earth when 3. 2. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 113–116, determine whether 113. When B 0 in an equation of the form Ax2 Bxy Cy2 Dx Ey F 0 the graph of the equation can be a parabola only if also. C 0 114. The graph of 1 4 x 2 y 4 1 is a hyperbola. 115. Only one set of parametric |
equations can represent the line y 3 2x. 116. There is a unique polar coordinate representation of each point in the plane. 117. Consider an ellipse with the major axis horizontal and 10 units in length. The number in the standard form of the equation of the ellipse must be less than what real number? Explain the change in the shape of the ellipse as b approaches this number. b 118. The graph of the parametric equations and is shown in the figure. How would the graph and x 2 sect equations the x 2 sec t y 3 tan t change for y 3 tant? x = 2 sec t y = 3 tan t x 4 y 4 2 −2 −4 FIGURE FOR 118 119. A moving object is modeled by the parametric equations is time (see figure). x 4 cos t y 3 sin t, How would the path change for the following? where and t (a) (b) x 4 cos 2t, x 5 cos t, y 3 sin 2t y 3 sin t y 4 2 −2 −2 −4 x 2 120. Identify the type of symmetry each of the following polar points has with the point in the figure. (a) 4, 6 (b) 4, 6 (c) 4, 6 π 2 π( 4, 6 ) 0 2 121. What is the relationship between the graphs of the rectan- gular and polar equations? (a) x2 y2 25, r 5 (b) x y 0, 4 122. Geometry The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis? (Hint: The area of an ellipse is A ab. ) y (0, 10) −a (, 0) x a (, 0) (0, 10)− 333202_100R.qxd 12/8/05 9:12 AM Page 805 10 Chapter Test Chapter Test 805 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Find the inclination of the line 2. Find the angle between the lines 2x 7y 3 0. 3x 2y 4 0 3. Find the distance between the point 7, 5 and the line and 4x y 6 0. y 5 x. In Exercises 4–7, classify the conic and write |
the equation in standard form. Identify the center, vertices, foci, and asymptotes (if applicable).Then sketch the graph of the conic. 4. 5. 6. 7. y 2 4x 4 0 x 2 4y 2 4x 0 9x2 16y2 54x 32y 47 0 2x2 2y2 8x 4y 9 0 8. Find the standard form of the equation of the parabola with vertex vertical axis, and passing through the point 0, 4. 3, 2, with a 9. Find the standard form of the equation of the hyperbola with foci 0, 0 and 0, 4 and asymptotes y ± 1 2x 2. 10. (a) Determine the number of degrees the axis must be rotated to eliminate the xy -term of the conic x 2 6xy y 2 6 0. (b) Graph the conic from part (a) and use a graphing utility to confirm your result. 11. Sketch the curve represented by the parametric equations and Eliminate the parameter and write the corresponding rectangular equation. y 2 sin. x 2 3 cos 12. Find a set of parametric equations of the line passing through the points 2, 3 and 13. Convert the polar coordinate to rectangular form. 6, 4. (There are many correct answers.) 5 6 2, 2 2, 14. Convert the rectangular coordinate representations of this point. to polar form and find two additional polar 15. Convert the rectangular equation x 2 y 2 4y 0 to polar form. In Exercises 16–19, sketch the graph of the polar equation. Identify the type of graph. r 4 1 cos r 2 3 sin 16. 18. 17. r 4 2 cos 19. r 3 sin 2 20. Find a polar equation of the ellipse with focus at the pole, eccentricity directrix y 4. e 1 4, and 21. A straight road rises with an inclination of 0.15 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 22. A baseball is hit at a point 3 feet above the ground toward the left field fence. The fence is 10 feet high and 375 feet from home plate. The path of the baseball can be modeled and the y 3 115 sin t 16t 2. Will the baseball go over the fence if it is hit at an angle of Will the baseball |
go over the fence if 35? x 115 cos t parametric 30? equations by 333202_100R.qxd 12/8/05 9:12 AM Page 806 Proofs in Mathematics Inclination and Slope If a nonvertical line has inclination and slope (p. 728) m, then m tan. (x 2, y2) y2 x (x1, 0) θ x2 − x1 (x1, y1) d (x2, y2 Proof m 0, If lines because m 0 tan 0. the line is horizontal and 0. So, the result is true for horizontal If the line has a positive slope, it will intersect the -axis. Label this point is a second point on the line, the slope as shown in the figure. If x x2, y2 x1, 0, is m y2 x2 0 x1 y2 x2 x1 tan. The case in which the line has a negative slope can be proved in a similar manner. Distance Between a Point and a Line The distance between the point and the line x1, y1 (p. 730) Ax By C 0 is d Ax1 By1 C A2 B2. Proof For simplicity’s sake, assume that the given line is neither horizontal nor vertical (see figure). By writing the equation in slope-intercept form x C B Ax By C 0 y A B you can see that the line has a slope of ing through BAx x1 y y1 is x1, y1 and perpendicular to the given line is. AC These two lines intersect at the point Ay1 A2 B2 ABx1 and y2 BA, So, the slope of the line passand its equation, x2, y2 where BC x BBx1 x2 Ay1 A2 B2. m AB. x2, y2 is Finally, the distance between 2 y2 ABy1 A2 B2 By1 d x2 x1 B2x1 A2Ax1 Ax1 By1 C A2 B2. and x1, y1 2 y1 x12 AC C2 B2Ax1 A2 B22 ABx1 A2y1 A2 B2 BC y12 By1 C2 y y 806 333202_100R.qxd 12/8/05 9:12 AM Page 807 Parabolic Paths There are many natural occurrences of parabolas |
in real life. For instance, the famous astronomer Galileo discovered in the 17th century that an object that is projected upward and obliquely to the pull of gravity travels in a parabolic path. Examples of this are the center of gravity of a jumping dolphin and the path of water molecules in a drinking fountain. Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at follows. (p. 736) h, k is as x h2 4py k, p 0 y k2 4px h, p 0 Vertical axis, directrix: y k p Horizontal axis, directrix: x h p The focus lies on the axis units (directed distance) from the vertex. If the the equation takes one of the following forms. vertex is at the origin p 0, 0, x2 4py y2 4px Vertical axis Horizontal axis Axis: =x h Focusx, y) Vertex: )h k (, Directrix: p− k =y Parabola with vertical axis Directrix: p− =x h p > 0 (x, y) Axis: y = k Focus: h p ( +, k ) Vertex: (, )h k Parabola with horizontal axis Proof For the case in which the directrix is parallel to the -axis and the focus lies x, y is any point on the parabola, above the vertex, as shown in the top figure, if h, k p then, by definition, it is equidistant from the focus and the directrix y k p. So, you have x x h2 y k p2 y k p x h2 y k p2 y k p2 x h2 y2 2yk p k p2 y2 2yk p k p2 x h2 y2 2ky 2py k2 2pk p2 y2 2ky 2py k2 2pk p2 x h2 2py 2pk 2py 2pk x h2 4py k. For the case in which the directrix is parallel to the -axis and the focus lies to x, y is any point on the the right of the vertex, as shown in the bottom figure, if h p, k parabola, then, by definition, it is equidistant from the focus and the directrix x h p. So, you have y x h p2 y k2 x h p x h p2 |
y k2 x h p2 x2 2xh p h p2 y k2 x2 2xh p h p2 x2 2hx 2px h2 2ph p2 y k2 x2 2hx 2px h2 2ph p2 2px 2ph y k2 2px 2ph y k2 4px h. Note that if a parabola is centered at the origin, then the two equations above would simplify to respectively. y 2 4px, x 2 4py and 807 333202_100R.qxd 12/8/05 9:12 AM Page 808 Polar Equations of Conics The graph of a polar equation of the form (p. 793) 1. r ep 1 ± e cos or 2. r ep 1 ± e sin is a conic, where the focus (pole) and the directrix. e > 0 is the eccentricity and p is the distance between π 2 p P r= (, )θ θ r F = (0, 0) x = r cos θ Directrix Q Proof A proof for other cases are similar. In the figure, consider a vertical directrix, right of the focus p is a point on the graph of r ep1 e cos F 0, 0. is shown here. The proofs of the units to the P r, p > 0 with If r ep 1 e cos 0 the distance between PQ p x P and the directrix is p r cos ep 1 e cos cos p 1 e cos p1 e cos 1 e cos e. r p P and the pole is simply PF r, the Moreover, because the distance between ratio of PF to PQ is PF PQ rr e e e and, by definition, the graph of the equation must be a conic. 808 333202_100R.qxd 12/8/05 9:12 AM Page 809 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Several mountain climbers are located in a mountain pass between two peaks. The angles of elevation to the two peaks are 0.84 radian and 1.10 radians. A range finder shows that the distances to the peaks are 3250 feet and 6700 feet, respectively (see figure). 5. A tour boat travels between two islands that are 12 miles apart (see figure). For a trip between the islands |
, there is enough fuel for a 20-mile trip. 6 7 0 0 f t 1.10 radians 3250 ft 0.84 radian (a) Find the angle between the two lines of sight to the peaks. (b) Approximate the amount of vertical climb that is necessary to reach the summit of each peak. 2. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long. Island 1 Island 2 12 mi Not drawn to scale (a) Explain why the region in which the boat can travel is bounded by an ellipse. (b) Let 0, 0 represent the center of the ellipse. Find the coordinates of each island. (c) The boat travels from one island, straight past the other island to the vertex of the ellipse, and back to the second island. How many miles does the boat travel? Use your answer to find the coordinates of the vertex. (d) Use the results from parts (b) and (c) to write an equation for the ellipse that bounds the region in which the boat can travel. 6. Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points 10, 2 2, 2 and is 6. 7. Prove that the graph of the equation (a) Find an equation that models the shape of the room. Ax2 Cy2 Dx Ey F 0 (b) How far apart are the two foci? (c) What is the area of the floor of the room? (The area of an ellipse is A ab. ) 3. Find the equation(s) of all parabolas that have the -axis as the axis of symmetry and focus at the origin. x 4. Find the area of the square inscribed in the ellipse below. y x 2 a 2 + y 2 b2 = 1 x is one of the following (except in degenerate cases). Conic (a) Circle (b) Parabola (c) Ellipse Condition A C A 0 |
or C 0 AC > 0 (but not both) (d) Hyperbola AC < 0 8. The following sets of parametric equations model projectile motion. x v0 cos t x v0 cos t y v0 sin t (a) Under what circumstances would you use each model? y h v0 sin t 16t2 (b) Eliminate the parameter for each set of equations. (c) In which case is the path of the moving object not affected by a change in the velocity? Explain. v 809 333202_100R.qxd 12/8/05 9:12 AM Page 810 9. As t equations increases, the ellipse given by the parametric 4 x cos t and y 2 sin t is traced out counterclockwise. Find a parametric representation for which the same ellipse is traced out clockwise. 10. A hypocycloid has the parametric equations x a b cos t b cosa b b t and y a b sin t b sina b b t. (a) Use a graphing utility to graph the hypocycloid for each value of and Describe each graph. a 3, b 1 a 10, b 1 a 4, b 3 a b. a 2, b 1 a 4, b 1 a 3, b 2 (b) (d) (c) (e) (f) 11. The curve given by the parametric equations x 1 t2 1 t2 and y t1 t2 1 t2 is called a strophoid. (a) Find a rectangular equation of the strophoid. (b) Find a polar equation of the strophoid. (c) Use a graphing utility to graph the strophoid. 12. The rose curves described in this chapter are of the form r a cos n or r a sin n n is a positive integer that is greater than or equal to and for some noninteger values of Describe the where 2. Use a graphing utility to graph r a sin n graphs. r a cos n n. −3 4 −4 12( r = e cos − 2 cos 4 + sin 5 θ θ θ ( FIGURE FOR 14 (a) The graph above was produced using 0 ≤ ≤ 2. Does this show the entire graph? Explain your reasoning. (b) Approximate the maximum -value of the graph. Does instead of r this value change if you use 0 ≤ ≤ 2? |
0 ≤ ≤ 4 Explain. 15. Use a graphing utility to graph the polar equation r cos 5 n cos 0 ≤ ≤ n 5 to n 5. for the integers for As you graph these equations, you should see the graph change shape from a heart to a bell. Write a short paragraph n explaining what values of produce the heart portion of the n curve and what values of produce the bell portion. 16. The planets travel in elliptical orbits with the sun at one focus. The polar equation of the orbit of a planet with one focus at the pole and major axis of length 2a is r 1 e 2a 1 e cos e where is the eccentricity. The minimum distance (perihelion) from the sun to a planet is and The the maximum distance (aphelion) is length of the major axis for the planet Neptune is a 9.000 109 kilometers and the eccentricity is e 0.0086. The length of the major axis for the planet Pluto is kilometers and the eccentricity is r a1 e r a1 e. a 10.813 109 e 0.2488. 13. What conic section is represented by the polar equation (a) Find the polar equation of the orbit of each planet. r a sin b cos? 14. The graph of the polar equation r ecos 2 cos 4 sin5 12 is called the butterfly curve, as shown in the figure. 810 (b) Find the perihelion and aphelion distances for each planet. (c) Use a graphing utility to graph the polar equation of each planet’s orbit in the same viewing window. (d) Do the orbits of the two planets intersect? Will the two planets ever collide? Why or why not? (e) Is Pluto ever closer to the sun than Neptune? Why is Pluto called the ninth planet and Neptune the eighth planet? 333202_0A01.qxd 12/6/05 2:09 PM Page A1 Appendix A Review of Fundamental Concepts of Algebra A.1 Real Numbers and Their Properties What you should learn • Represent and classify real numbers. • Order real numbers and use inequalities. • Find the absolute values of real numbers and find the distance between two real numbers. • Evaluate algebraic expressions. • Use the basic rules and properties of algebra. Why you should learn it Real numbers are used to represent many real-life quantities. For example, in Exercise 65 on page A9, you will use real numbers to represent |
the federal deficit. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Real numbers Irrational numbers Rational numbers Integers Noninteger fractions (positive and negative) Negative integers Whole numbers Natural numbers Zero FIGURE A.1 Subsets of real numbers Real Numbers Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as 5, 9, 0, 4 3, 0.666..., 28.21,,2, and 332. Here are some important subsets (each member of subset B is also a member of set A) of the real numbers. The three dots, called ellipsis points, indicate that the pattern continues indefinitely. 1, 2, 3, 4,... 0, 1, 2, 3, 4,......, 3, 2, 1, 0, 1, 2, 3,... Set of natural numbers Set of whole numbers Set of integers A real number is rational if it can be written as the ratio where For instance, the numbers q 0. pq of two integers, 1 3 0.3333... 0.3, 1 8 0.125, and 125 111 1.126126... 1.126 3.145 are rational. The decimal representation of a rational number either repeats as in. 173 A real number that cannot be written 55 as the ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers or terminates as in 0.5 1 2 2 1.4142135... 1.41 and 3.1415926... 3.14 are irrational. (The symbol means “is approximately equal to.”) Figure A.1 shows subsets of real numbers and their relationships to each other. Real numbers are represented graphically by a real number line. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure A.2. The term nonnegative describes a number that is either positive or zero. Negative direction −4 −3 −2 −1 0 1 2 3 4 Positive direction Origin FIGURE A.2 The real number line As illustrated in Figure A.3, there is a one-to-one correspondence between |
real numbers and points on the real number line. − 5 3 0.75 π −3 −2 −1 0 1 2 3 Every real number corresponds to exactly one point on the real number line. FIGURE A.3 One-to-one −2.4 2 −3 −2 −1 0 1 2 3 Every point on the real number line corresponds to exactly one real number. A1 333202_0A01.qxd 12/6/05 2:09 PM Page A2 A2 Appendix A Review of Fundamental Concepts of Algebra Ordering Real Numbers One important property of real numbers is that they are ordered. is less than is denoted by the inequality Definition of Order on the Real Number Line a b a If and are real numbers, a b of and is greater than b be described by saying that a ≤ b is less than or equal to a means that inequality b ≥ a b means that are inequality symbols. is greater than or equal to b if a < b. a a. b, b a is positive. The order This relationship can also and writing b > a. The and the inequality The symbols <, >, ≤, and ≥ a −1 0 1 b 2 FIGURE A.4 the left of b. a < b if and only if a lies to x ≤ 2 0 1 2 3 4 FIGURE A.5 −2 −1 FIGURE A. Geometrically, this definition implies that a < b if and only if a lies to the left of on the real number line, as shown in Figure A.4. b Example 1 Interpreting Inequalities Describe the subset of real numbers represented by each inequality. a. x ≤ 2 b. 2 ≤ x < 3 Solution a. The inequality x ≤ 2 shown in Figure A.5. denotes all real numbers less than or equal to 2, as b. The inequality 2 ≤ x < 3 inequality” denotes all real numbers between including 3, as shown in Figure A.6. means that x ≥ 2 2 and x < 3. and 3, including This “double but not 2 Now try Exercise 19. Inequalities can be used to describe subsets of real numbers called intervals. are the endpoints of In the bounded intervals below, the real numbers each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open interval are not included in the interval. and a b Bounded Intervals on the Real |
Number Line Notation a, b Interval Type Closed Inequality a ≤ x ≤ b The reason that the four types of intervals at the right are called bounded is that each has a finite length. An interval that does not have a finite length is unbounded (see page A3). a, b a, b a, b Open Graph 333202_0A01.qxd 12/6/05 2:09 PM Page A3 Appendix A.1 Real Numbers and Their Properties A3 or, Note that whenever you write intervals containing you always use a parenthesis and never a bracket. This is because these symbols are never an endpoint of an interval and therefore not included in the interval. The symbols negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the 1, unboundedness of an interval such as positive infinity, and, 3. or,, Unbounded Intervals on the Real Number Line Notation a, a,, b, b Interval Type Open Open Inequality, Entire real line < x < Graph a a b b x x x x x Example 2 Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. c. All x in the interval b. m is at least 3, 5 3. Solution a. The statement “c is at most 2” can be represented by b. The statement “m is at least 3, 5 c. “All x in the interval 3 ” can be represented by ” can be represented by c ≤ 2. m ≥ 3. 3 < x ≤ 5. Now try Exercise 31. Example 3 Interpreting Intervals Give a verbal description of each interval. a. 1, 0 b. 2, c., 0 Solution a. This interval consists of all real numbers that are greater than 1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 29. The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a b, a < b, or a > b. Law of Trichotomy 333202_0A01.qxd 12/6/05 2:09 PM Page A4 A4 Appendix A Review of Fundamental Concepts of Algebra Absolute Value and Distance The absolute value of a real |
number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Definition of Absolute Value If is a real number, then the absolute value of a a is a a, if a ≥ 0 a, if a < 0. Notice in this definition that the absolute value of a real number is never negative. For instance, if The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, 5 5 5. a 5, 0 0. then Evaluating the Absolute Value of a Number Example 4 x x Evaluate for (a) x > 0 and (b) x < 0. Solution a. If x > 0, then x x and b. If x < 0, then x x and x x x x x x 1. x x 1. Now try Exercise 47. Properties of Absolute Values a ≥ 0 ab ab 1. 3. 2. 4. a a b a a b, b 0 Absolute value can be used to define the distance between two points on the 3 and 4 is 7 real number line. For instance, the distance between −3 −2 −1 0 1 2 3 4 FIGURE A.7 and 4 is 7. The distance between 3 3 4 7 7 as shown in Figure A.7. Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is da, b b a a b. 333202_0A01.qxd 12/6/05 2:09 PM Page A5 Appendix A.1 Real Numbers and Their Properties A5 Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 5x, 2x 3, 4 x 2 2, 7x y Definition of an Algebraic Expression An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition. For example, x 2 5x 8 x 2 5x 8 x 2 are the variable terms and 8 is the constant term. has three terms: The numerical factor of a variable term is the coefficient of the variable term. For instance, the coefficient |
of and the coefficient of 5x 5, and is 1. x 2 is 5x To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. Here are two examples. Expression 3x 5 3x 2 2x 1 Value of Variable x 3 x 1 Substitute 33 5 312 21 1 Value of Expression 9 5 4 3 2 1 0 When an algebraic expression is evaluated, the Substitution Principle is then a can be replaced by b in any expression used. It states that “If involving a.” In the first evaluation shown above, for instance, 3 is substituted for x in the expression 3x 5. a b, Basic Rules of Algebra There are four arithmetic operations with real numbers: addition, multiplication, subtraction, and division, denoted by the symbols or /. Of these, addition and multiplication are the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively., and,, or Definitions of Subtraction and Division Subtraction: Add the opposite. Division: Multiply by the reciprocal. a b a b If b 0, then ab a1 b is the additive inverse (or opposite) of b, and In these definitions, is the multiplicative inverse (or reciprocal) of b. In the fractional form a is the numerator of the fraction and b is the denominator. b. a b 1b ab, 333202_0A01.qxd 12/6/05 2:09 PM Page A6 A6 Appendix A Review of Fundamental Concepts of Algebra Because the properties of real numbers below are true for variables and algebraic expressions as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the first property states that the order in which two real numbers are added does not affect their sum. Basic Rules of Algebra c Let b, a, and be real numbers, variables, or algebraic expressions. Property Commutative Property of Addition: Commutative Property of Multiplication: Associative Property of Addition: Associative Property of Multiplication: Distributive Properties: Additive Identity Property: Multiplicative Identity Property: Additive Inverse Property: Multiplicative Inverse Property: a b b a ab ba a b c a b c abc abc ab c ab ac a bc ac bc, Example 4x x 2 x 2 4x |
4 xx 2 x 24 x x 5 x 2 x 5 x 2 2x 3y8 2x3y 8 3x5 2x 3x 5 3x 2x y 8y y y 8 y 5y 2 0 5y2 4x 21 4x 2 5x3 5x3 0 x 2 4 1 1 x 2 4 Because subtraction is defined as “adding the opposite,” the Distributive Properties are also true for subtraction. For instance, the “subtraction form” of ab c ab ac is ab c ab ac. Properties of Negation and Equality Let a and b be real numbers, variables, or algebraic expressions. Notice the difference between the opposite of a number and a negative number. If is already a, negative, then its opposite, is positive. For instance, if a 5, then a a (5) 5. Property 1. 2. 3. 4. 5. 1a a a a ab ab ab ab ab a b a b 6. If 7. If 8. If 9. If then then a b, a b, a ± c b ± c, ac bc and a ± c b ± c. ac bc. then a b. c 0, then a b. Example 17 7 6 6 53 5 3 53 2x 2x x 8 x 8 x 8 1 3 0.5 3 2 42 2 16 2 1.4 1 7 5 3x 3 4 ⇒ x 4 1 ⇒ 1.4 7 5 333202_0A01.qxd 12/6/05 2:09 PM Page A7 Appendix A.1 Real Numbers and Their Properties A7 The “or” in the Zero-Factor Property includes the possibility that either or both factors may be zero. This is an inclusive or, and it is the way the word “or” is generally used in mathematics. Properties of Zero Let a and b be real numbers, variables, or algebraic expressions. 1. a 0 a and a 0 a 2. a 0 0 3. 0 a 0, a 0 4. a 0 is undefined. 5. Zero-Factor Property: If ab 0, then a 0 or b 0. Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that d 0. b 0 and 1. Equivalent Fractions: 2. Rules of Signs and a b a b if and only if ad |
bc. 3. Generate Equivalent Fractions: a b ac bc, c 0 4. Add or Subtract with Like Denominators. Add or Subtract with Unlike Denominators: a b ± c d ad ± bc bd 6. Multiply Fractions: a b c d ac bd 7. Divide Fractions: a b c d a b d c ad bc, c 0 Example 5 Properties and Operations of Fractions In Property 1 of fractions, the phrase “if and only if” implies two statements. One statement cd, ad bc. is: If The other statement is: If ad bc, where d 0, b 0 then ab cd. ab then and a. Equivalent fractions: 3 x 3 5 x 5 3x 15 b. Divide fractions: c. Add fractions with unlike denominators: x 3 2x 5 Now try Exercise 103. 3 2 7 x 5 x 3 2x 3 5 2 7 x 3 11x 15 14 3x ab c, If a, b, and c are integers such that then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factors — itself and 1—such as 2, 3, 5, 7, and 11. The numbers 4, 6, 8, 9, and 10 are composite because each can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers in precisely one way (disregarding order). For instance, the prime factorization of 24 is 24 2 2 2 3. 333202_0A01.qxd 12/6/05 2:09 PM Page A8 A8 Appendix A Review of Fundamental Concepts of Algebra A.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A real number is ________ if it can be written as the ratio of two integers, where p q q 0. 2. ________ numbers have infinite nonrepeating decimal representations. 3. The distance between a point on the real number line and the origin is the ________ ________ of the real number. 4. |
A number that can be written as the product of two or more prime numbers is called a ________ number. 5. An integer that has exactly two positive factors, the integer itself and 1, is called a ________ number. 6. An algebraic expression is a collection of letters called ________ and real numbers called ________. 7. The ________ of an algebraic expression are those parts separated by addition. 8. The numerical factor of a variable term is the ________ of the variable term. 9. The ________ ________ states that if ab 0, then a 0 or b 0. In Exercises 1– 6, determine which numbers in the set are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers. 1. 2. 3. 4. 5. 6. 3, 2, 0, 1, 4, 2, 11 3, 0, 3.12, 5 4, 3, 12, 5 2, 5, 2 9, 7 5, 7, 7 2.01, 0.666..., 13, 0.010110111..., 1, 6 2.3030030003..., 0.7575, 4.63, 10, 475,, 1 3, 1 2 25, 17, 12 2, 7.5, 1, 8, 22 5, 9, 3.12, 1 11.1, 3, 6, 13 7, 2 In Exercises 19–30, (a) give a verbal description of the subset of real numbers represented by the inequality or the interval, (b) sketch the subset on the real number line, and (c) state whether the interval is bounded or unbounded. 19. 21. 23. 25. 27. 29. x ≤ 5 x < 0 4, 5 20. 22. 24. 26. 28. 30, 2 In Exercises 7–10, use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. In Exercises 31–38, use inequality notation to describe the set. 7. 9. 5 8 41 333 8. 10. 1 3 6 11 31. All x in the interval 32. All y in the interval 2, 4 6, 0 33. y is nonnegative. 34. y is no more than 25. In Exercises |
11 and 12, approximate the numbers and place the correct symbol (< or >) between them. 35. 36. t k is at least 10 and at most 22. is less than 5 but no less than 11. 12. −2 −1 0 1 2 3 −7 −6 −5 −4 −3 −2 −1 4 0 In Exercises 13–18, plot the two real numbers on the real number line. Then place the appropriate inequality symbol (< or >) between them. 13. 15. 17. 4, 8 3 2, 7 6, 2 5 3 14. 16. 18. 3.5, 1 1, 16 3 8 7, 3 7 37. The dog’s weight W 3. is more than 65 pounds. r 38. The annual rate of inflation but no more than 5%. is expected to be at least 2.5% In Exercises 39–48, evaluate the expression. 39. 41. 43. 45. 47. 10 40. 42. 44. 46. 48. 0 4 1 3 3 33 x 1 x 1, x > 1 333202_0A01.qxd 12/6/05 2:09 PM Page A9 Appendix A.1 Real Numbers and Their Properties A9 In Exercises 49–54, place the correct symbol (<, >, or =) between the pair of real numbers. (a) Complete the Expenditures. table. Hint: Find Receipts – 49. 50. 51. 52. 53. 54. 33 44 55 66 22 (2)2 In Exercises 55–60, find the distance between a and b. 55. 56. 57. 58. 59. 60. a 126, b 75 a 126, b 75 a 5 2, b 0 4, b 11 a 1 5, b 112 a 16 a 9.34, b 5.65 75 4 Budget Variance In Exercises 61–64, the accounting department of a sports drink bottling company is checking to see whether the actual expenses of a department differ from the budgeted expenses by more than $500 or by more than 5%. Fill in the missing parts of the table, and determine whether each actual expense passes the “budget variance test.” Budgeted b Expense, Actual Expense, a 61. Wages $112,700 $113,356 62. Utilities $9,400 $9,772 63. Taxes $37,640 $37,335 |
64. Insurance $2,575 $2,613 a b 0.05b 65. Federal Deficit The bar graph shows the federal government receipts (in billions of dollars) for selected years from 1960 through 2000. (Source: U.S. Office of Management and Budget ( 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 2025.2 1032.0 192.8 92.5 1960 1970 517.1 1980 Year 1990 2000 Year 1960 1970 1980 1990 2000 Expenditures (in billions) $92.2 $195.6 $590.9 $1253.2 $1788.8 Surplus or deficit (in billions) (b) Use the table in part (a) to construct a bar graph showing the magnitude of the surplus or deficit for each year. 66. Veterans The table shows the number of living veterans (in thousands) in the United States in 2002 by age group. Construct a circle graph showing the percent of living veterans by age group as a fraction of the total number of (Source: Department of Veteran Affairs) living veterans. Age group Number of veterans Under 35 35– 44 45–54 55– 64 65 and older 2213 3290 4666 5665 9784 In Exercises 67–72, use absolute value notation to describe the situation. x 67. The distance between and 5 is no more than 3. 68. The distance between and x 10 is at least 6. 69. 70. y y is at least six units from 0. is at most two units from a. 71. While traveling on the Pennsylvania Turnpike, you pass milepost 326 near Valley Forge, then milepost 351 near Philadelphia. How many miles do you travel during that time period? 72. The temperature in Chicago, Illinois was 48 last night at midnight, then 82 at noon today. What was the change in temperature over the 12-hour period? 333202_0A01.qxd 12/6/05 2:09 PM Page A10 A10 Appendix A Review of Fundamental Concepts of Algebra In Exercises 73–78, identify the terms. Then identify the coefficients of the variable terms of the expression. (b) Use the result from part (a) to make a conjecture about the value of 5n as approaches 0. n 73. 75. 77. 7x 4 3x2 8x 11 4x3 x 2 5 74. 76. 78. 6x3 5x 33x2 1 3x4 x2 4 106 |
. (a) Use a calculator to complete the table. 1 10 100 10,000 100,000 n 5n In Exercises 79–84, evaluate the expression for each value of x. (If not possible, state the reason.) Expression 4x 6 9 7x x 2 3x 4 x 2 5x 4 x 1 x 1 x x 2 79. 80. 81. 82. 83. 84. Values (a) (a) (a) (ab) (b) (b) (ba) x 1 (b) x 1 (a) x 2 (b) x 2 In Exercises 85–96, identify the rule(s) of algebra illustrated by the statement. 85. x 9 9 x 86. 2 1 2 1 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. h 6 h 6 1, 1 h 6 x 3 x 3 0 2x 5x z x 5 x x y 10 x y 10 x3y x 3y 3xy 3t 4 3 t 3 4 7 12 1 1 7 7 712 1 12 12 In Exercises 97–104, perform the operation(s). (Write fractional answers in simplest form.) 97. 99. 101. 103. 1 6 5 3 16 16 5 5 12 8 12 1 4 2x x 3 4 98. 100. 102. 104. 6 7 10 11 4 7 6 33 6 4 5x 2 6 9 8 13 66 105. (a) Use a calculator to complete the table. 1 0.5 0.01 0.0001 0.000001 n 5n (b) Use the result from part (a) to make a conjecture n as increases without bound. about the value of 5n Synthesis True or False? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. 107. If a < b, 108. Because, < then where a b 0., then c a b u v and. c c b a u v, 109. Exploration Consider where u v 0. (a) Are the values of the expressions always equal? If not, under what conditions are they unequal? (b) If the two expressions are not equal for certain values is one of the expressions always greater and v, u of than the other? Explain. 110. Think About It Is there a difference between saying that a real number is positive and saying that a real number |
is nonnegative? Explain. 111. Think About It Because every even number is divisible is it possible that there exist any even prime by 2, numbers? Explain. 112. Writing Describe the differences among the sets of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. In Exercises 113 and 114, use the real numbers A, B, and C shown on the number line. Determine the sign of each expression. C B 0 A 113. (a) (b) A B A 114. (a) (b) C A C a a 115. Writing Can it ever be true that Explain. number a? for a real 333202_0A02.qxd 12/6/05 2:12 PM Page A11 A.2 Exponents and Radicals Appendix A.2 Exponents and Radicals A11 What you should learn • Use properties of exponents. • Use scientific notation to represent real numbers. • Use properties of radicals. • Simplify and combine radicals. • Rationalize denominators and numerators. • Use properties of rational exponents. Why you should learn it Real numbers and algebraic expressions are often written with exponents and radicals. For instance, in Exercise 105 on page A22, you will use an expression involving rational exponents to find the time required for a funnel to empty for different water heights. Te c h n o l o g y You can use a calculator to evaluate exponential expressions. When doing so, it is important to know when to use parentheses because the calculator follows the order of operations. For instance, evaluate 24 as follows Scientific: 2 y x 4 Graphing: 2 > 4 ENTER The display will be 16. If you omit the parentheses, the display will be 16. Integer Exponents Repeated multiplication can be written in exponential form. Repeated Multiplication a a a a a 444 2x2x2x2x Exponential Form a5 43 2x4 Exponential Notation n is a real number and If an a a a... a a is a positive integer, then n factors is the exponent and n where the th power.” n a is the base. The expression an is read “ to a An exponent can also be negative. In Property 3 below, be sure you see how to use a negative exponent. Properties of Exponents Let and be real numbers, variables, or algebraic expressions, and let and be integers. (All denominators and |
bases are nonzero.) a n b m Property aman amn Example 32 34 324 36 729 1. 2. 3. 4. 5. amn am an an 1 an a0 1, abm ambm n 1 a a 0 6. amn amn m a am bm b a2 a2 a2 7. 8. 4 x7 4 x3 x 7 x4 1 y4 1 y4 y x 2 10 1 5x3 53x3 125x3 y34 y3(4) y12 1 y12 3 23 x3 2 8 x3 x 22 22 22 4 333202_0A02.qxd 12/6/05 2:12 PM Page A12 A12 Appendix A Review of Fundamental Concepts of Algebra In and 24. It is important to recognize the difference between expressions such as the parentheses indicate that the exponent applies to the exponent applies 24 24, the negative sign as well as to the 2, but in only to the 2. So, 24 16. The properties of exponents listed on the preceding page apply to all integers 24 24, 24 16 and m and n, not just to positive integers as shown in the examples in this section. Example 1 Using Properties of Exponents Use the properties of exponents to simplify each expression. a. 3ab44ab3 b. 2xy 23 c. 3a4a20 d. 5x3 y 2 Solution a. 3ab44ab3 34aab4b3 12a2b 2xy 23 23x3y 23 8x3y6 3a4a20 3a1 3a, 52x32 5x3 25x. c. d. Now try Exercise 25. Example 2 Rewriting with Positive Exponents Rewrite each expression with positive exponents. a. x1 b. 1 3x2 c. 12a3b4 4a2b d. 3x2 y 2 c. a. b. 1 3x2 12a3b4 4a2b 2 3x2 y Solution x1 1 x 1x2 x 2 3 3 12a3 a2 4b b4 32x22 y2 32x4 y2 y2 32x4 y2 9x 4 d. Property 3 The exponent 2 does not apply to 3. 3a5 b5 Properties 3 and 1 Properties 5 and 7 Property 6 Property 3 Simplify. Now try Exercise 33. Rarely in algebra is there only one way |
to solve a problem. Don’t be concerned if the steps you use to solve a problem are not exactly the same as the steps presented in this text. The important thing is to use steps that you understand and, of course, steps that are justified by the rules of algebra. For instance, you might prefer the following steps for Example 2(d). 3x 2 y 2 y 3x 22 y 2 9x4 Note how Property 3 is used in the first step of this solution. The fractional form of this property is m a b b a m. 333202_0A02.qxd 12/6/05 2:12 PM Page A13 Appendix A.2 Exponents and Radicals A13 Scientific Notation Exponents provide an efficient way of writing and computing with very large (or very small) numbers. For instance, there are about 359 billion billion gallons of water on Earth—that is, 359 followed by 18 zeros. 359,000,000,000,000,000,000 It is convenient to write such numbers in scientific notation. This notation has is an integer. So, the number of the form gallons of water on Earth can be written in scientific notation as 1 ≤ c < 10 ± c 10n, where and n 3.59 100,000,000,000,000,000,000 3.59 1020. The positive exponent 20 indicates that the number is large (10 or more) and that the decimal point has been moved 20 places. A negative exponent indicates that the number is small (less than 1). For instance, the mass (in grams) of one electron is approximately 9.0 1028 0.0000000000000000000000000009. 28 decimal places Example 3 Scientific Notation Write each number in scientific notation. a. 0.0000782 b. 836,100,000 Solution a. 0.0000782 7.82 105 Now try Exercise 37. b. 836,100,000 8.361 108 Example 4 Decimal Notation Write each number in decimal notation. a. 9.36 106 b. 1.345 102 Solution a. 9.36 106 0.00000936 Now try Exercise 41. b. 1.345 102 134.5 Te c h n o l o g y Most calculators automatically switch to scientific notation when they are showing large (or small) numbers that exceed the display range. To enter numbers in scientific notation, your calculator should have an expo- nential entry key labeled EE |
or EXP. Consult the user’s guide for your calculator for instructions on keystrokes and how numbers in scientific notation are displayed. 333202_0A02.qxd 12/6/05 2:12 PM Page A14 A14 Appendix A Review of Fundamental Concepts of Algebra Radicals and Their Properties A square root of a number is one of its two equal factors. For example, 5 is a square root of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube root of a number is one of its three equal factors, as in 125 53. Definition of nth Root of a Number Let a and b be real numbers and let n ≥ 2 be a positive integer. If a bn then b is an nth root of a. If root is a cube root. n 2, the root is a square root. If n 3, the Some numbers have more than one nth root. For example, both 5 and 25, square roots of 25. The principal square root of 25, written as root, 5. The principal nth root of a number is defined as follows. 5 are is the positive Principal nth Root of a Number Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol na. Principal nth root The positive integer n is the index of the radical, and the number a is the radicand. If plural of index is indices.) omit the index and write rather than n 2, 2a. (The a A common misunderstanding is that the square root sign implies both negative and positive roots. This is not correct. The square root sign implies only a positive root. When a negative root is needed, you must use the negative sign with the square root sign. Incorrect: 4 ±2 Correct: 4 2 and 4 2 Example 5 Evaluating Expressions Involving Radicals a. b. c. d. e. because 62 36. 36 6 36 6 3125 5 4 64 532 2 481 to the fourth power to produce because because 36 62 6 6. because 5 3 53 125 43 64 4 25 32.. 81. is not a real number because there is no real number that can be raised Now try Exercise 51. 333202_0A02.qxd 12/6/05 2:12 PM Page A15 Appendix A |
.2 Exponents and Radicals A15 Here are some generalizations about the nth roots of real numbers. Generalizations About nth Roots of Real Numbers Real Number a Integer n a > 0 n > 0, is even. Root(s) of a na na, Example 481 3, 481 3 a > 0 or a < 0 n is odd. na 38 2 a < 0 a 0 n is even. No real roots 4 is not a real number. n is even or odd. n0 0 50 0 Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called perfect cubes because they have integer cube roots. Properties of Radicals Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers. Property nam nam na nb nab na na nb b m na mna nan a, 1. 2. 3. 4. 5. b 0 6. For n even, For n odd, nan a. nan a. 43 Example 382 382 22 4 5 7 5 7 35 427 427 49 9 310 610 32 3 122 12 12 3123 12 A common special case of Property 6 is a2 a. Example 6 Using Properties of Radicals Use the properties of radicals to simplify each expression. a. 8 2 35 3 b. c. 3x3 d. 6y6 Solution a. 8 2 8 2 16 4 353 5 3x3 x 6y6 y b. c. d. Now try Exercise 61. 333202_0A02.qxd 12/6/05 2:12 PM Page A16 A16 Appendix A Review of Fundamental Concepts of Algebra When you simplify a radical, it is important that both expressions are defined for the same values of the variable. For instance, in Example 7(b), 75x3 5x3x are both defined only for nonnegative values of Similarly, in Example 7(c), and are both defined for all real values of x. 45x4 5x and x. Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied. 1. All possible factors have been removed from the radical. 2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator). |
3. The index of the radical is reduced. To simplify a radical, factor the radicand into factors whose exponents are multiples of the index. The roots of these factors are written outside the radical, and the “leftover” factors make up the new radicand. Example 7 Simplifying Even Roots Perfect 4th power Leftover factor a. 448 416 3 424 3 2 43 Perfect square Leftover factor b. c. 75x3 25x 2 3x 5x2 3x 5x3x 45x4 5x 5x Find largest square factor. Find root of perfect square. Now try Exercise 63(a). Example 8 Simplifying Odd Roots Perfect cube Leftover factor a. 324 38 3 323 3 2 33 Perfect cube Leftover factor b. c. 324a4 38a3 3a 32a3 3a 2a 33a 340x6 38x6 5 32x23 5 2x2 35 Find largest cube factor. Find root of perfect cube. Find largest cube factor. Find root of perfect cube. Now try Exercise 63(b). 333202_0A02.qxd 12/6/05 2:12 PM Page A17 Appendix A.2 Exponents and Radicals A17 Radical expressions can be combined (added or subtracted) if they are like 2, are unlike radicals. To determine radicals—that is, if they have the same index and radicand. For instance, 32, 3 whether two radicals can be combined, you should first simplify each radical. are like radicals, but 1 and 2 and 2 2 Example 9 Combining Radicals a. 248 327 216 3 39 3 83 93 8 93 3 b. 316x 354x4 38 2x 327 x3 2x 2 32x 3x 32x 2 3x 32x Now try Exercise 71. Find square factors. Find square roots and multiply by coefficients. Combine like terms. Simplify. Find cube factors. Find cube roots. Combine like terms. Rationalizing Denominators and Numerators a bm To rationalize a denominator or numerator of the form multiply both numerator and denominator by a conjugate: a bm m is itself, perfect cube. a bm, and then the rationalizing factor for For cube roots, choose a rationalizing factor that generates a are conjugates of each other. If a bm a 0, m. or Example 10 Rationalizing Single-Term Denominators Rationalize the denominator of |
each expression. a. 5 23 b. 2 35 Solution 5 23 a. 5 23 53 23 53 6 3 3 3 is rationalizing factor. Multiply. Simplify. b. 2 35 2 35 2 352 353 2 325 5 352 352 352 is rationalizing factor. Multiply. Simplify. Now try Exercise 79. 333202_0A02.qxd 12/6/05 2:13 PM Page A18 A18 Appendix A Review of Fundamental Concepts of Algebra Example 11 Rationalizing a Denominator with Two Terms 23 7 33 37 73 77 23 7 32 72 23 7 9 7 23 7 2 3 7 Multiply numerator and denominator by conjugate of denominator. Use Distributive Property. Simplify. Square terms of denominator. Simplify. Now try Exercise 81. Sometimes it is necessary to rationalize the numerator of an expression. For instance, in Appendix A.4 you will use the technique shown in the next example to rationalize the numerator of an expression from calculus. Do not confuse the expression 5 7 with the expression 5 7. does not equal Similarly, equal x y. x y. x2 y2 In general, x y does not Example 12 Rationalizing a Numerator Multiply numerator and denominator by conjugate of numerator. Simplify. Square terms of numerator. 5 2 7 2 25 7 5 7 25 7 2 25 7 1 5 7 Simplify. Now try Exercise 85. Rational Exponents Definition of Rational Exponents If a is a real number and n is a positive integer such that the principal nth root of a exists, then a1n na, is the rational exponent of a. is defined as where a1n 1n Moreover, if m is a positive integer that has no common factor with n, then amn a1nm nam and amn am1n nam. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0A02.qxd 12/6/05 2:13 PM Page A19 Appendix A.2 Exponents and Radicals A19 The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken. Power Index bmn nbm nbm When you are working with rational exponents, the properties of integer exponents still apply. For instance, 212213 2(12)(13) 256 |
. Example 13 Changing from Radical to Exponential Form is not bmn is a real Rational exponents can be tricky, and you must remember that the expression nb defined unless number. This restriction produces some unusual-looking results. For instance, the number 813 is defined because 38 2, 826 68 but the number is undefined because is not a real number. a. b. c. 3 312 3xy5 23xy5 3xy(52) 2x 4x3 2xx34 2x1(34) 2x74 Now try Exercise 87. Te c h n o l o g y. For other There are four methods of evaluating radicals on most graphing calculators. For square roots, you can use the square root key. For cube roots, you can use the cube root key 3 roots, you can first convert the radical to exponential form and then use the exponential key or you can use the xth root key. Consult the user’s guide x for your calculator for specific keystrokes. >, Example 14 Changing from Exponential to Radical Form a. b. c. x2 y 232 x2 y 23 x 2 y 23 2y34z14 2y3z14 2 4y3z a32 1 1 a32 a3 d. x 0.2 x15 5x Now try Exercise 89. Rational exponents are useful for evaluating roots of numbers on a calculator, for reducing the index of a radical, and for simplifying expressions in calculus. Example 15 Simplifying with Rational Exponents x 0 Reduce index. a. b. c. d. e. f. 3245 5324 24 1 24 1 16 5x533x34 15x(53)(34) 15x1112, 9a3 a39 a13 3a 3125 6125 653 536 512 5 2x 1432x 113 2x 1(43)(13) x 1 2 2x 1, x 1 x 112 x 1 x 112 x 132 x 10 x 132, x 112 x 112 x 1 Now try Exercise 99. 333202_0A02.qxd 12/6/05 2:13 PM Page A20 A20 Appendix A Review of Fundamental Concepts of Algebra A.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In the exponential form nan, is the ________ and a is the ________. 2. A convenient way of writing very large or very small numbers is called |
________ ________. 3. One of the two equal factors of a number is called a __________ __________ of the number. 4. The ________ ________ ________ of a number is the th root that has the same sign as n a, and is denoted by 5. In the radical form, and the number a na. na is called the ________. the positive integer n is called the ________ of the radical 6. When an expression involving radicals has all possible factors removed, radical-free denominators, and a reduced index, it is in ________ ________. 7. The expressions a bm and a bm are ________ of each other. 8. The process used to create a radical-free denominator is know as ________ the denominator. 9. In the expression bmn, m denotes the ________ to which the base is raised and denotes n the ________ or root to be taken. In Exercises 1 and 2, write the expression as a repeated multiplication problem. In Exercises 17–24, evaluate the expression for the given value of x. 1. 85 2. 27 In Exercises 3 and 4, write the expression using exponential notation. 3. 4. 4.94.94.94.94.94.9 1010101010 In Exercises 5–12, evaluate each expression. 5. (a) 6. (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. (a) 12. (a) 32 3 55 52 330 23 322 3 44 34 41 4 32 22 31 21 31 31 22 (b) (b) (b) (b) 3 33 32 34 32 3 35 2 3 5 (b) 3225 (b) 20 (b) (b) 212 322 In Exercises 13–16, use a calculator to evaluate the expression. (If necessary, round your answer to three decimal places.) 4352 36 73 84103 43 34 13. 15. 16. 14. Expression 3x3 7x2 6x0 5x3 2x3 3x 4 4x2 5x3 17. 18. 19. 20. 21. 22. 23. 24. Value x 2 x 4 x 10 In Exercises 25–30, simplify each expression. 25. (a) 26. (a) 5z3 3x2 27. (a) 6y22y02 28 |
. (a) z33z4 29. (a) 30. (a) 7x 2 x3 r 4 r 6 (b) (b) (b) (b) (b) (b) 5x4x2 4x30 3x5 x3 25y8 10y4 12x y3 9x y 4 33 4 y y In Exercises 31–36, rewrite each expression with positive exponents and simplify. x 50, 2x50, (b) (b) z 23z 21 x 5 2x 22 32. (a) 31. (a) x 0 333202_0A02.qxd 12/6/05 2:13 PM Page A21 Appendix A.2 Exponents and Radicals A21 33. (a) 2x234x31 34. (a) 4y28y4 35. (a) 3n 32n 36. (a) x 2 xn x3 xn (b) (b) (b) (b) 1 x 10 x3y4 3 5 a2 b2b a3 b3a a b 3 3 In Exercises 37– 40, write the number in scientific notation. 37. Land area of Earth: 57,300,000 square miles 38. Light year: 9,460,000,000,000 kilometers 39. Relative density of hydrogen: 0.0000899 gram per cubic centimeter 40. One micron (millionth of a meter): 0.00003937 inch In Exercises 41– 44, write the number in decimal notation. 41. Worldwide daily consumption of Coca-Cola: (Source: The Coca-Cola Company) ounces 4.568 109 54. (a) 55. (a) 56. (a) 10032 1 64 125 27 13 13 (b) (b) (b) 12 9 4 1 32 1 125 25 43 In Exercises 57– 60, use a calculator to approximate the number. (Round your answer to three decimal places.) 57. (a) 58. (a) 59. (a) 60. (a) 57 3452 12.41.8 7 4.13.2 2 (b) (b) (b) (b) 5273 6125 532.5 13 32 3 133 3 2 In Exercises 61 and 62, use the properties of radicals to simplify each expression. 61. (a) 62. (a) 343 12 |
3 (b) (b) 596x5 43x24 42. Interior temperature of the sun: 1.5 107 degrees Celsius In Exercises 63–74, simplify each radical expression. 43. Charge of an electron: 44. Width of a human hair: 1.6022 1019 9.0 105 meter coulomb In Exercises 45 and 46, evaluate each expression without using a calculator. 45. (a) 25 108 46. (a) 1.2 1075 103 (b) (b) 38 1015 6.0 108 3.0 103 In Exercises 47–50, use a calculator to evaluate each expression. (Round your answer to three decimal places.) 47. (a) (b) 48. (a) (b) 49. (a) 50. (a) 800 7501 0.11 365 67,000,000 93,000,000 0.0052 9.3 10636.1 104 2.414 1046 1.68 1055 4.5 109 2.65 10413 (b) (b) 36.3 104 9 104 63. (a) 64. (a) 8 316 27 65. (a) 72x3 66. (a) 54xy4 67. (a) 68. (a) 69. (a) 70. (a) 71. (a) 72. (a) 73. (a) 74. (a) 316x5 43x4y2 250 128 427 75 5x 3x 849x 14100x 3x 1 10x 1 x 3 7 5x3 7 (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) z3 354 75 4 182 32a4 b2 75x2y4 5160x8z4 1032 618 316 3 354 29y 10y 348x2 7 75x2 780x 2125x 11245x 3 945x 3 In Exercises 75–78, complete the statement with <, =, or >. 75. 77. 5 3 5 3 532 22 3 3 11 532 42 11 76. 78. In Exercises 51–56, evaluate each expression without using a calculator. In Exercises 79–82, rationalize the denominator of the expression. Then simplify your answer. 51. (a) 52. |
(a) 53. (a) 9 2713 3235 (b) (b) (b) 327 8 3632 16 81 34 79. 1 3 80. 5 10 333202_0A02.qxd 12/6/05 2:13 PM Page A22 A22 Appendix A Review of Fundamental Concepts of Algebra 104. Erosion A stream of water moving at the rate of v feet inches. Find per second can carry particles of size the size of the largest particle that can be carried by a 3 stream flowing at the rate of 4 foot per second. 0.03v 105. Mathematical Modeling A funnel is filled with water to h a height of centimeters. The formula t 0.031252 12 h52, 0 ≤ h ≤ 12 represents the amount of time take for the funnel to empty. t (in seconds) that it will (a) Use the table feature of a graphing utility to find the times required for the funnel to empty for water h 12 heights of centimeters.... h 0, h 1, h 2, (b) What value does appear to be approaching as the height of the water becomes closer and closer to 12 centimeters? t 106. Speed of Light The speed of light is approximately 11,180,000 miles per minute. The distance from the sun to Earth is approximately 93,000,000 miles. Find the time for light to travel from the sun to Earth. Synthesis True or False? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. 107. x k1 x x k 108. ank ank 109. Verify that exponents a0 1, a 0. aman amn. ) (Hint: Use the property of 110. Explain why each of the following pairs is not equal. 3x1 3 x a2b34 a6b7 4x2 2x (a) (c) (e) (b) y3 y2 y6 (d) (f) a b2 a2 b2 2 3 5 111. Exploration List all possible digits that occur in the units place of the square of a positive integer. Use that list to determine whether is an integer. 5233 112. Think About It Square 25 and note that the radical is eliminated from the denominator. Is this equivalent to rationalizing the denominator? Why or why not? the real number 81. 2 5 3 82. 3 5 6 In Exercises |
83– 86, rationalize the numerator of the expression. Then simplify your answer. 83. 85. 8 2 5 3 3 84. 86. 2 3 7 3 4 In Exercises 87–94, fill in the missing form of the expression. Rational Exponent Form Radical Form 9 364 3215 14412 3216 24315 4813 1654 87. 88. 89. 90. 91. 92. 93. 94. In Exercises 95–98, perform the operations and simplify. 95. 97. 2x232 212x4 x3 x12 x32 x1 96. 98. x43y23 xy13 512 5x52 5x32 In Exercises 99 and 100, reduce the index of each radical. 99. (a) 100. (a) 432 6x3 (b) (b) 6(x 1)4 4(3x2)4 In Exercises 101 and 102, write each expression as a single radical. Then simplify your answer. 101. (a) 102. (a) 32 243x 1 (b) (b) 42x 310a7b T 103. Period of a Pendulum The period (in seconds) of a pendulum is T 2 L 32 L is the length of the pendulum (in feet). Find the where period of a pendulum whose length is 2 feet. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. 333202_0A03.qxd 12/6/05 2:14 PM Page A23 A.3 Polynomials and Factoring Appendix A.3 Polynomials and Factoring A23 What you should learn • Write polynomials in standard form. • Add, subtract, and multiply polynomials. • Use special products to multiply polynomials. • Remove common factors from polynomials. Polynomials 2x 5, The most common type of algebraic expression is the polynomial. Some examples are The first two are polynomials in x and the third is a polynomial in x and y. The terms of a polynomial in x have the form where a is the coefficient and k is the degree of the term. For instance, the polynomial 3x 4 7x 2 2x 4, 5x 2y 2 xy 3. |
ax k, and 2x 3 5x 2 1 2x 3 5x 2 0x 1 • Factor special polynomial has coefficients 2, 5, 0, and 1. forms. • Factor trinomials as the product of two binomials. • Factor polynomials by grouping. Why you should learn it Polynomials can be used to model and solve real-life problems. For instance, in Exercise 210 on page A34, a polynomial is used to model the stopping distance of an automobile. be real numbers and let n be a nonnegative integer. a0, a1, a2,..., an Definition of a Polynomial in x Let A polynomial in x is an expression of the form anx n an1x n1... a1x a 0 The polynomial is of degree 0. n, where a0 and an is the constant term. an is the leading coefficient, Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. In standard form, a polynomial is written with descending powers of x. Example 1 Writing Polynomials in Standard Form Polynomial 4x 2 5x 7 2 3x 4 9x 2 a. b. c. 8 Standard Form 5x 7 4x 2 3x 2 9x 2 4 8 8 8x 0 Degree 7 2 0 Now try Exercise 11. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polynomials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree is of its terms. For instance, the degree of the polynomial 11 because the sum of the exponents in the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest-degree term. Expressions are not polynomials if a variable is underneath a radical or if a polynomial expression (with degree greater than 0) is in the denominator of a term. The following expressions are not polynomials. 2x3y6 4xy x7y4 x3 3x x3 3x12 x2 5 x x2 5x1 The exponent “ 12 ” |
is not an integer. The exponent “ 1 ” is not a nonnegative integer. 333202_0A03.qxd 12/6/05 2:14 PM Page A24 A24 Appendix A Review of Fundamental Concepts of Algebra Operations with Polynomials You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms (terms having the same variables to the same powers) by adding their coefficients. For instance, 3xy 2 are like terms and their sum is 5xy 2 and 3xy 2 5xy 2 3 5xy 2 2xy 2. When an expression inside parentheses is preceded by a negative sign, remember to distribute the negative sign to each term inside the parentheses, as shown. x 2 x 3 x 2 x 3 Example 2 Sums and Differences of Polynomials a. 5x 3 7x 2 3 x 3 2x 2 x 8 5x 3 x 3 7x2 2x2 x 3 8 6x 3 5x 2 x 5 Group like terms. Combine like terms. b. 7x4 x 2 4x 2 3x4 4x 2 3x 7x4 x 2 4x 2 3x4 4x 2 3x 7x4 3x4 x2 4x2 4x 3x 2 4x4 3x 2 7x 2 Distributive Property Group like terms. Combine like terms. Now try Exercise 33. To find the product of two polynomials, use the left and right Distributive as a single quantity, you can 5x 7 Properties. For example, if you treat as follows. multiply 5x 7 3x 2 by 3x 25x 7 3x5x 7 25x 7 3x5x 3x7 25x 27 15x 2 21x 10x 14 Product of First terms Product of Outer terms Product of Inner terms Product of Last terms 15x 2 11x 14 Note in this FOIL Method (which can only be used to multiply two binomials) that the outer (O) and inner (I) terms are like terms and can be combined. Example 3 Finding a Product by the FOIL Method Use the FOIL Method to find the product of 2x 4 and x 5. Solution L F 2x 4x 5 2x2 10x 4x 20 O I 2x2 6x 20 Now try Exercise 47. 333202_0A03.qxd 12/6/05 2:14 PM |
Page A25 Appendix A.3 Polynomials and Factoring A25 Special Products Some binomial products have special forms that occur frequently in algebra. You do not need to memorize these formulas because you can use the Distributive Property to multiply. However, becoming familiar with these formulas will enable you to manipulate the algebra more quickly. Special Products v Let and be real numbers, variables, or algebraic expressions. u Special Product Example Sum and Difference of Same Terms u vu v u 2 v 2 Square of a Binomial u v2 u 2 2uv v 2 u v2 u 2 2uv v 2 Cube of a Binomial u v3 u 3 3u 2v 3uv 2 v 3 u v3 u 3 3u 2v 3uv2 v3 x 4x 4 x 2 42 x 2 16 x 32 x 2 2x3 32 x 2 6x 9 3x 22 3x2 23x2 22 9x 2 12x 4 x 23 x3 3x 22 3x22 23 x 3 6x 2 12x 8 x 13 x33x 213x1213 x 3 3x 2 3x 1 Example 4 Special Products Find each product. a. 5x 9 and 5x 9 b. x y 2 and x y 2 Solution a. The product of a sum and a difference of the same two terms has no middle u vu v u 2 v 2. 25x 2 81 5x 95x 9 5x2 9 2 term and takes the form b. By grouping x y as a special product. in parentheses, you can write the product of the trinomials Difference Sum x y 2x y 2 x y 2x y 2 x y 2 22 x 2 2xy y 2 4 Sum and difference of same terms Now try Exercise 67. 333202_0A03.qxd 12/6/05 2:14 PM Page A26 A26 Appendix A Review of Fundamental Concepts of Algebra Polynomials with Common Factors The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, then it is prime or irreducible over the integers. For instance, the polynomial is irreducible over the integers. Over the real numbers, this |
polynomial can be factored as x2 3 x2 3 x 3x 3. A polynomial is completely factored when each of its factors is prime. For instance x3 x2 4x 4 x 1x2 4 Completely factored is completely factored, but x 3 x2 4x 4 x 1x2 4 Not completely factored is not completely factored. Its complete factorization is x 3 x2 4x 4 x 1x 2x 2. The simplest type of factoring involves a polynomial that can be written as the product of a monomial and another polynomial. The technique used here is the Distributive Property, ab ac ab c ab c ab ac, in the reverse direction. a is a common factor. Removing (factoring out) any common factors is the first step in completely factoring a polynomial. Example 5 Removing Common Factors Factor each expression. a. b. c. 6x3 4x 4x2 12x 16 x 22x x 23 Solution a. 6x 3 4x 2x3x2 2x2 2x3x2 2 2x is a common factor. b. 4x2 12x 16 4x2 43x 44 4 is a common factor. 4x2 3x 4 c. x 22x x 23 x 22x 3 x 2 is a common factor. Now try Exercise 91. 333202_0A03.qxd 12/6/05 2:14 PM Page A27 Appendix A.3 Polynomials and Factoring A27 Factoring Special Polynomial Forms Some polynomials have special forms that arise from the special product forms on page A25. You should learn to recognize these forms so that you can factor such polynomials easily. Factoring Special Polynomial Forms Factored Form Difference of Two Squares u 2 v 2 u vu v Perfect Square Trinomial u 2 2uv v 2 u v2 u 2 2uv v 2 u v2 Sum or Difference of Two Cubes u 3 v3 u vu 2 uv v2 u3 v3 u vu2 uv v2 Example 9x 2 4 3x 2 2 2 3x 23x 2 x 2 6x 9 x 2 2x3 32 x 2 6x 9 x 2 2x3 32 x 32 x 32 x 3 8 x 3 23 27x3 1 3x 3 13 x 2x 2 2x 4 3x 19 |
x 2 3x 1 In Example 6, note that the first step in factoring a polynomial is to check for any common factors. Once the common factors are removed, it is often possible to recognize patterns that were not immediately obvious. One of the easiest special polynomial forms to factor is the difference of two squares. The factored form is always a set of conjugate pairs. u 2 v 2 u vu v Conjugate pairs Difference Opposite signs To recognize perfect square terms, look for coefficients that are squares of integers and variables raised to even powers. Example 6 Removing a Common Factor First 3 12x2 31 4x2 3 is a common factor. 312 2x2 31 2x1 2x Now try Exercise 105. Difference of two squares Example 7 Factoring the Difference of Two Squares a. x 22 y2 x 2 yx 2 y x 2 yx 2 y b. 16x 4 81 4x22 92 4x2 94x2 9 4x2 92x2 32 4x2 92x 32x 3 Now try Exercise 109. Difference of two squares Difference of two squares 333202_0A03.qxd 12/6/05 2:14 PM Page A28 A28 Appendix A Review of Fundamental Concepts of Algebra A perfect square trinomial is the square of a binomial, and it has the following form. u 2 2uv v 2 u v 2 or u 2 2uv v 2 u v 2 Like signs Like signs Note that the first and last terms are squares and the middle term is twice the product of and v. u Example 8 Factoring Perfect Square Trinomials Factor each trinomial. x2 10x 25 16x2 24x 9 b. a. Solution a. x 2 10x 25 x 2 2x5 5 2 16x2 24x 9 (4x2 24x3 32 4x 32 x 52 b. Now try Exercise 115. The next two formulas show the sums and differences of cubes. Pay special attention to the signs of the terms. Like signs Like signs u 3 v 3 u vu 2 uv v 2 u 3 v 3 u vu 2 uv v 2 Unlike signs Unlike signs Example 9 Factoring the Difference of Cubes Factor x 3 27. Solution x3 27 x3 33 Rewrite 27 as 33. x 3x 2 3x 9 Factor. Now try Exercise 123. Example 10 Factoring the Sum of Cubes a. y 3 8 y |
3 23 y 2y 2 2y 4 b. 3x 3 64 3x 3 43 Rewrite 8 as 23. Factor. Rewrite 64 as 43. 3x 4x 2 4x 16 Factor. Now try Exercise 125. 333202_0A03.qxd 12/6/05 2:14 PM Page A29 Appendix A.3 Polynomials and Factoring A29 Trinomials with Binomial Factors ax 2 bx c, To factor a trinomial of the form use the following pattern. Factors of a ax2 bx c x x Factors of c inner products add up to the middle term 6x 2 17x 5, one has outer and inner products that add up to a The goal is to find a combination of factors of and such that the outer and For instance, in the trinomial you can write all possible factorizations and determine which 17x. 2x 13x 5, 2x 53x 1 6x 5x 1, 6x 1x 5, bx. c 2x 53x 1 You can see that (O) and inner (I) products add up to is the correct factorization because the outer 17x. F O I L O I 2x 53x 1 6x 2 2x 15x 5 6x2 17x 5. Example 11 Factoring a Trinomial: Leading Coefficient Is 1 Factor x 2 7x 12. Solution The possible factorizations are x 2x 6, x 1x 12, and x 3x 4. Testing the middle term, you will find the correct factorization to be x 2 7x 12 x 3x 4. Now try Exercise 131. Example 12 Factoring a Trinomial: Leading Coefficient Is Not 1 Factoring a trinomial can involve trial and error. However, once you have produced the factored form, it is an easy matter to check your answer. For instance, you can verify the factorization in Example 11 by multiplying out the expression x 3x 4 to see that you obtain the original trinomial, x2 7x 12. Factor 2x 2 x 15. Solution The eight possible factorizations are as follows. 2x 1x 15 2x 3x 5 2x 5x 3 2x 15x 1 2x 1x 15 2x 3x 5 2x 5x 3 2x 15x 1 Testing the middle term, you will find the correct factorization to be 2x 2 x 15 2x 5x 3. O I 6x |
5x x Now try Exercise 139. 333202_0A03.qxd 12/6/05 2:14 PM Page A30 A30 Appendix A Review of Fundamental Concepts of Algebra Factoring by Grouping Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping. It is not always obvious which terms to group, and sometimes several different groupings will work. Example 13 Factoring by Grouping Use factoring by grouping to factor x 3 2x2 3x 6. Solution x 3 2x 2 3x 6 x 3 2x 2 3x 6 x 2x 2 3x 2 x 2x 2 3 Group terms. Factor each group. Distributive Property Now try Exercise 147. Factoring a trinomial can involve quite a bit of trial and error. Some of this trial and error can be lessened by using factoring by grouping. The key to this method of factoring is knowing how to rewrite the middle term. In general, to ac factor a trinomial b that add up to and use these factors to rewrite the middle term. This technique is illustrated in Example 14. by grouping, choose factors of the product ax2 bx c Example 14 Factoring a Trinomial by Grouping Use factoring by grouping to factor 2x 2 5x 3. Another way to factor the polynomial in Example 13 is to group the terms as follows. x3 2x2 3x 6 x3 3x 2x2 6 xx2 3 2x2 3 x2 3x 2 As you can see, you obtain the same result as in Example 13. Solution 2x 2 5x 3, In the trinomial 6. 6 ac Now, product is rewrite the middle term as factors as 5x 6x x. 2x 2 5x 3 2x 2 6x x 3 a 2 and 61 c 3, and which implies that the So, you can 6 1 5 b. This produces the following. 2x 2 6x x 3 2xx 3 x 3 x 32x 1 Rewrite middle term. Group terms. Factor groups. Distributive Property So, the trinomial factors as 2x 2 5x 3 x 32x 1. Now try Exercise 153. Guidelines for Factoring Polynomials 1. Factor out any common factors using the Distributive Property. 2. Factor according to one of the special polynomial forms. ax2 bx c mx rnx s. 3 |
. Factor as 4. Factor by grouping. 333202_0A03.qxd 12/6/05 2:14 PM Page A31 Appendix A.3 Polynomials and Factoring A31 A.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. For the polynomial anxn an1xn1... a1x a0, ________, and the constant term is ________. the degree is ________, the leading coefficient is 2. A polynomial in x in standard form is written with ________ powers of x. 3. A polynomial with one term is called a ________, while a polynomial with two terms is called a ________, and a polynomial with three terms is called a ________. 4. To add or subtract polynomials, add or subtract the ________ ________ by adding their coefficients. 5. The letters in “FOIL” stand for the following. F ________ O ________ I ________ L ________ 6. The process of writing a polynomial as a product is called ________. 7. A polynomial is ________ ________ when each of its factors is prime. In Exercises 1–6, match the polynomial with its description. [The polynomials are labeled (a), (b), (c), (d), (e), and (f).] In Exercises 23–28, determine whether the expression is a polynomial. If so, write the polynomial in standard form. (a) (c) (e) 3x 2 x 3 3x 2 3x 1 3x 5 2x 3 x (b) (d) (f) 1 2x 3 12 2 3x 4 x 2 10 1. A polynomial of degree 0 2. A trinomial of degree 5 3. A binomial with leading coefficient 2 4. A monomial of positive degree 5. A trinomial with leading coefficient 2 3 6. A third-degree polynomial with leading coefficient 1 In Exercises 7–10, write a polynomial that fits the description. (There are many correct answers.) 7. A third-degree polynomial with leading coefficient 2 8. A fifth-degree polynomial with leading coefficient 6 9. A fourth-degree binomial with a negative leading coefficient 10. A third-degree bin |
omial with an even leading coefficient In Exercises 11–22, (a) write the polynomial in standard form, (b) identify the degree and leading coefficient of the polynomial, and (c) state whether the polynomial is a monomial, a binomial, or a trinomial. 11. 13. 15. 17. 19. 21. 14x 1 2x5 3x 4 2x 2 5 x 5 1 3 1 6x4 4x5 4x3y 12. 14. 16. 18. 20. 22. 2x 2 x 1 7x y 25y2 1 t2 9 3 2x x5y 2x2y2 xy 4 23. 25. 27. 28. 2x 3x 3 8 3x 4 x y2 y4 y 3 y 2 y4 24. 26. 2x 3 x 3x1 x2 2x 3 2 33. 34. 30. 32. 29. 36. 35. 31. In Exercises 29– 46, perform the operation and write the result in standard form. 6x 5 8x 15 2x 2 1 x 2 2x 1 x 3 2 4x 3 2x 5x 2 1 3x 2 5 15x 2 6 8.3x 3 14.7x 2 17 15.2x4 18x 19.1 13.9x4 9.6x 15 5z 3z 10z 8 y 3 1 y 2 1 3y 7 3xx 2 2x 1 y24y2 2y 3 5z3z 1 3x5x 2 1 x 34x 4x3 x 3 2.5x2 33x 2 3.5y2y3 4x1 8x 3 45. 8 y 46. 2y4 7 41. 44. 43. 38. 37. 42. 40. 39. 333202_0A03.qxd 12/6/05 2:14 PM Page A32 A32 Appendix A Review of Fundamental Concepts of Algebra In Exercises 47–84, multiply or find the special product. 47. 48. 49. 50. 51. 52. 53. 55. 57. 59. 61. 63. 65. 67. 68. 69. 71. 73. 75. 77. 79. 81. 82. 83. 84. x 3x 4 x 5x 10 3x 52x 1 7x 24x 3 x 2 x 1x 2 x 1 |
x 2 3x 2x 2 3x 2 x 10x 10 x 2yx 2y 2x 3 2 2x 5y2 x 13 2x y3 4x3 32 m 3 nm 3 n x y 1x y 1 x 3 y2 2r 2 52r 2 5 1 2x 32 1 3x 21 1.2x 32 1.5x 41.5x 4 5xx 1 3xx 1 2x 1x 3 3x 3 u 2u 2u 2 4 x yx yx 2 y 2 3x 2 54. 56. 58. 60. 62. 64. 66. 70. 72. 74. 76. 78. 80. 2x 32x 3 2x 3y2x 3y 4x 52 5 8x 2 x 23 3x 2y3 8x 32 x 1 y2 3a3 4b23a3 4b2 3t 52 2 2x 1 5 1.5y 32 2.5y 32.5y 3 2x 1 5 In Exercises 85–88, find the product. (The expressions are not polynomials, but the formulas can still be used.) 85. 87. x yx y x 5 2 86. 88. 5 x5 x x 32 In Exercises 89–96, factor out the common factor. 89. 91. 93. 95. 3x 6 2x 3 6x xx 1 6x 1 x 32 4x 3 90. 92. 94. 96. 5y 30 4x 3 6x 2 12x 3xx 2 4x 2 3x 12 3x 1 In Exercises 103 –112, completely factor the difference of two squares. 103. 104. 105. 106. 107. 108. 109. 111. x2 81 x 2 49 32y2 18 4 36y2 16x2 1 9 4 25 y2 64 x 1 2 4 9u2 4v2 110. 112. 25 z 5 2 25x2 16y2 In Exercises 113 –122, factor the perfect square trinomial. 113. 115. 117. 119. 121. x 2 4x 4 4t 2 4t 1 25y 2 10y 1 9u2 24uv 16v2 x2 4 3x 4 9 114. 116. 118. 120. 122. x 2 10x 25 9x 2 12x 4 36y2 108y 81 4x2 4xy y2 z 2 z 1 4 In |
Exercises 123 –130, factor the sum or difference of cubes. 123. 125. 127. 129. x 3 8 y 3 64 8t 3 1 u3 27v3 124. 126. 128. 130. x 3 27 z 3 125 27x 3 8 64x3 y3 In Exercises 131–144, factor the trinomial. 131. 133. 135. 137. 139. 141. 143. x 2 x 2 s 2 5s 6 20 y y 2 x 2 30x 200 3x 2 5x 2 5x 2 26x 5 9z 2 3z 2 132. 134. 136. 138. 140. 142. 144. x 2 5x 6 t 2 t 6 24 5z z 2 x 2 13x 42 2x 2 x 1 12x 2 7x 1 5u 2 13u 6 In Exercises 145–152, factor by grouping. 145. 147. 149. 151. x 3 x 2 2x 2 2x 3 x 2 6x 3 6 2x 3x3 x4 6x3 2x 3x 2 1 146. 148. 150. 152. x 3 5x 2 5x 25 5x 3 10x 2 3x 6 x 5 2x 3 x 2 2 8x5 6x2 12x3 9 In Exercises 97–102, find the greatest common factor such that the remaining factors have only integer coefficients. In Exercises 153–158, factor the trinomial by grouping. 97. 99. 101. 1 1 2x 4 2 x3 2x2 5x 3 xx 3 4x 3 2 98. 100. 102. 1 1 3 y 5 3 y 4 5y2 2y 5 yy 1 2 y 1 4 153. 155. 157. 3x 2 10x 8 6x 2 x 2 15x 2 11x 2 154. 2x 2 9x 9 6x 2 x 15 156. 158. 12x2 13x 1 333202_0A03.qxd 12/6/05 2:14 PM Page A33 In Exercises 159–192, completely factor the expression. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 1 |
16 6x 2 54 12x 2 48 x 3 4x 2 x 3 9x x 2 2x 1 16 6x x 2 1 4x 4x 2 9x 2 6x 1 2x 2 4x 2x 3 2y 3 7y 2 15y 9x 2 10x 1 13x 6 5x 2 1 81x2 2 9 x 8 96x 1 8x2 1 3x 3 x 2 15x 5 5 x 5x 2 x 3 x 4 4x 3 x 2 4x 3u 2u2 6 u3 1 4 x3 3x2 3 4 x 9 5 x3 x2 x 5 t 1 2 49 x 2 1 2 4x 2 x 2 8 2 36x 2 2t 3 16 5x 3 40 4x2x 1 2x 1 2 53 4x 2 83 4x5x 1 2x 1x 3 2 3x 1 2x 3 73x 2 21 x 2 3x 21 x3 7x2x 2 12x x2 1 27 3x 22x 14 x 2 34x 1 3 2xx 54 x 24x 5 3 5x6 146x53x 23 33x 223x6 15 x2 2 x2 14 x 2 15 1 In Exercises 193–196, find all values of b for which the trinomial can be factored. 193. 194. 195. 196. x 2 bx 15 x2 bx 50 x 2 bx 12 x2 bx 24 Appendix A.3 Polynomials and Factoring A33 In Exercises 197–200, find two integer values of c such that the trinomial can be factored. (There are many correct answers.) 197. 199. 2x 2 5x c 3x2 x c 198. 200. 3x 2 10x c 2x2 9x c 201. Cost, Revenue, and Profit An electronics manufacturer C radios per week. The total cost x can produce and sell (in dollars) of producing x radios is C 73x 25,000 and the total revenue R (in dollars) is R 95x. (a) Find the profit P in terms of x. (b) Find the profit obtained by selling 5000 radios per week. 202. Cost, Revenue, and Profit An artisan can produce and (in dollars) of hats per month. The total cost C x sell producing hats is x C 460 12x and the total revenue R (in dollars) is R 36x. |
(a) Find the profit P in terms of x. (b) Find the profit obtained by selling 42 hats per month. 203. Compound Interest After 2 years, an investment of $500 compounded annually at an interest rate will yield an amount of 5001 r2. r (a) Write this polynomial in standard form. (b) Use a calculator to evaluate the polynomial for the values of shown in the table. r r 5001 r2 21 2 % 3% 4% 41 2 % 5% (c) What conclusion can you make from the table? 204. Compound Interest After 3 years, an investment of $1200 compounded annually at an interest rate will yield an amount of 12001 r3. r (a) Write this polynomial in standard form. (b) Use a calculator to evaluate the polynomial for the values of shown in the table. r r 12001 r3 2% 3% 31 2 % 4% 41 2 % (c) What conclusion can you make from the table? 333202_0A03.qxd 12/6/05 2:14 PM Page A34 A34 A34 Appendix A Appendix A Review of Fundamental Concepts of Algebra Review of Fundamental Concepts of Algebra 205. Volume of a Box A take-out fast-food restaurant is constructing an open box by cutting squares from the corners of a piece of cardboard that is 18 centimeters by 26 centimeters (see figure). The edge of each cut-out square is centimeters. x (a) Find the volume of the box in terms of x. (b) Find the volume when x 1, x 2, and x 3. x x 2 − 8 2− x 1 x x 18 cm x x 26 26 cm 26 2− x 18 2− x 206. Volume of a Box An overnight shipping company is designing a closed box by cutting along the solid lines and folding along the broken lines on the rectangular piece of corrugated cardboard shown in the figure. The length and width of the rectangle are 45 centimeters and 15 centimeters, respectively. (a) Find the volume of the shipping box in terms of x. (b) Find the volume when x 3, x 5, and x 7. 45 cm x m c 5 1 Geometry In Exercises 207 and 208, find a polynomial that represents the total number of square feet for the floor plan shown in the figure. 207. x x 14 ft 209. Geometry Find the area of the sh |
aded region in each figure. Write your result as a polynomial in standard form. 2 + 6 x x + 4 2x x (a) (b) 12x 8x 6x 9x 210. Stopping Distance The stopping distance of an automobile is the distance traveled during the driver’s reaction time plus the distance traveled after the brakes are applied. In an experiment, these distances were measured (in feet) when the automobile was traveling at a speed of x miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time R was R 1.1x and the braking distance was B B 0.0475x 2 0.001x 0.23. (a) Determine the polynomial that represents the total stopping distance T. (b) Use the result of part (a) to estimate the total stopping miles per x 30, x 40, x 55 and distance when hour. (c) Use the bar graph to make a statement about the total stopping distance required for increasing speeds. Reaction time distance Braking distance 22 ft ) 250 225 200 175 150 125 100 75 50 25 20 30 40 50 60 Speed (in miles per hour) x 208. 14 ft x x 18 ft x 333202_0A03.qxd 12/6/05 2:14 PM Page A35 Geometric Modeling In Exercises 211–214, draw a “geometric factoring model” to represent the factorization. For instance, a factoring model for 2x2 3x 1 2x 1x 1 is shown in the figure 211. 212. 213. 214. 3x 2 7x 2 3x 1x 2 x 2 4x 3 x 3x 1 2x 2 7x 3 2x 1x 3 x 2 3x 2 x 2x 1 1 x x x 1 Geometry In Exercises 215–218, write an expression in factored form for the area of the shaded portion of the figure. 215. 216. Appendix A.3 Polynomials and Factoring A35 (a) Factor the expression for the volume. (b) From the result of part (a), show that the volume of concrete is 2 (average radius)(thickness of the tank) h. 220. Chemistry The rate of change of an autocatalytic kQx kx 2, is the amount of x is the amount of substance is a constant of proportionality. Factor |
the chemical reaction is the original substance, formed, and expression. where Q k Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 221–224, determine whether 221. The product of two binomials is always a second-degree polynomial. 222. The sum of two binomials is always a binomial. 223. The difference of two perfect squares can be factored as the product of conjugate pairs. 224. The sum of two perfect squares can be factored as the binomial sum squared. 225. Find the degree of the product of two polynomials of degrees m n and. 226. Find the degree of the sum of two polynomials of degrees r r + 2 18 217. x x 8 x x 218x + 3) 219. Geometry The volume V cylindrical concrete storage figure is r radius, storage tank. V R 2h r 2h is the inside radius, and of concrete used to make the the is the outside is the height of the tank shown R where h in R h r r m and n if m < n. 227. Think About It When the polynomial x3 3x2 2x 1 is subtracted from an unknown polynomial, the difference is x + 3 5x 2 8. If it is possible, find the unknown polynomial. 228. Logical Reasoning Verify that by letting x2 y2 expressions. Are there any values of x y2 x2 y2? Explain. x 3 y 4 and x y2 is not equal to and evaluating both for which and y x 229. Factor 230. Factor 231. Factor x2n y2n x3n y3n x 3n y2n completely. completely. completely. 232. Writing Explain what is meant when it is said that a polynomial is in factored form. 233. Give an example of a polynomial that is prime with respect to the integers. 333202_0A04.qxd 12/6/05 2:16 PM Page A36 A36 Appendix A Review of Fundamental Concepts of Algebra A.4 Rational Expressions What you should learn • Find domains of algebraic expressions. • Simplify rational expressions. • Add, subtract, multiply, and divide rational expressions. • Simplify complex fractions and rewrite difference quotients. Why you should learn it Rational expressions can be used to solve real-life problems. For instance |
, in Exercise 84 on page A45, a rational expression is used to model the projected number of households banking and paying bills online from 2002 through 2007. Domain of an Algebraic Expression The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, x 1 x 2 are equivalent because 2x 3 and 2x 3. Example 1 Finding the Domain of an Algebraic Expression a. The domain of the polynomial 2x 3 3x 4 is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted. b. The domain of the radical expression x 2 is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number. c. The domain of the expression x 2 x 3 is the set of all real numbers except zero, which is undefined. Now try Exercise 1. x 3, which would result in division by The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as 1 x, 2x 1 x 1, or x 2 1 x 2 1 is a rational expression. Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from To write a fraction in simplest form, divide out common factors. ±1 The key to success in simplifying rational expressions lies in your ability to factor polynomials. 333202_0A04.qxd 12/6/05 2:16 PM Page A37 In Example 2, do not make the mistake of trying to simplify further by dividing out terms. x 6 3 x 6 3 x 2 Remember that to simplify fractions, divide out common factors, not terms. Appendix A.4 Rational Expressions A37 Simplifying Rational Expressions When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of that must |
be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. x Example 2 Simplifying a Rational Expression Write x 2 4x 12 3x 6 Solution in simplest form. x2 4x 12 3x 6 x 6x 2 3x 2 x 6 3, x 2 Factor completely. Divide out common factors. (because division by Note that the original expression is undefined when zero is undefined). To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x 2. x 2 Now try Exercise 19. Sometimes it may be necessary to change the sign of a factor to simplify a rational expression, as shown in Example 3. Example 3 Simplifying Rational Expressions Write 12 x x2 2x2 9x 4 Solution in simplest form. 12 x x2 2x2 9x 4 4 x3 x 2x 1x 4 x 43 x 2x 1x 4 Factor completely. 4 x x 4 3 x 2x 1, x 4 Divide out common factors. Now try Exercise 25. 333202_0A04.qxd 12/6/05 2:16 PM Page A38 A38 Appendix A Review of Fundamental Concepts of Algebra Operations with Rational Expressions To multiply or divide rational expressions, use the properties of fractions discussed in Appendix A.1. Recall that to divide fractions, you invert the divisor and multiply. Example 4 Multiplying Rational Expressions 2x2 x 6 x2 4x 5 x3 3x2 2x 4x2 6x 2x 3x 2 x 5x 1 x 2x 2 2x 5, xx 2x 1 2x2x 3 x 0, x 1, x 3 2 Now try Exercise 39. In Example 4 the restrictions x 0, are listed with the simplified expression in order to make the two domains agree. Note that the value x 5 is excluded from both domains, so it is not necessary to list this value. x 1, and x 3 2 Example 5 Dividing Rational Expressions x 3 8 x 2 4 x 2 2x 2x 4 x 2 4 x 2x2 2x 4 x 2x 2 Invert and multiply. x 2x2 2x 4 x2 2x 4 Divide out common factors. x ±2 x2 2x 4, Now try Exercise 41. To add or subtract rational expressions, you can use the LCD (least common denominator) method or the basic |
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