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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... |
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 No... |
. 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 Tr... |
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... |
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... |
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 ... |
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 Exerc... |
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 classifi... |
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 ... |
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 Exercis... |
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.... |
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 ... |
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 I... |
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... |
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 equati... |
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 s... |
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 orientati... |
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 parametri... |
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: pa... |
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 cur... |
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 p... |
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 eq... |
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 ... |
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 represent... |
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, ... |
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 ... |
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... |
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... |
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 poin... |
.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... |
. 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_... |
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 ... |
, 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 Chapt... |
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 ________ _____... |
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.... |
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... |
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 ... |
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... |
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 denominato... |
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 afte... |
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 un... |
(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 s... |
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 ... |
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. Writi... |
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 Ex... |
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 ... |
, 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 p... |
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... |
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... |
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... |
. 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 s... |
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 chan... |
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 th... |
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 li... |
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. Stan... |
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 ... |
, 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 ... |
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 ... |
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 ... |
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.... |
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... |
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 ... |
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 nega... |
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 expre... |
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 ... |
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... |
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. Th... |
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 s... |
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 1... |
. (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 I... |
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... |
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 Append... |
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 E... |
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 fac... |
.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 numb... |
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... |
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 Alg... |
. 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 undefin... |
________ ________. 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 ... |
. (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 Expone... |
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) 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 tha... |
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 o... |
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 A3... |
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 ... |
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 manipu... |
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 2... |
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 fac... |
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... |
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 seve... |
. 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 pol... |
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... |
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 4b2... |
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... |
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... |
(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 polynom... |
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, thes... |
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 bino... |
, 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 expression... |
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 N... |
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