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0 (a) Vr 4 3 V3 (a) ht t2 2t h2 (a) f y 3 y (a) f x x 8 2 (a) f 8 f 4 (b) (b) (b) (b) (b) (b) f 3 (cc) gs 2 (c) V2r 41. h1.5 (c) hx 2 f 0.25 (c) f 4x2 f 1 (c gx ht 1 2t 3 5 t ht 4 3 2 1 42 333202_0104.qxd 12/7/05 8:35 AM Page 50 50 43. Chapter 1 Functions and Their Graphs f x 1 2x 4, x โค 0 x 22 44. f x 9 x 2, x < 3 x 3... |
In Exercises 75โ78, match the data with one of the following functions f x cx, gx cx2, hx cx, and rx c x and determine the value of the constant the function fit the data in the table. c that will make 75. 76. 77. 78 32 1 2 0 0 1 4 2 32 32 1 1 4 0 4 1 1 Undef. 32 In Exercises 79โ86, find the difference quotient and si... |
ordered, Use the table to estimate the maximum profit. P Units, x 110 120 130 140 Profit, P 3135 3240 3315 3360 Units, x 150 160 170 Profit, P 3375 3360 3315 x, P (b) Plot the points from the table in part (a). Does the relation defined by the ordered pairs represent as a function of x? P (c) If P is a function of wri... |
United States from 1990 to 2002 (see figure) can be approximated by the model p pt 0.182t2 0.57t 27.3, 2.50t 21.3, 0 โค t โค 7 8 โค t โค 12 t represents the year, with where corresponding to 1990. Use this model to find the average price of a mobile (Source: U.S. home in each year from 1990 to 2002. Census Bureau) t 0 p 5... |
. 97. Cost, Revenue, and Profit A company produces a product for which the variable cost is $12.30 per unit and the fixed costs are $98,000. The product sells for $17.98. Let be the number of units produced and sold. x (a) The total cost for a business is the sum of the variable as a cost and the fixed costs. Write the... |
or false. Justify your answer. 103. The domain of the function given by f x x 4 1 is,, and the range of 104. The set of ordered pairs 0, 4, 2, 2 2, 2, 0,. f x is 8, 2, represents a function. 6, 0, 4, 0, 105. Writing In your own words, explain the meanings of domain and range. 106. Think About gx 3x 2. It Consider and ... |
5 FIGURE 1.53 The Graph of a Function In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective. f x, f x such is in the domain of As you study this section, remember that The graph of a function is the collection of ordered pairs x x y... |
function of and only if no vertical line intersects the graph at more than one point. y if Example 2 Vertical Line Test for Functions Use the Vertical Line Test to decide whether the graphs in Figure 1.54 represent y as a function of x. y 4 2 x 1 โ3 โ2 โ1 (a) FIGURE 1.54 y 4 3 2 1 (b) y 4 3 1 โ1 โ1 (c Solution a. This... |
Zero of h: FIGURE 1.57 t 3 2 x t 3x2 x 10 0 3x 5x 2 0 3x 5 0 x 2 0 Set f x equal to 0. Factor. x 5 3 x 2 Set 1st factor equal to 0. Set 2nd factor equal to 0. x 5 3 as its -intercepts. x 2. and x 2, 0 In Figure 1.55, note that the graph of f a. b. f and The zeros of are 3, 0 5 has 10 x2 0 10 x2 0 10 x2 ยฑ 10 x Set gx e... |
graphs in Figure 1.59 to describe the increasing or decreasing behavior of each function. Solution a. This function is increasing over the entire real line. b. This function is increasing on the interval interval 1, 1, and increasing on the interval, 1, 1,. decreasing on the c. This function is increasing on the inter... |
Relative Minimum Use a graphing utility to approximate the relative minimum of the function given by f x 3x2 4x 2. Solution is shown in Figure 1.61. By using the zoom and trace features or The graph of the minimum feature of a graphing utility, you can estimate that the function has a relative minimum at the point f 0... |
secant line, and the slope of this line is denoted as x2, fx2 x1, fx1 and msec. Average rate of change of f from x1 to x2 fx2 fx1 x1 x2 change in y change in x msec Example 6 Average Rate of Change of a Function Find the average rates of change of 0 and (b) from 1 to x1 x2 (see Figure 1.63). fx x3 3x (a) from x1 2 to ... |
x x 5 2x3 x jx 2 x6 x8 kx x 5 2x 4 x 2 px x9 3x5 x3 x What do you notice about the equations of functions that are odd? What do you notice about the equations of functions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a functio... |
and x1 f x 0. in the interval, 5. A function value is a relative ________ of f if there exists an interval containing f a f a โฅ f x. implies implies x1 < x2 x1, x2 f x1 a. > f x2 x1 < x < x2 such that 6. The ________ ________ ________ ________ between any two points through the two points, and this line is called the ... |
. x y 2 1 y 4 2 โ2 4 6 y 4 2 โ2 โ4 โ4 12. x2 y 2 25 x 2 4 y 6 4 2 โ 333202_0105.qxd 12/7/05 8:36 AM Page 62 62 Chapter 1 Functions and Their Graphs 13. x2 2xy 1 y 4 2 โ2 โ4 โ4 x 2 4 142 โ4 โ6 In Exercises 15โ24, find the zeros of the function algebraically. 15. 17. 19. 20. 21. 22. 23. 24. 16. 18. f x 3x2 22x 16 f x x2 ... |
12/7/05 8:36 AM Page 63 In Exercises 39โ 48, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you ... |
x. h 77. y 781, 2) (3, 2) 4 3 2 1 792, 4) h y x= 2 x1 1, 3 80. y 4 h โ2 (8, 2) 2 x 8 4 y 6 x= 3 x x In Exercises 81โ 84, write the length of the rectangle as a function of y. L 81. y 82. y x 6 4 y โ2 L (8, 4) x = 21 = 23 y (2, 4) L 1 2 3 4 83. y 84. y 4 3 2 y y= 2 x (4, 21, 2 85. Electronics The number of lumens (time... |
be approximated by the model. F F 0.004t 4 0.46t2 431.6, 2 โค t โค 12 t where 1992. represents the year, with (Source: Institute of International Education) corresponding to t 2 (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the... |
the function (d) interpret your answer to part (c) in the from context of the problem, (e) find the equation of the secant line through and (f) graph the secant line in the same viewing window as your position function. and to t2, t2, t1 t1 91. An object is thrown upward from a height of 6 feet at a velocity of 64 fee... |
f x gx f x gx f x 2 gx f x 2 (d) (c) 100. Think About It Does the graph in Exercise 11 represent Explain. as a function of y? x 107. 108. 109. 110. x2 10x 0 100 x 52 0 x3 x 0 16x2 40x 25 0 In Exercises 111โ114, evaluate the function at each specified value of the independent variable and simplify. 111. 112. 113. 114. ... |
Write the linear function f for which f 1 3 and f 4 0. Solution To find the equation of the line that passes through x2, y2 m y2 x2 4, 0, y1 x1 first find the slope of the line. 0 3 3 4 1 3 1 x1, y1 1, 3 and Next, use the point-slope form of the equation of a line. mx x1 y y1 y 3 1x 1 y x 4 f x x 4 Point-slope form Su... |
(x3 โ2 โ1 โ1 2 1 (0, 0) 3 x FIGURE 1.68 333202_0106.qxd 12/7/05 8:40 AM Page 68 68 Chapter 1 Functions and Their Graphs Cubic, Square Root, and Reciprocal Functions The basic characteristics of the graphs of the cubic, square root, and reciprocal functions are summarized below. 1. The graph of the cubic function f x x3... |
x x the greatest integer less than or equal to x. Some values of the greatest integer function are as follows. 1 greatest integer โค 1 1 1 2 greatest integer โค 1 1 10 greatest integer โค 1 1 1.5 greatest integer โค 1.5 1 0 10 2 The graph of the greatest integer function f x x y 3 2 1 โ4 โ3 โ2 โ1 1 2 3 4 x ( ) = [[ ]] f x... |
linear functions. At and to the left of and to the right of the graph is the line 1, 5 f1 21 3 5. x 1 x 1 as shown in Figure 1.74. Notice that the point is an open dot. This is because is a solid dot and the point the graph is the line y x 4, y 2x 3, x 1 1, 3 Now try Exercise 43. Parent Functions The eight graphs show... |
โ8, (a) write the linear function such that it has the indicated function values and (b) sketch the graph of the function. f 1 4, f 5 4, f 5 1, f 10 12, f 1 6, 15 f 2 2, f 2 17 f 5 1 f 16 1 f 4 3 f 4 11 f 1 2 f 1 11 f 3 8, f 3 9, 4. 2. 8. 6. 7. 5. 2 3 In Exercises 9โ28, use a graphing utility to graph the function. Be ... |
d) g 14.5 (c) g4 (d) g3 2 In Exercises 37โ42, sketch the graph of the function. 37. 39. 41. gx x gx x 2 gx x 1 38. 40. 42. gx 4 x gx x 1 gx x 3 In Exercises 43โ50, graph the function. 43. 44. 45. 46. 47 2x 3, 3 x, gx x 6, 2x 4, f x 4 x, 4 x, f x 1 x 12, x 2, f x x 2 5, x 2 4x 3 333202_0106.qxd 12/7/05 8:40 AM Page 72 7... |
the cost of a call lasting 12 minutes and 30 seconds. 62. Communications The cost of using a telephone calling card is $1.05 for the first minute and $0.38 for each additional minute or portion of a minute. (a) A customer needs a model for the cost of using a calling card for a call lasting minutes. Which of the follo... |
for the final hour. Write and graph a piecewise-defined function that gives the depth of the snow during the snowstorm. How many inches of snow accumulated from the storm? Model It V 100 75 50 25 ) 60, 100) (10, 75) (20, 75) (45, 50) (5, 50) (50, 50) (30, 25) (40, 25) 20 30 50 Time (in minutes) 40 t 60 (0, 0) 10 FIGUR... |
a flow rate of 10 gallons per minute, and two drainpipes have flow rates of 5 gallons per minute each. The figure shows t. the volume Determine the combination of the input pipe and drain pipes in which the fluid is flowing in specific subintervals of the 1 hour of time shown on the graph. (There are many correct answ... |
1 2 x โ1 1 2 3 x FIGURE 1.76 FIGURE 1.77 ยฉ Ken Fisher/Getty Images The following list summarizes this discussion about horizontal and vertical shifts. Vertical and Horizontal Shifts Let be a positive real number. Vertical and horizontal shifts in the graph of are represented as follows. c y f x In items 3 and 4, be su... |
gx x 12, hx x 12 2 b. c. f x x 2, gx x 42, hx x 42 2 333202_0107.qxd 12/7/05 8:41 AM Page 76 76 Chapter 1 Functions and Their Graphs y 2 1 โ1 โ2 โ2 โ1 FIGURE 1.80 Reflecting Graphs The second common type of transformation is a reflection. For instance, if you consider the -axis to be a mirror, the graph of x f (x) = x... |
gebraic Solution a. The graph of g is a reflection of the graph of f x in the -axis because gx x f x. b. The graph of h is a reflection of the graph of f y in the -axis because hx x f x. c. The graph of is a left shift of two units x followed by a reflection in the -axis because k kx x 2 f x 2. Graphical Solution f a. ... |
id transformations are those that cause a distortionโa change in the shape of the original graph. For is represented instance, a nonrigid transformation of the graph of gx cf x, by and a vertical shrink if Another nonrigid transformation of the graph of y f x where the transformation is a horizontal shrink if where the... |
and reflections are called ________ transformations. 2. A reflection in the -axis of x is represented by y f x hx 3. Transformations that cause a distortion in the shape of the graph of is represented by ________. y f x hx of y f x are ________, while a reflection in the -axis y called ________ transformations. 4. A n... |
(0, 2)โ 8 4. For each function, sketch (on the same set of coordinate axes) a graph of each function for c 3, 1, 1, and 3. FIGURE FOR 5 FIGURE FOR 6 (a) (b) f x x2 c, x2 c, f x x c2, x c2 In Exercises 5โ8, use the graph of to sketch each graph.To print an enlarged copy of the graph go to the website www.mathgraphs.com... |
f x x to write an equation for each function whose graph is shown. (a) y 4 2 โ2 โ4 โ6 โ8 x 6 8 10 (b) y 2 โ2 โ4 โ8 โ10 2 4 6 8 10 (c) y (d) 8 6 4 2 โ2 โ4 2 4 6 8 10 x y 2 โ4 โ2 2 4 6 โ4 โ8 โ10 x x In Exercises 13โ18, identify the parent function and the transformation shown in the graph. Write an equation for the func... |
in the -axis y but moved 10 units upward and 48. The shape of f x x, seven units downward but moved one unit to the left and Section 1.7 Transformations of Functions 81 49. The shape of f x x, but moved six units to the left and 50. The shape of x reflected in both the -axis and the -axis f x x, but moved nine units d... |
โ2 โ1 2 1 โ1 โ2 Graphical Analysis In Exercises 61โ64, use the viewing window shown to write a possible equation for the transformation of the parent function. 61. โ4 63. โ4 6 โ2 1 โ7 62. โ10 64. โ4 8 8 7 โ1 5 โ3 2 8 Graphical Reasoning In Exercises 65 and 66, use the graph of f to sketch the graph of g. To print an e... |
Board of Governors of the Federal t 0 t where 1990. Reserve System) (a) Describe the transformation of the parent function Then sketch the graph over the specified f x x2. domain. (b) Rewrite the function so that t 0 represents 2000. Explain how you got your answer. Synthesis True or False? the statement is true or fa... |
x 3 5 x x2 3x 28 x2 3x x2 5x 4 In Exercises 83 and 84, evaluate the function at the specified values of the independent variable and simplify. 83. 84. f 3 f x x2 6x 11 (a) f x x 10 3 (a) f 10 (b) (b) f 1 2 f 26 (c) f x 3 (c) f x 10 In Exercises 85โ88, find the domain of the function. 85. f x 2 11 x 87. f x 81 x2 86. f... |
defined as follows. g g f 1. Sum: 2. Difference: 3. Product: 4. Quotient: f gx f x gx f gx f x gx fgx f x gx x f x f gx, g gx 0 Example 1 Finding the Sum of Two Functions Given f x 2x 1 and gx x 2 2x 1, find f gx. Solution f gx f x gx 2x 1 x 2 2x 1 x 2 4x Now try Exercise 5(a). 333202_0108.qxd 12/7/05 8:43 AM Page 85 ... |
86 86 Chapter 1 Functions and Their Graphs Composition of Functions f ห g x g(x) g f f(g(x)) Domain of g FIGURE 1.90 Domain of f The following tables of values help illustrate the composition f gx given in Example 4. x gx gx f gx x f gx Note that the first two tables can be combined (or โcomposedโ) to produce the valu... |
Section 1.8 Combinations of Functions: Composite Functions 87 Example 5 Finding the Domain of a Composite Function f x x2 9 Given find the domain of and f g. gx 9 x2, find the composition f gx. Then Solution f gx f gx f 9 x2 9 x22 9 9 x2 9 x2 From this, it might appear that the domain of the composition is the set of ... |
804t 2 500 2016t 2 16t 4 320t 160 500 320t 2 320t 80 320t 160 500 320t 2 420 The composite function as a function of the amount of time the food has been out of refrigeration. represents the number of bacteria in the food NTt b. The bacterial count will reach 2000 when Solve this hours. When you equation to find that ... |
f /gx. f gx, f /g? fgx, and (d) f x x 2, f x 2x 5, f x x 2, f x 2x 5, f x x 2 6, 5. 6. 7. 8. 9. 10. f x x2 4, 11. 12, gx x 2 gx 2 x gx 4x 5 gx 4 gx 1 x gx x2 x2 1 gx 1 x2 gx x3 16. 17. 15. 13. 14. and gx x 4. In Exercises 13 โ24, evaluate the indicated function for fx x 2 1 f g2 f g0 f g3t fg6 5 f g f g f g1 f g1 f gt... |
f x 3x 5, f x x2 1, f x x23, f x x, f x x 4, f x 1 x, f x 3, x2 1 gx x 2 gx x3 1 gx x gx x6 gx x 6 gx 3 x gx x 3 gx x 1 In Exercises 43โ46, use the graphs of and to evaluate the functions. g f y y = f(x) 4 3 2 1 y = g(x 43. (a) 44. (a) 45. (a) 46. (a) f g3 f g1 f g2 f g1 (b) (b) (b) (b) fg2 fg4 g f 2 g f 3 In Exercise... |
parent company. (b) Use a graphing utility to graph R1, R2, and R3 in the same viewing window. bt t, 57. Vital Statistics Let United States in year of deaths in the United States in year where corresponds to 2000. be the number of births in the represent the number t 0 and let dt t, (a) If pt is the population of the ... |
2003 corresponding to represents the year, with (Source: National Sporting Goods Association, t 7 t where 1997. U.S. Census Bureau) (a) Find and interpret ht Et Pt. (b) Evaluate the function in part (a) for t 7, 10, and 12. Model It 61. Health Care Costs The table shows the total amounts (in billions of dollars) spent... |
b) Write the area A of the circular base of the tank as a function of the radius (c) Find and interpret A rx. r. 333202_0108.qxd 12/7/05 8:43 AM Page 92 92 Chapter 1 Functions and Their Graphs 64. Physics A pebble is dropped into a calm pond, causing ripples in the form of concentric circles (see figure). The t radius ... |
Form the composite functions R Sp and S Rp and interpret each. R S20,500 (d) Find and S R20,500. Which yields the lower cost for the hybrid car? Explain. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. If f x x 1 and f g)x g f )x. gx 6x, then ... |
x x 4 can be written as follows. 1, 5, 2, 6, 3, 7, 4, 8 In this case, by interchanging the first and second coordinates of each of these f 1. ordered pairs, you can form the inverse function of which is denoted by It is a function from the set f 1x x 4: to the set 5, 1, 6, 2, 7, 3, 8, 4 and can be written as follows. ... |
f 1x x g f 1 f 1 f x x. and g is the inverse function of the func- (read โ -inverseโ). So, f f The domain of must be equal to the range of be equal to the domain of f 1. f 1, and the range of must f In this is written, it always refers to the inverse function of the func- to denote the inverse function f 1. 1 Donโt be... |
in the following way. If the point must lie on the graph of a reflection of the graph of are related to each other b, a then the point f 1 is and vice versa. This means that the graph of in the line as shown in Figure 1.93. lies on the graph of f 1, f y x, a, b f, f 1 Example 3 Finding Inverse Functions Graphically x ... |
determining whether a function has an inverse function. This test is called the Horizontal Line Test for inverse functions. Horizontal Line Test for Inverse Functions A function has an inverse function if and only if no horizontal line f intersects the graph of at more than one point. f If no horizontal line intersect... |
y2 Original function Replace f(x) by y. Interchange x and y. Isolate y-term. For simple functions (such as the one in Example 1), you can find inverse functions by inspection. For more complicated functions, however, it is best to use the following guidelines. The key step in these guidelines is Step 3โinterchanging t... |
98 98 Chapter 1 Functions and Their Graphs Example 7 Finding an Inverse Function Find the inverse function of f x 3x 13 โ2 1 2 3 x โ1 โ2 โ3 FIGURE 1.99 Solution The graph of Horizontal Line Test, you know that f is a curve, as shown in Figure 1.99. Because this graph passes the is one-to-one and has an inverse functio... |
. 4. A function f is ________ if each value of the dependent variable corresponds to exactly one value of the independent variable. 5. A graphical test for the existence of an inverse function of f is called the _______ Line Test. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at... |
2 โ2 2 4 6 31. y 2 โ2 2 โ2 6 2 x x โ4 โ2 โ2 2 4 32. y 4 2 โ2 2 4 6 x x In Exercises 33โ38, use a graphing utility to graph the function, and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. 34. f x 10 gx 4 x 6 hx x 4 x 4 gx x 53 f x 2x16 x2 f x 1 x 22 1 8 33.... |
functions given by to find the indicated value or 69. 71. 73. f 1 g11 f 1 f 16 ( f g)1 70. 72. 74. g1 f 13 g1 g14 g1 f 1 80. Digital Camera Sales The factory sales (in millions of dollars) of digital cameras in the United States from 1998 through 2003 are shown in the table. The time (in years) is given by with (Sourc... |
Traveled The total numbers (in billions) of miles traveled by motor vehicles in the United States from 1995 through 2002 are shown in the table. The time t, (in years) is given by with corresponding to 1995. (Source: U.S. Federal Highway Administration) t 5 f Year, t Miles traveled, f t 5 6 7 8 9 10 11 12 2423 2486 25... |
is $73. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether 85. If f is an even function, exists. 86. If the inverse function of exists and the graph of f f 1. is an -intercept of y y -intercept, the -intercept of x f has a f 1 f In Exercises 89โ 92, ... |
nth power. โข Write mathematical models for inverse variation. โข Write mathematical models for joint variation. Why you should learn it You can use functions as models to represent a wide variety of real-life data sets. For instance, in Exercise 71 on page 113, a variation model can be used to model the water temperatu... |
a โbest-fittingโ line is discussed on the next page. 333202_0110.qxd 12/7/05 2:49 PM Page 104 104 Chapter 1 Functions and Their Graphs Least Squares Regression and Graphing Utilities So far in this text, you have worked with many different types of mathematical models that approximate real-life data. In some instances... |
model is a good fit. p* r 0.991, p p Now try Exercise 7. Indianapolis 500 p 11 10 FIGURE 1.101 5 6 7 8 9 10 11 12 13 14 Year (5 โ 1995) t t 5 6 7 8 9 10 11 12 13 14 p p* 8.06 8.11 8.61 8.72 9.05 9.48 9.61 10.03 10.15 10.25 8.00 8.27 8.54 8.80 9.07 9.34 9.61 9.88 10.14 10.41 333202_0110.qxd 12/7/05 8:47 AM Page 105 Sec... |
, 46.05) 100 80 60 40 20 ) 1000 2000 3000 4000 Gross income (in dollars) FIGURE 1.102 333202_0110.qxd 12/7/05 8:47 AM Page 106 106 Chapter 1 Functions and Their Graphs Direct Variation as an nth Power Another type of direct variation relates one variable to a power of another variable. For example, in the formula for t... |
10 Mathematical Modeling and Variation 107 Inverse Variation Inverse Variation The following statements are equivalent. 1. 3. x. 2. y is inversely proportional to x. y varies inversely as y k x for some constant k. x y If and are related by an equation of the form as the th power of (or n y x is inversely proportional ... |
2. 3. If x, and are related by an equation of the form z y, z kx ny m then varies jointly as the th power of and the n x z m th power of y. Example 6 Joint Variation The simple interest for a certain savings account is jointly proportional to the time and the principal. After one quarter (3 months), the interest on a ... |
and inverse variation are said to have ________ variation. 9. The joint variation model z kxy x y. is ________ ________ to and can be described as varies jointly as and x z y, or z PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. 1998, 137,673 1999, 139,368 20... |
and Their Graphs 7. Sports The lengths (in feet) of the winning menโs discus throws in the Olympics from 1912 to 2004 are listed below. (Source: The World Almanac and Book of Facts) 1912 148.3 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 ... |
of Outback Steakhouse in 2005. (f) Use your schoolโs library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e). 9. Data Analysis: Broadway Shows The table shows the S annual gross ticket sales (in millions of dollars) for Broadway shows in New York City from 1995 through... |
graph to determine whether varies directly as some power of or inversely as some power of Explain. x. y x 11. y 12 In Exercises 13โ16, use the given value of k y kx2. table for the direct variation model on a rectangular coordinate system. to complete the Plot the points x 2 4 6 8 10 y kx2 13. 15. k 1 k 1 2 14. 16. k ... |
gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Then use this information to find a linear model that relates gallons to liters. Then use the model to find the numbers of liters in 5 gallons and 25 gallons. 333202_0110.qxd 12/7/05 8:47 AM Page 112 112 Chapter... |
figure). A force of 15 pounds is required to stretch each spring 1 foot. Because of a pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet, and the springs are at their natural length when the door is open. Find the combined lifting force applied to the door ... |
proportional to z 64 r s 3. is jointly proportional to and r 11 x varies directly as y 9. x 42 and when varies directly as the square of when y. 3 F F 4158 P P 28 z z 6 v v 1.5 x and inversely as y. and when y 4. q varies jointly as and and inversely as the square of q 6.3, x 6 p p 4.1, s 1.2. when and s. and inversel... |
best buy? 69. Fluid Flow The velocity of a fluid flowing in a conduit v is inversely proportional to the cross-sectional area of the conduit. (Assume that the volume of the flow per unit of time is held constant.) Determine the change in the velocity of water flowing from a hose when a person places a finger over the ... |
ises 35โ38.) (c) Use the model in part (b) to approximate the force required to compress the spring 9 centimeters. 73. Data Analysis: Light Intensity A light probe is located x (in centimeters from a light source, and the intensity microwatts per square centimeter) of the light is measured. The results are shown as ord... |
3 (c) f 4 f x x2 10, 6x2 1, f 1 (b) f 2 (a) x โฅ 2 x < 2 (c) f 8 76. In the equation for kinetic energy, E 1 is directly proportional to the mass the amount m 2m v2, of kinetic energy of an object and the square of its velocity E v. 77. If the correlation coefficient for a least squares regression the regression line c... |
average rate of change of a function (p. 59). Identify even and odd functions (p. 60). Chapter Summary 115 Review Exercises 1โ4 5โ8 5โ8 9โ14 15โ24 25โ28 29โ36 37โ 44 45, 46 47โ50 51โ54 55โ62 63, 64 65, 66 67โ70 71, 72 73โ76 77, 78 79, 80 81โ84 85โ88 89โ94 95โ98 99โ102 333202_010R.qxd 12/7/05 8:49 AM Page 116 116 Chapt... |
Exercises In Exercises 1 and 2, plot the points in the Cartesian 1.1 plane. 1. 2. 2, 2, 0, 4, 3, 6, 1, 7 5, 0, 8, 1, 4, 2, 3, 3 In Exercises 3 and 4, determine the quadrant(s) in which x, y is located so that the condition(s) is (are) satisfied. 3. x > 0 and y 2 4. y > 0 In Exercises 5โ8, (b) find the distance between... |
Cheesecake Factory, Inc.) 12. Meteorology The apparent temperature is a measure of relative discomfort to a person from heat and high x (in humidity. The table shows the actual temperatures y degrees Fahrenheit) versus the apparent temperatures (in degrees Fahrenheit) for a relative humidity of 75%. x y 70 70 75 77 80... |
graph to estimate the force necessary to stretch the spring 10 inches. 46. Number of Stores The numbers of Target stores for the years 1994 to 2003 can be approximated by the model N N 3.69t2 939, 4 โค t โค 13 t where 1994. is the time (in years), with (Source: Target Corp.) (a) Sketch a graph of the model. t 4 correspo... |
per year $5.15 increase per year 333202_010R.qxd 12/7/05 8:49 AM Page 119 Review Exercises 119 In Exercises 67โ70, determine whether the equation 1.4 represents as a function of y x. 67. 68. 69. 70. 16x y 4 0 2x y 3 0 y 1 x y x 2 In Exercises 71 and 72, evaluate the function at each specified value of the independent ... |
80, find the difference quotient and simplify your answer. 79. f x 2x2 3x 1, 80. f x x3 5x2 x In Exercises 85โ 88, find the zeros of the function algebraically. 85. 86. 87. fx 3x2 16x 21 fx 5x2 4x 1 f x 8x 3 11 x 88. fx x3 x 2 25x 25 In Exercises 89 and 90, determine the intervals over which the function is increasing... |
, 5, 8x 5 In Exercises 115 and 116, the figure shows the graph of a transformed parent function. Identify the parent function. 115. y 116. y 10 8 6 4 2 8 6 4 2 โ8 โ4 โ2 2 x โ2 โ2 2 4 6 8 x f f. h. h. 117. 118. 121. 120. 119. In Exercises 117โ130, (c) Sketch the graph of in terms of 1.7 h is related to one of the parent... |
to t 7 (a) Find and interpret v dt. (b) Use a graphing utility to graph vt, dt, and the function from part (a) in the same viewing window. (c) Find v d10. Use the graph in part (b) to verify your result. 138. Bacteria Count The number N of bacteria in a refriger- ated food is given by NT 25T 2 50T 300, 2 โค T โค 20 T wh... |
. A linear model that approximates these data is I 2.09t 37.2 t where 1995. t 5 represents the year, with (Source: U.S. Census Bureau) corresponding to Year 1995 1996 1997 1998 1999 2000 2001 2002 Median income, I 47.1 49.7 51.6 54.2 56.5 59.1 60.3 61.1 (a) Plot the actual data and the model on the same set of coordina... |
Time The travel time between two cities is inversely proportional to the average speed. A train travels between the cities in 3 hours at an average speed of 65 miles per hour. How long would it take to travel between the cities at an average speed of 80 miles per hour? 160. Cost The cost of constructing a wooden box w... |
the circle shown at the left. In Exercises 7 and 8, find an equation of the line passing through the points. 7. 2, 3, 4, 9 8. 3, 0.8, 7, 6 9. Find equations of the lines that pass through the point 3, 8 and are (a) parallel to and (b) perpendicular to the line 4x 7y 5. 10. Evaluate x 9 x2 81 11. Determine the domain o... |
you? In mathematics, the word proof is used to mean simply a valid argument. When you are proving a statement or theorem, you must use facts, definitions, and accepted properties in a logical order. You can also use previously proved theorems in your proof. For instance, the Distance Formula is used in the proof of th... |
a linear equation for the monthly wage S. new job offer in terms of the monthly sales of your (c) Use a graphing utility to graph both equations in the same viewing window. Find the point of intersection. What does it signify? (d) You think you can sell $20,000 per month. Should you change jobs? Explain. 2. For the nu... |
of change of the function from fx x2 4x 3. x2. to Find x1 1.5 (b) x1 1, x2 (a) (c) (d) (e) x1 x1 x1 x1 1, x2 1, x2 1, x2 1, x2 2 1.25 1.125 1.0625 (f) Does the average rate of change seem to be approaching one value? If so, what value? (g) Find the equations of the secant lines through the points x1, fx1 and x2, fx2 f... |
graphs of function values. f and f1 to complete each table of y 4 2 โ2 โ2 โ1 โ2 โ2 โ4 (a) x 4 2 0 4 f f 1x (b) x 3 2 0 1 (c) (d) f f 1x x f f 1x x f 1x 3 2 0 1 4 3 0 4 2 mi x 3 โ x 1 mi 3 mi Q Not drawn to scale. (a) Write the total time of the trip as a function of T x. (b) Determine the domain of the function. (c) U... |
Advertising Cost, Exercise 105, page 181 โข Athletics, Exercise 109, page 182 โข Data Analysis: Cable Television, โข Recycling, Exercise 74, page 161 Exercise 112, page 195 โข Average Speed, Exercise 79, page 196 โข Height of a Projectile, Exercise 67, page 205 127 333202_0201.qxd 12/7/05 9:10 AM Page 128 128 Chapter 2 Pol... |
ROM and Eduspaceยฎ for this text contain additional resources related to the concepts discussed in this chapter. The graph of a quadratic function is a special type of โUโ-shaped curve called a parabola. Parabolas occur in many real-life applicationsโespecially those involving reflective properties of satellite dishes a... |
a 2, 1, a for y x h2 h 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? h for y x2 k k 4, 2, and 4. How does chang- Graph 2, ing the value of affect the graph? k 333202_0201.qxd 12/7/05 9:10 AM Page 130 130 Chapter 2 Polynomial and Rational Functions Example 1 Sketching Graphs of Quadratic Func... |
atic function identifies four basic transformations of the graph of y x2. a. The factor a produces a vertical stretch or shrink. b. If a < 0, x in the -axis. the graph is reflected c. The factor x h2 a horizontal shift of units. represents h d. The term represents a k vertical shift of units. k f x ( ) = 2( + 2= 2 2 x ... |
Simplify. Write in standard form. is a parabola that opens From this form, you can see that the graph of This corresponds to a left shift of two upward and has its vertex at units and a downward shift of one unit relative to the graph of as shown in Figure 2.6. In the figure, you can see that the axis of the parabola ... |
, f(x) = โ(x โ 3)2 + 1 (3, 1) (2, 0) (4, 0) 1 3 5 x y = โx2 (1, 2) y = f(x) โ1 y 2 1 โ1 โ2 โ3 โ4 FIGURE 2.7 y 2 1 (0, 0) 1 x 0 a0 12 2 a 2 Substitute 0 for x; solve for a. which implies that the equation in standard form is f x 2x 12 2. FIGURE 2.8 Now try Exercise 43. 333202_0201.qxd 12/7/05 9:10 AM Page 133 Section 2.... |
units should be produced each day to yield a minimum cost? is the total cost (in dollars) and where x Baseball f(x) = โ0.0032x 2 + x + 3 (156.25, 81.125) x 200 100 Distance (in feet) 300 y 100 ) 80 60 40 20 FIGURE 2.9 Solution Use the fact that the function has a minimum when function you can see that 120 2(0.075 x b ... |
โ6 (g) y 2 4 6 (3, 2)โ 6 4 2 (2, 0) โ4 โ2 (0, โ2) 2 4 (4, 0) 2 4 6 8 (d) y โ2 โ4 โ6 (2, 4) x 6 (f) y 4 2 โ2 (h) 2 y 4 (0, 3) x 4 โ4 โ2 โ4 1. 3. 5. 7. f x x 22 f x x 2 2 f x 4 (x 2)2 f x x 32 2 2. 4. 6. 8. f x x 42 f x 3 x 2 f x x 12 2 f x x 42 In Exercises 9โ12, graph each function. Compare the graph of each function ... |
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