CONCEPTS
The following list includes terms that are defined and theorems that are stated in this chapter. For each term or theorem, (1) give a precise definition or statement, (2) state in general terms what it means and (3) describe the types of problems with which it is associated.
Slope of line Parallel lines Perpendicular lines
Domain Intercepts Zeros of function
Graphing window Local maximum Vertical asymptote Inverse function One-to-one function Periodic function Sine function Cosine function Arcsine function
e Exponential function Logarithm
Composition Parametric equations Polar coordinates
REVIEW EXERCISES
TRUE OR FALSE
State whether each statement is true or false and briefly explain why. If the statement is false, try to “fix it” by modifying the given statement to a new statement that is true.
1. For a graph, you can compute the slope using any two points and get the same value.
2. All graphs must pass the vertical line test.
3. A cubic function has a graph with one local maximum and one local minimum.
4. If a function has no local maximum or minimum, then it is one-to-one.
5. The graph of the inverse of f can be obtained by reflecting the graph of f across the diagonal y= x.
6. If f is a trigonometric function, then the solution of the equa-tion f (x)= 1 is f−1(1).
7. Exponential and logarithmic functions are inverses of each other.
8. All quadratic functions have graphs that look like the parabola y= x2.
9. Polar coordinates are a specific example of parametric equa-tions.
10. Every curve has an infinite number of parametric representa-tions.
In exercises 1 and 2, find the slope of the line through the given points.
1. (2, 3), (0, 7) 2. (1, 4), (3, 1)
In exercises 3 and 4, determine if the lines are parallel, perpen-dicular or neither.
3. y= 3x + 1 and y = 3(x − 2) + 4 4. y= −2(x + 1) − 1 and y =12x+ 2
5. Determine if the points (1, 2), (2, 4) and (0, 6) form the vertices of a right triangle.
6. The data represents populations at various times. Plot the points, discuss any patterns and predict the population at the next time: (0, 2100), (1, 3050), (2, 4100) and (3, 5050).
7. Find an equation of the line through the given points and com-pute the y-coordinate corresponding to x= 4.
y
x
2 4 6
2 4
8. For f (x)= x2− 3x − 4, compute f (0), f (2) and f (4).
In exercises 9 and 10, find an equation of the line with given slope and point.
9. m= −13, (−1, −1) 10. m= 14, (0, 2)
In exercises 11 and 12, use the vertical line test to determine if the curve is the graph of a function.
11. y
x
12. y
x
In exercises 13 and 14, find the domain of the given function.
13. f (x)=√
4− x2 14. f (x)= x− 2 x2− 2
REVIEW EXERCISES
In exercises 15–28, sketch a graph of the function showing ex-trema, intercepts and asymptotes.
15. f (x)= x2+ 2x − 8 16. f (x)= x3− 6x + 1 17. f (x)= x4− 2x2+ 1 18. f (x)= x5− 4x3+ x − 1 19. f (x)= 4x
x+ 2 20. f (x)= x− 2
x2− x − 2 21. f (x)= sin 3x 22. f (x)= tan 4x 23. f (x)= sin x + 2 cos x 24. f (x)= sec 2x 25. f (x)= 4e2x 26. f (x)= 3e−4x 27. f (x)= ln 3x 28. f (x)= eln 2x
29. Determine all intercepts of y= x2+ 2x − 8 (see exercise 15).
30. Determine all intercepts of y= x4− 2x2+ 1 (see exercise 17).
31. Find all vertical asymptotes of y= 4x x+ 2. 32. Find all vertical asymptotes of y= x− 2
x2− x − 2.
In exercises 33–36, find or estimate all zeros of the given function.
33. f (x)= x2− 3x − 10 34. f (x)= x3+ 4x2+ 3x 35. f (x)= x3− 3x2+ 2 36. f (x)= x4− 3x − 2 In exercises 37 and 38, determine the number of solutions.
37. sin x= x3 38. √
x2+ 1 = x2− 1 39. A surveyor stands 50 feet from a telephone pole and measures
an angle of 34◦to the top. How tall is the pole?
40. Find sinθ given that 0 < θ <π2 and cosθ = 15. 41. Convert to fractional or root form: (a) 5−1/2(b) 3−2. 42. Convert to exponential form: (a) 2
√x (b) 3 x2. 43. Rewrite ln 8− 2 ln 2 as a single logarithm.
44. Solve the equation for x: eln 4x= 8.
In exercises 45 and 46, solve the equation for x.
45. 3e2x= 8 46. 2 ln 3x= 5.
In exercises 47 and 48, find f◦g and g◦ f and identify their respective domains.
47. f (x)= x2, g(x) =√ x− 1 48. f (x)= x2, g(x) = 1
x2− 1
In exercises 49 and 50, identify functions f (x) and g(x) such that ( f◦g)(x) equals the given function.
49. e3x2+2 50. √
sin x+ 2
In exercises 51 and 52, complete the square and explain how to transform the graph of y x2 into the graph of the given function.
51. f (x)= x2− 4x + 1 52. f (x)= x2+ 4x + 6 In exercises 53–56, determine if the function is one-to-one. If so, find its inverse.
53. x3− 1 54. e−4x
55. e2x2 56. x3− 2x + 1
In exercises 57–60, graph the inverse without solving for the inverse.
57. x5+ 2x3− 1 58. x3+ 5x + 2 59. √
x3+ 4x 60. ex3+2x
In exercises 61–64, evaluate the quantity using the unit circle.
61. sin−11 62. cos−1
−12 63. tan−1(−1) 64. csc−1(−2)
In exercises 65–68, simplify the expression using a triangle.
65. sin(sec−12) 66. tan(cos−1(4/5)) 67. sin−1(sin(3π/4)) 68. cos−1(sin(−π/4)) In exercises 69 and 70, find all solutions of the equation.
69. sin 2x = 1 70. cos 3x= 12
In exercises 71–74, sketch the plane curve defined by the para-metric equations and find a corresponding x-y equation for the curve.
71.
x= −1 + 3 cos t
y= 2 + 3 sin t 72.
x= 2 − t y= 1 + 3t
REVIEW EXERCISES
73.
x= t2+ 1
y= t4 74.
x= cos t y= cos2t− 1
In exercises 75–78, sketch the plane curves defined by the para-metric equations.
75.
x= cos 2t
y= sin 6t 76.
x= cos 6t y= sin 2t 77.
x= cos 2t cos t
y= cos 2t sin t 78.
x= cos 2t cos 3t y= cos 2t sin 3t
In exercises 79–82, match the parametric equations with the corresponding plane curve.
79.
x= t2− 1
y= t3 80.
x= t3 y= t2− 1 81.
x= cos 2t cos t
y= cos 2t sin t 82.
x= cos(t + cos t) y= cos(t + sin t)
y
x 8
8 4 4
1 1 2 3
y
x
1 1
1 1
FIGURE A FIGURE B
y
x 3 2
1 1
8
4 4 8
y
x 0.5 1
0.5
1
0.5
1 0.5
1
FIGURE C FIGURE D
In exercises 83 and 84, find parametric equations for the given curve.
83. The line segment from (2, 1) to (4, 7)
84. The portion of the parabola y= x2+ 1 from (1, 2) to (3, 10)
In exercises 85 and 86, sketch the graph of the polar equation and find a corresponding x-y equation.
85. r = 3 cos θ 86. r= 2 sec θ
In exercises 87–94, sketch the graph and identify all values ofθ where r 0 and a range of values of θ that produces one copy of the graph.
87. r = 2 sin θ 88. r= 2 − 2 sin θ 89. r = 2 − 3 sin θ 90. r= cos 3θ + sin 2θ 91. r2= 4 sin 2θ 92. r= ecosθ− 2 cos 4θ
93. r = 2
1+ 2 sin θ 94. r= 2
1+ 2 cos θ
In exercises 95 and 96, find a polar equation corresponding to the rectangular equation.
95. x2+ y2= 9 96. (x− 3)2+ y2= 9
CONNECTIONS
1. Sketch a graph of any function y= f (x) that has an in-verse. (Your choice.) Sketch the graph of the inverse function y= f−1(x). Then sketch the graph of y= g(x) = f (x + 2).
Sketch the graph of y= g−1(x) and use the graph to deter-mine a formula for g−1(x) in terms of f−1(x). Repeat this for h(x)= f (x) + 3 and k(x) = f (x − 4) + 5.
2. In tennis, a serve must clear the net and then land inside of a box drawn on the other side of the net. In this exercise, you will explore the margin of error for successfully serving. First, consider a straight serve (this essentially means a serve hit infinitely hard) struck 9 feet above the ground. Call the start-ing point (0, 9). The back of the service box is 60 feet away, at (60, 0). The top of the net is 3 feet above the ground and 39 feet from the server, at (39, 3). Find the service angleθ (i.e., the angle as measured from the horizontal) for the trian-gle formed by the points (0, 9), (0, 0) and (60, 0). Of course, most serves curve down due to gravity. Ignoring air resistance, the path of the ball struck at angleθ and initial speed v ft/s is y= − 16
(v cos θ)2x2− (tan θ)x + 9. To hit the back of the service line, you need y= 0 when x = 60. Substitute in these values along withv = 120. Multiply by cos2θ and replace sinθ with√
1− cos2θ. Replacing cos θ with z gives you an algebraic equation in z. Numerically estimate z. Similarly, sub-stitute x= 39 and y = 3 and find an equation for w = cos θ.