Periodic Functions: Get Stronger Answer Key

Graphs of the Sine and Cosine Function

1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[/latex] for a certain P. This means that the function values repeat for every P units on the x-axis.

3. The absolute value of the constant A (amplitude) increases the total range and the constant D (vertical shift) shifts the graph vertically.

7. amplitude: [latex]\frac{2}{3}[/latex]; period: 2π; midline: [latex]y=0[/latex]; maximum: [latex]y=23[/latex] occurs at [latex]x=0[/latex]; minimum: [latex]y=−23[/latex] occurs at [latex]x=\pi[/latex]; for one period, the graph starts at 0 and ends at 2π
A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].

9. amplitude: 4; period: 2π; midline: [latex]y=0[/latex]; maximum [latex]y=4[/latex] occurs at [latex]x=\frac{\pi}{2}[/latex]; minimum: [latex]y=−4[/latex] occurs at [latex]x=\frac{3\pi}{2}[/latex]; one full period occurs from [latex]x=0[/latex] to [latex]x=2π[/latex]
A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].

11. amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at [latex]x=\pi[/latex]; minimum: [latex]y=−1[/latex] occurs at [latex]x=\frac{\pi}{2}[/latex]; one full period is graphed from [latex]x=0[/latex] to [latex]x=\pi[/latex]
A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].

13. amplitude: 4; period: 2; midline: [latex]y=0[/latex]; maximum: [latex]y=4[/latex] occurs at [latex]x=0[/latex]; minimum: [latex]y=−4[/latex] occurs at [latex]x=1[/latex]
A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].

15. amplitude: 3; period: [latex]\frac{\pi}{4}[/latex]; midline: [latex]y=5[/latex]; maximum: [latex]y=8[/latex] occurs at [latex]x=0.12[/latex]; minimum: [latex]y=2[/latex] occurs at [latex]x=0.516[/latex]; horizontal shift: −4; vertical translation 5; one period occurs from [latex]x=0[/latex] to [latex]x=\frac{\pi}{4}[/latex]
A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.

17. amplitude: 5; period: [latex]\frac{2\pi}{5}; midline: [latex]y=−2[/latex]; maximum: [latex]y=3[/latex] occurs at [latex]x=0.08[/latex]; minimum: [latex]y=−7[/latex] occurs at [latex]x=0.71[/latex]; phase shift:−4; vertical translation:−2; one full period can be graphed on [latex]x=0[/latex] to [latex]x=\frac{2\pi}{5}[/latex]
A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].

19. amplitude: 1; period: 2π; midline: y=1; maximum:[latex]y=2[/latex] occurs at [latex]x=2.09[/latex]; maximum:[latex]y=2[/latex] occurs at[latex]t=2.09[/latex]; minimum:[latex]y=0[/latex] occurs at [latex]t=5.24[/latex]; phase shift: [latex]−\frac{\pi}{3}[/latex]; vertical translation: 1; one full period is from [latex]t=0[/latex] to [latex]t=2π[/latex]
A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.

21. amplitude: 1; period: 4π; midline: [latex]y=0[/latex]; maximum: [latex]y=1[/latex] occurs at [latex]t=11.52[/latex]; minimum: [latex]y=−1[/latex] occurs at [latex]t=5.24[/latex]; phase shift: −[latex]\frac{10\pi}{3}[/latex]; vertical shift: 0
A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.

23. amplitude: 2; midline: [latex]y=−3[/latex]; period: 4; equation: [latex]f(x)=2\sin\left(\frac{\pi}{2}x\right)−3[/latex]

25. amplitude: 2; period: 5; midline: [latex]y=3[/latex]; equation: [latex]f(x)=−2\cos\left(\frac{2\pi}{5}x\right)+3[/latex]

27. amplitude: 4; period: 2; midline: [latex]y=0[/latex]; equation: [latex]f(x)=−4\cos\left(\pi\left(x−\frac{\pi}{2}\right)\right)[/latex]

29. amplitude: 2; period: 2; midline [latex]y=1[/latex]; equation: [latex]f(x)=2\cos\left(\frac{\pi}{x}\right)+1[/latex]

31. [latex]\frac{\pi}{6},\frac{5\pi}{6}[/latex]

33. [latex]\frac{\pi}{4},\frac{3\pi}{4}[/latex]

35. [latex]\frac{3\pi}{2}[/latex]

37. [latex]\frac{\pi}{2},\frac{3\pi}{2}[/latex]

39. [latex]\frac{\pi}{2},\frac{3\pi}{2}[/latex]

41. [latex]\frac{\pi}{6},\frac{11\pi}{6}[/latex]

47.
a. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[/latex];
b. [latex]h(t)=12.5\sin\left(\frac{\pi}{5}\left(t−2.5\right)\right)+13.5;[/latex]
c. 26 ft

Graphs of Other Trigonometric Functions

1.  Since [latex]y=\csc x[/latex] is the reciprocal function of [latex]y=\sin x[/latex], you can plot the reciprocal of the coordinates on the graph of [latex]y=\sin x[/latex] to obtain the y-coordinates of [latex]y=\csc x[/latex]. The x-intercepts of the graph [latex]y=\sin x[/latex] are the vertical asymptotes for the graph of [latex]y=\csc x[/latex].

3. Answers will vary. Using the unit circle, one can show that [latex]\tan(x+\pi)=\tan x[/latex].

5. The period is the same: 2π.

6. I

7. IV

8. II

9. III

11. period: 8; horizontal shift: 1 unit to left

13. 1.5

15. 5

17. [latex]−\cot x\cos x−\sin x[/latex]

19. stretching factor: 2; period: [latex]\frac{\pi}{4}[/latex]; asymptotes: [latex]x=\frac{1}{4}\left(\frac{\pi}{2}+\pi k\right)+8[/latex], where k is an integer
A graph of two periods of a modified tangent function. There are two vertical asymptotes.

21. stretching factor: 6; period: 6; asymptotes: [latex]x=3k[/latex], where k is an integer
A graph of two periods of a modified cosecant function. Vertical Asymptotes at x= -6, -3, 0, 3, and 6.

23. stretching factor: 1; period: π; asymptotes: [latex]x=πk[/latex], where k is an integer
A graph of two periods of a modified tangent function. Vertical asymptotes at multiples of pi.

25. Stretching factor: 1; period: π; asymptotes: [latex]x=\frac{\pi}{4}+{\pi}k[/latex], where k is an integer
A graph of two periods of a modified tangent function. Three vertical asymptiotes shown.

27. stretching factor: 2; period: 2π; asymptotes: [latex]x=πk[/latex], where k is an integer
A graph of two periods of a modified cosecant function. Vertical asymptotes at multiples of pi.

29. stretching factor: 4; period: [latex]\frac{2\pi}{3}[/latex]; asymptotes: [latex]x=\frac{\pi}{6}k[/latex], where k is an odd integer
A graph of two periods of a modified secant function. Vertical asymptotes at x=-pi/2, -pi/6, pi/6, and pi/2.

31. stretching factor: 7; period: [latex]\frac{2\pi}{5}[/latex]; asymptotes: [latex]x=\frac{\pi}{10}k[/latex], where k is an odd integer
A graph of two periods of a modified secant function. There are four vertical asymptotes all pi/5 apart.

33. stretching factor: 2; period: 2π; asymptotes: [latex]x=−\frac{\pi}{4}+\pi k[/latex], where k is an integer
A graph of two periods of a modified cosecant function. Three vertical asymptotes, each pi apart.

35. stretching factor: [latex]\frac{7}{5}[/latex]; period: 2π; asymptotes: [latex]x=\frac{\pi}{4}+\pi[/latex]k, where k is an integer
A graph of a modified cosecant function. Four vertical asymptotes.

37. [latex]y=\tan\left(3\left(x−\frac{\pi}{4}\right)\right)+2[/latex]
A graph of two periods of a modified tangent function. Vertical asymptotes at x=-pi/4 and pi/12.

39. [latex]f(x)=\csc(2x)[/latex]

41. [latex]f(x)=\csc(4x)[/latex]

43. [latex]f(x)=2\csc x[/latex]

45. [latex]f(x)=\frac{1}{2}\tan(100\pi x)[/latex]

49. [latex]f(x)=\frac{\csc(x)}{\sec(x)}[/latex]

55. a. [latex](−\frac{\pi}{2}\text{,}\frac{\pi}{2})[/latex];
b.
A graph of a half period of a secant function. Vertical asymptotes at x=-pi/2 and pi/2.
c. [latex]x=−\frac{\pi}{2}[/latex] and [latex]x=\frac{\pi}{2}[/latex]; the distance grows without bound as |x| approaches [latex]\frac{\pi}{2}[/latex]—i.e., at right angles to the line representing due north, the boat would be so far away, the fisherman could not see it;
d. 3; when [latex]x=−\frac{\pi}{3}[/latex], the boat is 3 km away;
e. 1.73; when [latex]x=\frac{\pi}{6}[/latex], the boat is about 1.73 km away;
f. 1.5 km; when [latex]x=0[/latex].

57. a. [latex]h(x)=2\tan\left(\frac{\pi}{120}x\right)[/latex];
b.
An exponentially increasing function with a vertical asymptote at x=60.
c. [latex]h(0)=0:[/latex] after 0 seconds, the rocket is 0 mi above the ground; [latex]h(30)=2:[/latex] after 30 seconds, the rockets is 2 mi high;
d. As x approaches 60 seconds, the values of [latex]h(x)[/latex] grow increasingly large. The distance to the rocket is growing so large that the camera can no longer track it.