Solutions 48: Graphs of the Sine and Cosine Function

Solutions to Odd-Numbered Exercises

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.

5. At the point where the terminal side of t intersects the unit circle, you can determine that the sin t equals the y-coordinate of the point.

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]

43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.
A sinusoidal graph that increases like the function y=x, shown from 0 to 100.

45. The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.
A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.

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