Solutions to Odd-Numbered Exercises

1. The sine and cosine functions have the property that $f(x+P)=f(x)$ 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: $\frac{2}{3}$; period: 2π; midline: $y=0$; maximum: $y=23$ occurs at $x=0$; minimum: $y=−23$ occurs at $x=\pi$; for one period, the graph starts at 0 and ends at 2π

9. amplitude: 4; period: 2π; midline: $y=0$; maximum $y=4$ occurs at $x=\frac{\pi}{2}$; minimum: $y=−4$ occurs at $x=\frac{3\pi}{2}$; one full period occurs from $x=0$ to $x=2π$

11. amplitude: 1; period: π; midline: y=0; maximum: y=1 occurs at $x=\pi$; minimum: $y=−1$ occurs at $x=\frac{\pi}{2}$; one full period is graphed from $x=0$ to $x=\pi$

13. amplitude: 4; period: 2; midline: $y=0$; maximum: $y=4$ occurs at $x=0$; minimum: $y=−4$ occurs at $x=1$

15. amplitude: 3; period: $\frac{\pi}{4}$; midline: $y=5$; maximum: $y=8$ occurs at $x=0.12$; minimum: $y=2$ occurs at $x=0.516$; horizontal shift: −4; vertical translation 5; one period occurs from $x=0$ to $x=\frac{\pi}{4}$

17. amplitude: 5; period: $\frac{2\pi}{5}; midline: [latex]y=−2$; maximum: $y=3$ occurs at $x=0.08$; minimum: $y=−7$ occurs at $x=0.71$; phase shift:−4; vertical translation:−2; one full period can be graphed on $x=0$ to $x=\frac{2\pi}{5}$

19. amplitude: 1; period: 2π; midline: y=1; maximum:$y=2$ occurs at $x=2.09$; maximum:$y=2$ occurs at$t=2.09$; minimum:$y=0$ occurs at $t=5.24$; phase shift: $−\frac{\pi}{3}$; vertical translation: 1; one full period is from $t=0$ to $t=2π$

21. amplitude: 1; period: 4π; midline: $y=0$; maximum: $y=1$ occurs at $t=11.52$; minimum: $y=−1$ occurs at $t=5.24$; phase shift: −$\frac{10\pi}{3}$; vertical shift: 0

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

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

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

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

31. $\frac{\pi}{6},\frac{5\pi}{6}$

33. $\frac{\pi}{4},\frac{3\pi}{4}$

35. $\frac{3\pi}{2}$

37. $\frac{\pi}{2},\frac{3\pi}{2}$

39. $\frac{\pi}{2},\frac{3\pi}{2}$

41. $\frac{\pi}{6},\frac{11\pi}{6}$

43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.

45. The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.

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