Solutions to Graph’s of the Sine and Cosine Function

Solutions to Try Its

1. 6π

2. [latex]\frac{1}{2}[/latex] compressed

3. [latex]\frac{π}{2}[/latex]; right

4. 2 units up

5. midline: [latex]y=0[/latex]; amplitude: |A|=[latex]\frac{1}{2}[/latex]; period: P=[latex]\frac{2π}{|B|}=6\pi[/latex]; phase shift:[latex]\frac{C}{B}=\pi[/latex]

6. [latex]f(x)=\sin(x)+2[/latex]

7. two possibilities: [latex]y=4\sin(\frac{π}{5}x−\frac{π}{5})+4[/latex] or [latex]y=−4sin(\frac{π}{5}x+4\frac{π}{5})+4[/latex]

8. midline: y=0; amplitude: |A|=0.8; period: P=[latex]\frac{2π}{|B|}=\pi[/latex]; phase shift: [latex]\frac{C}{B}=0[/latex] or none
A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).

9. [latex]\text{midline:}y=0;\text{amplitude:}|A|=2;\text{period:}\text{P}=\frac{2\pi}{|B|}=6;\text{phase shift:}\text{C}{B}=−\text{1}{2}[/latex]
A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.

10. 7
A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].

11. [latex]y=3\cos(x)−4[/latex]
A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).

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(\frac{\pi}{2}x)−3[/latex]

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

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

29. amplitude: 2; period: 2; midline [latex]y=1[/latex]; equation: [latex]f(x)=2\cos(\frac{\pi}{x})+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(\frac{\pi}{5}(t−2.5))+13.5;[/latex]
c. 26 ft