Sinusoidal Functions - AP Precalculus

1. What is the range of $y = 4 \, \text{sin}(x) - 2$?

   [-6, 2]. Range is $[D-|A|, D+|A|] = [-2-4, -2+4]$.

2. What is the range of $y = 2 \, \text{cos}(x) - 6$?

   $[-8, -4]$. Range is $[-6-2, -6+2] = [-8, -4]$

3. Identify the sinusoidal axis for $y = 4 \, \text{sin}(x) - 1$.

   $y = -1$. The sinusoidal axis is at $y = D = -1$.

4. Determine the vertical shift for $y = \text{cos}(x) - 4$.

   -4. The constant term gives the vertical shift downward.

5. What is the general form of a sinusoidal function?

   $y = A \, \text{sin}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.

6. Identify the phase shift for $y = 5 \, \text{sin}(x - \frac{\pi}{6})$.

   $\frac{\pi}{6}$. Phase shift is $\frac{\pi}{6}$ units to the right.

7. Find the phase shift of $y = \text{cos}(x - \frac{\pi}{2})$.

   $\frac{\pi}{2}$. Phase shift is $\frac{\pi}{2}$ units to the right.

8. What does the parameter $B$ affect in the function $y = A \, \text{sin}(B(x - C)) + D$?

   Period. $B$ determines horizontal compression/stretch, affecting how often the function repeats.

9. What is the period of $y = 5 \, \text{cos}(\frac{x}{3})$?

   $6\pi$. Period = $\frac{2\pi}{\frac{1}{3}} = 6\pi$.

10. Calculate the period of $y = 2 \, \text{sin}(3x - \pi)$.

    $\frac{2\pi}{3}$. Period = $\frac{2\pi}{3}$ when $B = 3$.

11. State the maximum value of $y = 8 + 5 \, \text{sin}(x)$.

    13. Maximum occurs when $\sin(x) = 1$: $8 + 5(1) = 13$.

12. What is the period of $y = \text{cos}(\frac{x}{4})$?

    $8\pi$. Period = $\frac{2\pi}{\frac{1}{4}} = 8\pi$.

13. How is the period of a sinusoidal function calculated?

    $\frac{2\pi}{B}$. Period equals $\frac{2\pi}{B}$ where $B$ is the frequency parameter.

14. What is the phase shift of the function $y = A \, \text{sin}(B(x - C)) + D$?

    $C$. The horizontal shift is $C$ units to the right when positive.

15. Identify the phase shift for $y = \text{sin}(x + \pi)$.

    $-\pi$. Rewrite as $\sin(x - (-\pi))$, so phase shift is $-\pi$.

16. What does the parameter $A$ represent in the sinusoidal function $y = A \, \text{sin}(B(x - C)) + D$?

    Amplitude. $A$ controls the maximum distance from the sinusoidal axis.

17. Determine the amplitude of $y = -4 \, \text{sin}(x)$.

    4. Amplitude is the absolute value: $|-4| = 4$.

18. Determine the sinusoidal axis for $y = 3 \, \text{sin}(x) + 4$.

    $y = 4$. The sinusoidal axis is at the vertical shift $y = 4$.

19. What does the sinusoidal axis represent?

    Average value of max and min. The horizontal line around which the function oscillates.

20. State the vertical shift of $y = \text{cos}(x) + 5$.

    5. The constant term added to the function shifts it vertically.

21. Find the maximum value of $y = 3 \, \cos(x) + 1$.

    4. Maximum occurs when cosine equals 1: $3(1) + 1 = 4$.

22. Identify the amplitude of $y = 3 \, \text{cos}(2x)$.

    3. The coefficient of cosine gives the amplitude.

23. Calculate the period of $y = \text{sin}(\frac{1}{2}x)$.

    $4\pi$. Period = $\frac{2\pi}{\frac{1}{2}} = 4\pi$.

24. Find the minimum value of $y = 2 \, \text{sin}(x) - 3$.

    -5. Minimum occurs when sine equals -1: $2(-1) - 3 = -5$.

25. What is the phase shift of $y = 4 \, \text{cos}(x + \frac{\pi}{3})$?

    $-\frac{\pi}{3}$. Rewrite as $\cos(x - (-\frac{\pi}{3}))$, so phase shift is $-\frac{\pi}{3}$.

26. Find the maximum value of $y = 3 \, \text{cos}(x) + 1$.

    4. Maximum occurs when cosine equals 1: $3(1) + 1 = 4$.

27. What is the range of $y = -3 \, \text{sin}(x) + 2$?

    [-1, 5]. Range is $[2-3, 2+3] = [-1, 5]$ since amplitude is 3.

28. What does the parameter $D$ represent in a sinusoidal function?

    Vertical shift. $D$ moves the entire graph up or down from the x-axis.

29. What is the period of $y = 6 \, \text{sin}(\frac{x}{6})$?

    $12\pi$. Period = $\frac{2\pi}{\frac{1}{6}} = 12\pi$.

30. What is the equation for a sinusoidal function with amplitude 5 and period $\pi$?

    $y = 5 \, \text{sin}(2x)$. Amplitude 5 and period $\pi$ means $B = \frac{2\pi}{\pi} = 2$.

31. Calculate the period for $y = 3 \, \text{sin}(5x)$.

    $\frac{2\pi}{5}$. Period = $\frac{2\pi}{5}$ when $B = 5$.

32. Find the maximum value of $y = 3 \, \text{sin}(x) - 2$.

    1. Maximum occurs when $\sin(x) = 1$: $3(1) - 2 = 1$.

33. Determine the amplitude of $y = -2 \, \text{cos}(x)$.

    2. Amplitude is the absolute value of the coefficient: $|-2| = 2$.

34. Determine the period of $y = \text{sin}(4x)$.

    $\frac{\pi}{2}$. Period = $\frac{2\pi}{4} = \frac{\pi}{2}$.

35. What is the frequency of $y = \text{cos}(\frac{x}{2})$?

    $\frac{1}{4\pi}$. Frequency = $\frac{B}{2\pi} = \frac{\frac{1}{2}}{2\pi} = \frac{1}{4\pi}$.

36. Find the phase shift of $y = 2 \, \text{sin}(x - \frac{\pi}{4})$.

    $\frac{\pi}{4}$. Phase shift is the value subtracted inside the function.

37. Find the amplitude of $y = 7 \, \text{cos}(x) + 3$.

    7. The coefficient 7 in front of cosine is the amplitude.

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