Periodic Phenomena - AP Precalculus

1. Identify the amplitude of $y = 0.5 \, \text{sin}(x)$.

   0.5. Amplitude is the absolute value of the coefficient.

2. Identify the vertical shift in $y = \text{sin}(x) - 3$.

   Down 3. Vertical shift is the constant term added to the function.

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

   5. Amplitude is the absolute value of the coefficient of cosine.

4. Determine the phase shift of $y = \text{sin}(x + \frac{\pi}{4})$.

   $\frac{\pi}{4}$ to the left. Phase shift is $\frac{\pi}{4}$ left for $(x + \frac{\pi}{4})$.

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

   $\frac{\pi}{2}$. Period formula: $\frac{2\pi}{B}$ where $B = 4$.

6. Determine the vertical shift in $y = \text{cos}(x) + 2$.

   Up 2. Vertical shift is the constant term added to the function.

7. What is the vertical shift of $y = -\text{sin}(x) + 4$?

   Up 4. Vertical shift is the constant term added to the function.

8. State the general form of the cosine function.

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

9. What is the period of $y = \text{sin}(5x)$?

   $\frac{2\pi}{5}$. Period formula: $\frac{2\pi}{B}$ where $B = 5$.

10. What is the general form of the sine function?

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

11. What is the frequency of $y = \text{sin}(10x)$?

    10. Frequency is the coefficient $B$ of $x$.

12. What is the vertical shift in $y = \text{sin}(x) + 6$?

    Up 6. Vertical shift is the constant term added to the function.

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

    $4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{2}$.

14. Identify the period of $y = \cos(\frac{x}{3})$.

    $6\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{3}$.

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

    $8\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{4}$.

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

    $[-1, 3]$. Range is $[D - |A|, D + |A|]$ where $A = 2$ and $D = 1$.

17. What is the period of the cosine function $y = \cos(2x)$?

    $\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.

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

    2. Amplitude is the absolute value of the coefficient.

19. Identify the amplitude of $y = 6 \, \text{sin}(x)$.

    6. Amplitude is the absolute value of the coefficient of sine.

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

    $\frac{\pi}{4}$ to the right. Phase shift is $C$ in the form $(x - C)$.

21. Determine the frequency of $y = \text{cos}(5x)$.

    5. Frequency is the coefficient $B$ of $x$.

22. Find the range of $y = -4 \, \text{cos}(x) + 2$.

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

23. What is the phase shift of $y = \text{cos}(x + \frac{\pi}{2})$?

    $\frac{\pi}{2}$ to the left. Phase shift is $\frac{\pi}{2}$ left for $(x + \frac{\pi}{2})$.

24. Find the range of $y = 2 \, \text{sin}(x) - 4$.

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

25. Find the range of $y = 3 \, \text{sin}(x) - 1$.

    $[-4, 2]$. Range is $[D - |A|, D + |A|]$ where $A = 3$ and $D = -1$.

26. Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{6})$.

    $\frac{\pi}{6}$ to the right. Phase shift is $C$ in the form $(x - C)$.

27. What is the vertical shift of $y = -\text{cos}(x) + 3$?

    Up 3. Vertical shift is the constant term added to the function.

28. What is the vertical shift of $y = \text{cos}(x) - 5$?

    Down 5. Vertical shift is the constant term added to the function.

29. What is the frequency of $y = \text{cos}(3x)$?

    3. Frequency is the coefficient $B$ of $x$.

30. Find the frequency of $y = \sin(7x)$.

    7. Frequency is the coefficient $B$ of $x$.

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

    $\frac{\pi}{2}$ to the right. Phase shift is $C$ in the form $(x - C)$.

32. What is the period of the function $y = \sin(3x)$?

    $\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{B}$ where $B = 3$.

33. What is the period of the function $y = \text{sin}(3x)$?

    $\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{B}$ where $B = 3$.

34. Determine the phase shift of $y = \text{sin}(x + \pi)$.

    $\pi$ to the left. Phase shift is $\pi$ left for $(x + \pi)$.

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

    $[-2, 2]$. Range is $[-|A|, |A|]$ for standard cosine function.

36. What is the phase shift of $y = \text{sin}(x + \frac{\pi}{3})$?

    $\frac{\pi}{3}$ to the left. Phase shift is $\frac{\pi}{3}$ left for $(x + \frac{\pi}{3})$.

37. What is the frequency of $y = \text{cos}(8x)$?

    8. Frequency is the coefficient $B$ of $x$.

38. State the period of the function $y = \sin(0.5x)$.

    $4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 0.5$.

39. What is the period of $y = \sin(5x)$?

    $\frac{2\pi}{5}$. Period formula: $\frac{2\pi}{B}$ where $B = 5$.

40. What is the period of the cosine function $y = \text{cos}(2x)$?

    $\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.

41. State the amplitude of $y = 4 \, \text{cos}(x)$.

    4. Amplitude is the absolute value of the coefficient of cosine.

42. What is the amplitude of $y = -3 \, \text{sin}(x)$?

    3. Amplitude is the absolute value of the coefficient.

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