Sinusoidal Function Context and Data Modeling - AP Precalculus

1. What is the frequency of $y = 6 \cos(7x)$?

   Frequency = $\frac{7}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=7.

2. State the phase shift of the function $y = \text{sin}(3x - \text{π})$.

   Phase shift = $\frac{\text{π}}{3}$. Phase shift = $\frac{C}{B}$ where C=$\pi$ and B=3.

3. Calculate the phase shift for $y = \text{sin}(x + \frac{\text{π}}{3})$.

   Phase shift = $-\frac{\text{π}}{3}$. For $\sin(x+C)$, phase shift = $-C$.

4. Determine the period of $y = 3 \sin(5x) - 2$.

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

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

   Period = $2\text{π}$. Standard sine and cosine functions complete one cycle in $2\pi$.

6. Determine the period of $y = 3 \text{sin}(5x) - 2$.

   Period = $\frac{2\text{π}}{5}$. Period = $\frac{2\pi}{|B|}$ where B=5.

7. What is the frequency of a function with period $2\text{π}$?

   Frequency = $\frac{1}{2\text{π}}$. Frequency is the reciprocal of period.

8. Identify the phase shift of $y = 8 \text{cos}(x - \frac{\text{π}}{4})$.

   Phase shift = $\frac{\text{π}}{4}$. Phase shift = C when in the form $(x-C)$.

9. Identify the vertical shift in $y = 3 \text{cos}(x) + 5$.

   Vertical shift = 5. D value shows vertical displacement from x-axis.

10. Find the midline of $y = 2 \text{sin}(x) + 3$.

    Midline = $y = 3$. Midline is determined by the vertical shift D.

11. Convert $180^\text{o}$ to radians.

    $\text{π}$. $180° \times \frac{\pi}{180} = \pi$ radians.

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

    Frequency = $\frac{3}{2\text{π}}$. Frequency = $\frac{B}{2\pi}$ where B=3.

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

    Period = $2\text{π}$. Basic cosine has the same period as basic sine.

14. Identify the phase shift of $y = 5 \text{cos}(x + \frac{\text{π}}{2})$.

    Phase shift = $-\frac{\text{π}}{2}$. Phase shift = $-\frac{C}{B}$ for the form $\cos(x+C)$.

15. What is the formula for converting degrees to radians?

    Radians = Degrees $\times \frac{\pi}{180}$. Multiply degrees by $\frac{\pi}{180}$ to convert to radians.

16. State the phase shift for $y = \text{cos}(x + \text{π})$.

    Phase shift = $-\text{π}$. For $\cos(x+\pi)$, phase shift = $-\pi$.

17. Convert $60^\text{o}$ to radians.

    $\frac{\text{π}}{3}$. $60° \times \frac{\pi}{180} = \frac{\pi}{3}$ radians.

18. Find the amplitude of $y = -4 \text{cos}(3x)$.

    Amplitude = 4. Amplitude equals $|A|$ regardless of sign.

19. Determine the period of $y = 4 \cos(2x)$.

    Period = $π$. Period = $\frac{2\pi}{|B|}$ where B=2.

20. Identify the vertical shift in $y = 2 \text{sin}(x) + 6$.

    Vertical shift = 6. D represents the upward shift from the x-axis.

21. Convert $90^\text{o}$ to radians.

    $\frac{\text{π}}{2}$. $90° \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.

22. Calculate the period of $y = 2 \text{cos}(0.5x) + 1$.

    Period = $4\text{π}$. Period = $\frac{2\pi}{|B|}$ where B=0.5.

23. Convert $45^\text{o}$ to radians.

    $\frac{\text{π}}{4}$. $45° \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.

24. What is the frequency of $y = \sin(3x)$?

    Frequency = $\frac{3}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=3.

25. Which parameter determines the vertical shift in $y = A \text{sin}(B(x - C)) + D$?

    Vertical shift = $D$. D shifts the function up or down from the x-axis.

26. Find the amplitude of $y = -2 \text{cos}(x)$.

    Amplitude = 2. Amplitude is absolute value of coefficient A.

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

    $y = A \times \sin(B(x - C)) + D$. Standard form where A=amplitude, B affects period, C=phase shift, D=vertical shift.

28. Identify the amplitude of $y = 3 \text{cos}(2x - \frac{\text{π}}{4}) + 1$.

    Amplitude = 3. Amplitude is the absolute value of the coefficient A.

29. Identify the midline for $y = 5 \text{sin}(x) - 3$.

    Midline = $y = -3$. Midline is y equals the vertical shift value.

30. What is the effect of parameter $C$ in $y = A \sin(B(x - C)) + D$?

    $C$ affects the phase shift. $C$ determines horizontal shift left or right.

31. What is the effect of parameter $D$ in $y = A \text{sin}(B(x - C)) + D$?

    $D$ affects the vertical shift. D moves the entire graph up or down vertically.

32. What does the amplitude represent in a sinusoidal function?

    Amplitude is the peak vertical distance from the midline. Maximum deviation from the center line of oscillation.

33. What is the effect of parameter $B$ in $y = A \text{sin}(B(x - C)) + D$?

    $B$ affects the period. B determines how many cycles occur in $2\pi$ units.

34. Identify the midline for $y = -3 \sin(2x) + 5$.

    Midline = $y = 5$. Midline equals the vertical shift $D=5$.

35. Determine the vertical shift of $y = \text{sin}(x) - 4$.

    Vertical shift = $-4$. D value directly gives the vertical displacement.

36. Which parameter affects the amplitude in $y = A \text{sin}(B(x - C)) + D$?

    Amplitude = $|A|$. A controls the vertical stretch and amplitude.

37. Find the amplitude of $y = 7 \text{sin}(x)$.

    Amplitude = 7. Coefficient A directly determines amplitude magnitude.

38. Identify the midline for $y = -3 \text{sin}(2x) + 5$.

    Midline = $y = 5$. Midline equals the vertical shift D=5.

39. State the midline of $y = \text{cos}(x) + 2$.

    Midline = $y = 2$. Midline is the horizontal line y=D.

40. What is the effect of parameter $C$ in $y = A \text{sin}(B(x - C)) + D$?

    $C$ affects the phase shift. C determines horizontal shift left or right.

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