Sinusoidal Function Transformations - AP Precalculus

1. Find the phase shift of $y = 4 \, \text{cos}(2(x - \frac{\text{π}}{6}))$.

   Phase shift is $\frac{\text{π}}{6}$. Phase shift equals $C = \frac{\pi}{6}$ (right shift).

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

   Period is $4\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi$.

3. What is the minimum value of $y = A \, \text{cos}(B(x - C)) + D$?

   Minimum value is $D - A$. Minimum occurs when sinusoidal reaches its lowest point.

4. What is the effect of $C = \text{π}$ in $y = A \, \text{cos}(B(x - C)) + D$?

   Shift right $\text{π}$ units. Positive $C$ creates rightward horizontal shift.

5. Calculate the period of $y = \sin\left(\frac{x}{2}\right)$.

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

6. Identify the period of $y = \text{cos}(3x) - 4$.

   Period is $\frac{2\text{π}}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ regardless of other parameters.

7. Determine the frequency of $y = 2 \sin(5x)$.

   Frequency is $\frac{5}{2\pi}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.

8. What is the effect of $B = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

   Period is $2\text{π}$. When $B = 1$, the period remains the natural $2\pi$.

9. Calculate the period of $y = 2 \, \text{sin}(x - \frac{\text{π}}{6})$.

   Period is $2\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi$ since $B = 1$.

10. Calculate the phase shift of $y = \text{sin}(2(x - \frac{\text{π}}{2}))$.

    Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).

11. What transformation occurs if $A < 0$ in $y = A \, \text{sin}(B(x - C)) + D$?

    Reflection across the x-axis. Negative $A$ flips the graph upside down.

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

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

13. Identify the maximum value of $y = 3 \, \text{sin}(x) + 2$.

    Maximum value is 5. Maximum = amplitude + vertical shift = $3 + 2 = 5$.

14. What does a phase shift of $-\text{π}$ indicate in $y = A \, \text{cos}(B(x - C)) + D$?

    Shift left $\text{π}$ units. Negative phase shift means leftward horizontal translation.

15. Determine the period of $y = 4 \, \text{sin}(3x) - 1$.

    Period is $\frac{2\text{π}}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ from coefficient of $x$.

16. What transformation occurs if $D < 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

    Downward vertical shift. Negative $D$ moves the entire graph downward.

17. What is the maximum value of $y = A \sin(B(x - C)) + D$?

    Maximum value is $D + A$. Maximum occurs when sinusoidal reaches its highest point.

18. What is the period formula for $y = A \, \text{sin}(B(x - C)) + D$?

    Period = $\frac{2\text{π}}{B}$. Derived from dividing the natural period $2\pi$ by frequency factor $B$.

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

    Vertical shift is 5. Constant term $D = 5$ raises the midline.

20. What is the phase shift if $C = -\frac{\text{π}}{3}$?

    Phase shift is $\frac{\text{π}}{3}$. Phase shift = $-C$ where $C = -\frac{\pi}{3}$.

21. Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.

    Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).

22. Find the phase shift in $y = \text{cos}(x + \frac{\text{π}}{4})$.

    Phase shift is $-\frac{\text{π}}{4}$. Rewrite as $\cos(x - (-\frac{\pi}{4}))$, so $C = -\frac{\pi}{4}$.

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

    $A$ represents the amplitude. The coefficient $A$ determines the vertical stretch and amplitude.

24. Identify the amplitude in $y = 3 \, \text{sin}(2x - \frac{\text{π}}{3}) + 4$.

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

25. Find the amplitude of $y = -7 \, \text{cos}(x)$.

    Amplitude is 7. Amplitude is $|A| = |-7| = 7$ (absolute value of coefficient).

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

    Vertical shift is -2. Constant term $D = -2$ shifts the graph down 2 units.

27. What is the vertical shift in $y = A \, \text{sin}(B(x - C)) + D$?

    Vertical shift is $D$. The constant $D$ moves the midline up or down.

28. What does $B$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

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

29. What is the frequency formula for $y = A \, \text{sin}(B(x - C)) + D$?

    Frequency = $\frac{B}{2\text{π}}$. Frequency measures cycles per $2\pi$ units of input.

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

    Amplitude is 6. Amplitude is the coefficient $A = 6$.

31. What transformation occurs if $A = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

    No change in amplitude. When $A = 1$, amplitude remains at standard value 1.

32. Determine the amplitude of $y = -2 \, \text{sin}(x) + 3$.

    Amplitude is 2. Amplitude is $|A| = |-2| = 2$ (absolute value).

33. Determine the frequency of $y = 2 \, \text{sin}(5x)$.

    Frequency is $\frac{5}{2\text{π}}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.

34. Find the period of $y = 5 \, \text{cos}(4x)$.

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

35. What transformation occurs if $D > 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

    Upward vertical shift. Positive $D$ moves the entire graph upward.

36. Determine the period of $y = 4 \, \sin(3x) - 1$.

    Period is $\frac{2\pi}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ from coefficient of $x$.

37. What does $C$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

    $C$ represents the phase shift. The value $C$ determines horizontal translation of the graph.

38. What is the maximum value of $y = A \, \text{sin}(B(x - C)) + D$?

    Maximum value is $D + A$. Maximum occurs when sinusoidal reaches its highest point.

39. Find the vertical shift of $y = 5 \, \text{sin}(x) + 6$.

    Vertical shift is 6. Constant $D = 6$ shifts the midline up 6 units.

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