AP Precalculus Flashcards: Sinusoidal Function Transformations

What this deck covers

This deck focuses on Sinusoidal Function Transformations, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the period formula for $y = A \, \text{sin}(B(x - C)) + D$?

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

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

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

Flashcard 3: Calculate the period of $y = \text{sin}(\frac{x}{2})$.

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

Flashcard 4: What is the minimum value of $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 5: What is the effect of $C = \text{π}$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 6: Calculate the period of $y = \sin\left(\frac{x}{2}\right)$.

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

Flashcard 7: What transformation occurs if $D < 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 8: Identify the period of $y = \text{cos}(3x) - 4$.

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

Flashcard 9: Determine the frequency of $y = 2 \sin(5x)$.

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

Flashcard 10: What is the general form of a sinusoidal function?

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

Flashcard 11: What is the effect of $B = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 12: Calculate the period of $y = 2 \, \text{sin}(x - \frac{\text{π}}{6})$.

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

Flashcard 13: Calculate the phase shift of $y = \text{sin}(2(x - \frac{\text{π}}{2}))$.

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

Flashcard 14: Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.

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

Flashcard 15: What does $C$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

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

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

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

Flashcard 17: Find the phase shift in $y = \text{cos}(x + \frac{\text{π}}{4})$.

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

Flashcard 18: What is the general form of a sinusoidal function?

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

Flashcard 19: Identify the maximum value of $y = 3 \, \text{sin}(x) + 2$.

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

Flashcard 20: What does a phase shift of $-\text{π}$ indicate in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 21: Determine the period of $y = 4 \, \text{sin}(3x) - 1$.

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

Flashcard 22: What transformation occurs if $D < 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 23: Identify the maximum value of $y = 3 \, \text{sin}(x) + 2$.

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

Flashcard 24: Identify the period of $y = \text{cos}(3x) - 4$.

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

Flashcard 25: What does $B$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 26: Determine the amplitude of $y = 6 \, \text{sin}(x)$.

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

Flashcard 27: Find the period of $y = 5 \, \text{cos}(4x)$.

Answer: Period is $\frac{\text{π}}{2}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{4} = \frac{\pi}{2}$.

Flashcard 28: What is the maximum value of $y = A \sin(B(x - C)) + D$?

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

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

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

Flashcard 30: Calculate the period of $y = 2 \, \text{sin}(x - \frac{\text{π}}{6})$.

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

Flashcard 31: What is the minimum value of $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 32: Identify the vertical shift in $y = 3 \, \text{cos}(x) + 5$.

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

Flashcard 33: What does $A$ represent in the sinusoidal function $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 34: What is the phase shift if $C = -\frac{\text{π}}{3}$?

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

Flashcard 35: What is the effect of $B = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 36: Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.

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

Flashcard 37: Find the phase shift in $y = \text{cos}(x + \frac{\text{π}}{4})$.

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

Flashcard 38: Determine the amplitude of $y = -2 \, \text{sin}(x) + 3$.

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

Flashcard 39: What does $A$ represent in the sinusoidal function $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 40: Identify the amplitude in $y = 3 \, \text{sin}(2x - \frac{\text{π}}{3}) + 4$.

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

Flashcard 41: Find the amplitude of $y = -7 \, \text{cos}(x)$.

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

Flashcard 42: Identify the vertical shift in $y = \text{sin}(x) - 2$.

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

Flashcard 43: Identify the vertical shift in $y = 3 \, \text{cos}(x) + 5$.

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

Flashcard 44: What transformation occurs if $A = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 45: Find the vertical shift of $y = 5 \, \text{sin}(x) + 6$.

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

Flashcard 46: Identify the vertical shift in $y = \text{sin}(x) - 2$.

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

Flashcard 47: Calculate the phase shift of $y = \text{sin}(2(x - \frac{\text{π}}{2}))$.

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

Flashcard 48: What is the vertical shift in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 49: What is the effect of $C = \text{π}$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 50: Find the amplitude of $y = -7 \, \text{cos}(x)$.

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

Flashcard 51: Find the phase shift of $y = 4 \, \text{cos}(2(x - \frac{\text{π}}{6}))$.

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

Flashcard 52: What does $B$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 53: What is the frequency formula for $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 54: Determine the amplitude of $y = 6 \, \text{sin}(x)$.

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

Flashcard 55: What transformation occurs if $A = 1$ in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 56: Determine the amplitude of $y = -2 \, \text{sin}(x) + 3$.

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

Flashcard 57: What is the vertical shift in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 58: Determine the frequency of $y = 2 \, \text{sin}(5x)$.

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

Flashcard 59: What is the phase shift if $C = -\frac{\text{π}}{3}$?

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

Flashcard 60: Find the period of $y = 5 \, \text{cos}(4x)$.

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

Flashcard 61: What transformation occurs if $A < 0$ in $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 62: What transformation occurs if $D > 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 63: Identify the amplitude in $y = 3 \, \text{sin}(2x - \frac{\text{π}}{3}) + 4$.

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

Flashcard 64: Determine the period of $y = 4 \, \sin(3x) - 1$.

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

Flashcard 65: What does $C$ represent in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 66: What transformation occurs if $D > 0$ in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 67: What does a phase shift of $-\text{π}$ indicate in $y = A \, \text{cos}(B(x - C)) + D$?

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

Flashcard 68: What is the maximum value of $y = A \, \text{sin}(B(x - C)) + D$?

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

Flashcard 69: Find the vertical shift of $y = 5 \, \text{sin}(x) + 6$.

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