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: 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: Identify the maximum value of $y = 3 \, \text{sin}(x) + 2$.

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

Flashcard 14: 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 15: 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 16: 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 17: 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 18: 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 19: Identify the vertical shift in $y = 3 \, \text{cos}(x) + 5$.

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

Flashcard 20: 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 21: 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 22: 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 23: 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 24: 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 25: Find the amplitude of $y = -7 \, \text{cos}(x)$.

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

Flashcard 26: 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 27: 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 28: 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 29: 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 30: Determine the amplitude of $y = 6 \, \text{sin}(x)$.

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

Flashcard 31: 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 32: Determine the amplitude of $y = -2 \, \text{sin}(x) + 3$.

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

Flashcard 33: 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 34: 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 35: 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 36: 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 37: 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 38: 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 39: 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.