AP Precalculus Flashcards: Sinusoidal Functions

What this deck covers

This deck focuses on Sinusoidal Functions, 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 range of $y = 4 \, \text{sin}(x) - 2$?

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

Flashcard 2: What is the range of $y = 2 \, \text{cos}(x) - 6$?

Answer: $[-8, -4]$. Range is $[-6-2, -6+2] = [-8, -4]$

Flashcard 3: Identify the sinusoidal axis for $y = 4 \, \text{sin}(x) - 1$.

Answer: $y = -1$. The sinusoidal axis is at $y = D = -1$.

Flashcard 4: Determine the vertical shift for $y = \text{cos}(x) - 4$.

Answer: -4. The constant term gives the vertical shift downward.

Flashcard 5: What is the general form of a sinusoidal function?

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

Flashcard 6: Identify the phase shift for $y = 5 \, \text{sin}(x - \frac{\pi}{6})$.

Answer: $\frac{\pi}{6}$. Phase shift is $\frac{\pi}{6}$ units to the right.

Flashcard 7: Find the phase shift of $y = \text{cos}(x - \frac{\pi}{2})$.

Answer: $\frac{\pi}{2}$. Phase shift is $\frac{\pi}{2}$ units to the right.

Flashcard 8: What does the parameter $B$ affect in the function $y = A \, \text{sin}(B(x - C)) + D$?

Answer: Period. $B$ determines horizontal compression/stretch, affecting how often the function repeats.

Flashcard 9: What is the period of $y = 5 \, \text{cos}(\frac{x}{3})$?

Answer: $6\pi$. Period = $\frac{2\pi}{\frac{1}{3}} = 6\pi$.

Flashcard 10: Calculate the period of $y = 2 \, \text{sin}(3x - \pi)$.

Answer: $\frac{2\pi}{3}$. Period = $\frac{2\pi}{3}$ when $B = 3$.

Flashcard 11: State the maximum value of $y = 8 + 5 \, \text{sin}(x)$.

Answer:

1. Maximum occurs when $\sin(x) = 1$: $8 + 5(1) = 13$.

Flashcard 12: What is the period of $y = \text{cos}(\frac{x}{4})$?

Answer: $8\pi$. Period = $\frac{2\pi}{\frac{1}{4}} = 8\pi$.

Flashcard 13: How is the period of a sinusoidal function calculated?

Answer: $\frac{2\pi}{B}$. Period equals $\frac{2\pi}{B}$ where $B$ is the frequency parameter.

Flashcard 14: What is the phase shift of the function $y = A \, \text{sin}(B(x - C)) + D$?

Answer: $C$. The horizontal shift is $C$ units to the right when positive.

Flashcard 15: Identify the phase shift for $y = \text{sin}(x + \pi)$.

Answer: $-\pi$. Rewrite as $\sin(x - (-\pi))$, so phase shift is $-\pi$.

Flashcard 16: What does the parameter $A$ represent in the sinusoidal function $y = A \, \text{sin}(B(x - C)) + D$?

Answer: Amplitude. $A$ controls the maximum distance from the sinusoidal axis.

Flashcard 17: Determine the amplitude of $y = -4 \, \text{sin}(x)$.

Answer:

1. Amplitude is the absolute value: $|-4| = 4$.

Flashcard 18: Determine the sinusoidal axis for $y = 3 \, \text{sin}(x) + 4$.

Answer: $y = 4$. The sinusoidal axis is at the vertical shift $y = 4$.

Flashcard 19: What does the sinusoidal axis represent?

Answer: Average value of max and min. The horizontal line around which the function oscillates.

Flashcard 20: State the vertical shift of $y = \text{cos}(x) + 5$.

Answer:

1. The constant term added to the function shifts it vertically.

Flashcard 21: Find the maximum value of $y = 3 \, \cos(x) + 1$.

Answer:

1. Maximum occurs when cosine equals 1: $3(1) + 1 = 4$.

Flashcard 22: Identify the amplitude of $y = 3 \, \text{cos}(2x)$.

Answer:

1. The coefficient of cosine gives the amplitude.

Flashcard 23: Calculate the period of $y = \text{sin}(\frac{1}{2}x)$.

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

Flashcard 24: Find the minimum value of $y = 2 \, \text{sin}(x) - 3$.

Answer: -5. Minimum occurs when sine equals -1: $2(-1) - 3 = -5$.

Flashcard 25: What is the phase shift of $y = 4 \, \text{cos}(x + \frac{\pi}{3})$?

Answer: $-\frac{\pi}{3}$. Rewrite as $\cos(x - (-\frac{\pi}{3}))$, so phase shift is $-\frac{\pi}{3}$.

Flashcard 26: Find the maximum value of $y = 3 \, \text{cos}(x) + 1$.

Answer:

1. Maximum occurs when cosine equals 1: $3(1) + 1 = 4$.

Flashcard 27: What is the range of $y = -3 \, \text{sin}(x) + 2$?

Answer: [-1, 5]. Range is $[2-3, 2+3] = [-1, 5]$ since amplitude is 3.

Flashcard 28: What does the parameter $D$ represent in a sinusoidal function?

Answer: Vertical shift. $D$ moves the entire graph up or down from the x-axis.

Flashcard 29: What is the period of $y = 6 \, \text{sin}(\frac{x}{6})$?

Answer: $12\pi$. Period = $\frac{2\pi}{\frac{1}{6}} = 12\pi$.

Flashcard 30: What is the equation for a sinusoidal function with amplitude 5 and period $\pi$?

Answer: $y = 5 \, \text{sin}(2x)$. Amplitude 5 and period $\pi$ means $B = \frac{2\pi}{\pi} = 2$.

Flashcard 31: Calculate the period for $y = 3 \, \text{sin}(5x)$.

Answer: $\frac{2\pi}{5}$. Period = $\frac{2\pi}{5}$ when $B = 5$.

Flashcard 32: Find the maximum value of $y = 3 \, \text{sin}(x) - 2$.

Answer:

1. Maximum occurs when $\sin(x) = 1$: $3(1) - 2 = 1$.

Flashcard 33: Determine the amplitude of $y = -2 \, \text{cos}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient: $|-2| = 2$.

Flashcard 34: Determine the period of $y = \text{sin}(4x)$.

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

Flashcard 35: What is the frequency of $y = \text{cos}(\frac{x}{2})$?

Answer: $\frac{1}{4\pi}$. Frequency = $\frac{B}{2\pi} = \frac{\frac{1}{2}}{2\pi} = \frac{1}{4\pi}$.

Flashcard 36: Find the phase shift of $y = 2 \, \text{sin}(x - \frac{\pi}{4})$.

Answer: $\frac{\pi}{4}$. Phase shift is the value subtracted inside the function.

Flashcard 37: Find the amplitude of $y = 7 \, \text{cos}(x) + 3$.

Answer:

1. The coefficient 7 in front of cosine is the amplitude.