AP Precalculus Flashcards: Periodic Phenomena

What this deck covers

This deck focuses on Periodic Phenomena, 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: Identify the amplitude of $y = 0.5 \, \text{sin}(x)$.

Answer: 0.5. Amplitude is the absolute value of the coefficient.

Flashcard 2: Identify the vertical shift in $y = \text{sin}(x) - 3$.

Answer: Down 3. Vertical shift is the constant term added to the function.

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

Answer:

1. Amplitude is the absolute value of the coefficient of cosine.

Flashcard 4: Determine the phase shift of $y = \text{sin}(x + \frac{\pi}{4})$.

Answer: $\frac{\pi}{4}$ to the left. Phase shift is $\frac{\pi}{4}$ left for $(x + \frac{\pi}{4})$.

Flashcard 5: What is the period of $y = \text{cos}(4x)$?

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

Flashcard 6: Determine the vertical shift in $y = \text{cos}(x) + 2$.

Answer: Up 2. Vertical shift is the constant term added to the function.

Flashcard 7: What is the vertical shift of $y = -\text{sin}(x) + 4$?

Answer: Up 4. Vertical shift is the constant term added to the function.

Flashcard 8: State the general form of the cosine function.

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

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

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

Flashcard 10: What is the general form of the sine 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 11: What is the frequency of $y = \text{sin}(10x)$?

Answer:

1. Frequency is the coefficient $B$ of $x$.

Flashcard 12: What is the vertical shift in $y = \text{sin}(x) + 6$?

Answer: Up 6. Vertical shift is the constant term added to the function.

Flashcard 13: Find the period of $y = \text{sin}(\frac{1}{2}x)$.

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

Flashcard 14: Identify the period of $y = \cos(\frac{x}{3})$.

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

Flashcard 15: What is the period of $y = \text{cos}(\frac{x}{4})$?

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

Flashcard 16: What is the range of the function $y = 2 \, \text{sin}(x) + 1$?

Answer: $[-1, 3]$. Range is $[D - |A|, D + |A|]$ where $A = 2$ and $D = 1$.

Flashcard 17: What is the period of the cosine function $y = \cos(2x)$?

Answer: $\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.

Flashcard 18: State the amplitude of $y = -2 \, \text{cos}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 19: Identify the amplitude of $y = 6 \, \text{sin}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient of sine.

Flashcard 20: Find the phase shift of $y = \text{sin}(x - \frac{\pi}{4})$.

Answer: $\frac{\pi}{4}$ to the right. Phase shift is $C$ in the form $(x - C)$.

Flashcard 21: Determine the frequency of $y = \text{cos}(5x)$.

Answer:

1. Frequency is the coefficient $B$ of $x$.

Flashcard 22: Find the range of $y = -4 \, \text{cos}(x) + 2$.

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

Flashcard 23: What is the phase shift of $y = \text{cos}(x + \frac{\pi}{2})$?

Answer: $\frac{\pi}{2}$ to the left. Phase shift is $\frac{\pi}{2}$ left for $(x + \frac{\pi}{2})$.

Flashcard 24: Find the range of $y = 2 \, \text{sin}(x) - 4$.

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

Flashcard 25: Find the range of $y = 3 \, \text{sin}(x) - 1$.

Answer: $[-4, 2]$. Range is $[D - |A|, D + |A|]$ where $A = 3$ and $D = -1$.

Flashcard 26: Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{6})$.

Answer: $\frac{\pi}{6}$ to the right. Phase shift is $C$ in the form $(x - C)$.

Flashcard 27: What is the vertical shift of $y = -\text{cos}(x) + 3$?

Answer: Up 3. Vertical shift is the constant term added to the function.

Flashcard 28: What is the vertical shift of $y = \text{cos}(x) - 5$?

Answer: Down 5. Vertical shift is the constant term added to the function.

Flashcard 29: What is the frequency of $y = \text{cos}(3x)$?

Answer:

1. Frequency is the coefficient $B$ of $x$.

Flashcard 30: Find the frequency of $y = \sin(7x)$.

Answer:

1. Frequency is the coefficient $B$ of $x$.

Flashcard 31: Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{2})$.

Answer: $\frac{\pi}{2}$ to the right. Phase shift is $C$ in the form $(x - C)$.

Flashcard 32: What is the period of the function $y = \sin(3x)$?

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

Flashcard 33: What is the period of the function $y = \text{sin}(3x)$?

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

Flashcard 34: Determine the phase shift of $y = \text{sin}(x + \pi)$.

Answer: $\pi$ to the left. Phase shift is $\pi$ left for $(x + \pi)$.

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

Answer: $[-2, 2]$. Range is $[-|A|, |A|]$ for standard cosine function.

Flashcard 36: What is the phase shift of $y = \text{sin}(x + \frac{\pi}{3})$?

Answer: $\frac{\pi}{3}$ to the left. Phase shift is $\frac{\pi}{3}$ left for $(x + \frac{\pi}{3})$.

Flashcard 37: What is the frequency of $y = \text{cos}(8x)$?

Answer:

1. Frequency is the coefficient $B$ of $x$.

Flashcard 38: State the period of the function $y = \sin(0.5x)$.

Answer: $4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 0.5$.

Flashcard 39: What is the period of $y = \sin(5x)$?

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

Flashcard 40: What is the period of the cosine function $y = \text{cos}(2x)$?

Answer: $\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.

Flashcard 41: State the amplitude of $y = 4 \, \text{cos}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient of cosine.

Flashcard 42: What is the amplitude of $y = -3 \, \text{sin}(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient.