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: 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 2: 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 3: Identify the amplitude of $y = 0.5 \, \text{sin}(x)$.

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

Flashcard 4: Identify the amplitude of $y = 0.5 \, \text{sin}(x)$.

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

Flashcard 5: Determine the phase shift of $y = \text{sin}(x + \pi)$.

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

Flashcard 6: Identify the vertical shift in $y = \text{sin}(x) - 3$.

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

Flashcard 7: What is the amplitude of $y = 5 \, \text{cos}(x)$?

Answer:

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

Flashcard 8: State the amplitude of $y = 4 \, \text{cos}(x)$.

Answer:

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

Flashcard 9: Identify the vertical shift in $y = \text{sin}(x) - 3$.

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

Flashcard 10: 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 11: 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 12: What is the period of $y = \text{cos}(4x)$?

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

Flashcard 13: Determine the vertical shift in $y = \text{cos}(x) + 2$.

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

Flashcard 14: 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 15: 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 16: What is the range of $y = -2 \, \text{cos}(x)$?

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

Flashcard 17: 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 18: 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 19: What is the period of $y = \text{sin}(5x)$?

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

Flashcard 20: 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 21: What is the frequency of $y = \text{sin}(10x)$?

Answer:

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

Flashcard 22: 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 23: What is the amplitude of $y = 5 \, \text{cos}(x)$?

Answer:

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

Flashcard 24: 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 25: Determine the vertical shift in $y = \text{cos}(x) + 2$.

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

Flashcard 26: 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 27: Identify the period of $y = \cos(\frac{x}{3})$.

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

Flashcard 28: 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 29: 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 30: 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 31: 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 32: 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 33: What is the period of the cosine function $y = \cos(2x)$?

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

Flashcard 34: State the amplitude of $y = -2 \, \text{cos}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 35: Identify the amplitude of $y = 6 \, \text{sin}(x)$.

Answer:

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

Flashcard 36: 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 37: Determine the frequency of $y = \text{cos}(5x)$.

Answer:

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

Flashcard 38: 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 39: 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 40: 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 41: 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 42: 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 43: 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 44: 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 45: 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 46: 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 47: What is the frequency of $y = \text{cos}(3x)$?

Answer:

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

Flashcard 48: 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 49: Find the frequency of $y = \sin(7x)$.

Answer:

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

Flashcard 50: What is the frequency of $y = \text{cos}(3x)$?

Answer:

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

Flashcard 51: 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 52: State the amplitude of $y = -2 \, \text{cos}(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 53: State the period of the function $y = \sin(0.5x)$.

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

Flashcard 54: 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 55: What is the frequency of $y = \text{cos}(8x)$?

Answer:

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

Flashcard 56: 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 57: 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 58: Determine the phase shift of $y = \text{sin}(x + \pi)$.

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

Flashcard 59: 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 60: What is the range of $y = -2 \, \text{cos}(x)$?

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

Flashcard 61: Identify the amplitude of $y = 6 \, \text{sin}(x)$.

Answer:

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

Flashcard 62: 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 63: What is the frequency of $y = \text{sin}(10x)$?

Answer:

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

Flashcard 64: What is the frequency of $y = \text{cos}(8x)$?

Answer:

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

Flashcard 65: State the period of the function $y = \sin(0.5x)$.

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

Flashcard 66: What is the period of $y = \text{cos}(4x)$?

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

Flashcard 67: What is the period of $y = \sin(5x)$?

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

Flashcard 68: What is the period of the cosine function $y = \text{cos}(2x)$?

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

Flashcard 69: State the amplitude of $y = 4 \, \text{cos}(x)$.

Answer:

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

Flashcard 70: 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 71: What is the amplitude of $y = -3 \, \text{sin}(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient.