AP Precalculus Flashcards: Sine And Cosine Function Graphs

What this deck covers

This deck focuses on Sine And Cosine Function Graphs, 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 amplitude of the function $y = -4 \, \cos(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient, so $|-4| = 4$.

Flashcard 2: Identify the phase shift in $y = \cos(x + \frac{\pi}{3})$.

Answer: $\frac{\pi}{3}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/3}{1} = -\frac{\pi}{3}$ (left).

Flashcard 3: What is the vertical shift in $y = \sin(x) + 3$?

Answer: 3 units up. The constant $+3$ shifts the entire graph 3 units upward.

Flashcard 4: What is the frequency of the function $y = \cos(4x)$?

Answer:

1. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 4$.

Flashcard 5: What is the range of $y = -5 \, \sin(x) + 2$?

Answer: [-3, 7]. Range is $[2-5, 2+5] = [-3, 7]$ with amplitude 5 and vertical shift 2.

Flashcard 6: What is the range of $y = 4 \, \cos(x) + 1$?

Answer: [-3, 5]. Range is $[1-4, 1+4] = [-3, 5]$ with amplitude 4 and vertical shift 1.

Flashcard 7: What is the vertical shift in $y = \sin(x) + 3$?

Answer: 3 units up. The constant $+3$ shifts the entire graph 3 units upward.

Flashcard 8: Identify the amplitude of $y = 10 \, \cos(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 9: What is the standard form of the sine function?

Answer: $y = a \, \sin(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

Flashcard 10: What is the phase shift in $y = \cos(3x - \pi)$?

Answer: $\frac{\pi}{3}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi)}{3} = \frac{\pi}{3}$ to the right.

Flashcard 11: What is the maximum value of $y = 5 \, \sin(x)$?

Answer:

1. Maximum value equals the amplitude for positive coefficient.

Flashcard 12: Identify the period of $y = \sin(\frac{2}{3}x)$.

Answer: $3\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2/3} = 3\pi$.

Flashcard 13: Identify the amplitude of $y = \frac{1}{2} \cos(x)$.

Answer: $\frac{1}{2}$. Amplitude is the absolute value of the coefficient.

Flashcard 14: What is the period of $y = \cos(2x)$?

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

Flashcard 15: What is the period of $y = \sin(\frac{x}{2})$?

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

Flashcard 16: What is the vertical shift in $y = \cos(x) - 2$?

Answer: 2 units down. The constant $-2$ shifts the entire graph 2 units downward.

Flashcard 17: What is the range of $y = 4 \, \cos(x) + 1$?

Answer: [-3, 5]. Range is $[1-4, 1+4] = [-3, 5]$ with amplitude 4 and vertical shift 1.

Flashcard 18: What is the frequency of the function $y = \cos(4x)$?

Answer:

1. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 4$.

Flashcard 19: What is the minimum value of $y = -6 \, \cos(x)$?

Answer: -6. For negative amplitude, minimum equals the amplitude value.

Flashcard 20: What is the amplitude of $y = 7 \, \sin(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 21: What is the standard form of the cosine function?

Answer: $y = a \, \cos(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

Flashcard 22: What effect does a negative coefficient have on $y = -\sin(x)$?

Answer: Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.

Flashcard 23: Identify the phase shift in $y = \sin(x - \frac{\pi}{4})$.

Answer: $\frac{\pi}{4}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/4)}{1} = \frac{\pi}{4}$ to the right.

Flashcard 24: State the phase shift of $y = \sin(x + \frac{\pi}{6})$.

Answer: $\frac{\pi}{6}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/6}{1} = -\frac{\pi}{6}$ (left).

Flashcard 25: What is the effect of $b$ in $y = a \, \cos(bx + c) + d$?

Answer: Affects the period. Parameter $b$ determines how many cycles occur in $2\pi$ units.

Flashcard 26: Identify the amplitude of $y = 10 \, \cos(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 27: Identify the vertical shift in $y = \cos(x) + 4$.

Answer: 4 units up. The constant $+4$ shifts the entire graph 4 units upward.

Flashcard 28: Identify the amplitude of $y = \frac{1}{2} \cos(x)$.

Answer: $\frac{1}{2}$. Amplitude is the absolute value of the coefficient.

Flashcard 29: What is the maximum value of $y = -2 \sin(x)$?

Answer:

1. For negative amplitude, maximum equals the negative of the amplitude.

Flashcard 30: State the period of $y = \cos(\frac{1}{3}x)$.

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

Flashcard 31: What is the amplitude of $y = 3 \, \sin(x)$?

Answer:

1. The coefficient of sine gives the amplitude (distance from center to peak).

Flashcard 32: Identify the vertical shift in $y = \cos(x) + 4$.

Answer: 4 units up. The constant $+4$ shifts the entire graph 4 units upward.

Flashcard 33: What is the phase shift in $y = \sin(2x + \pi)$?

Answer: $-\frac{\pi}{2}$. Phase shift is $-\frac{c}{b} = -\frac{\pi}{2} = -\frac{\pi}{2}$.

Flashcard 34: What is the maximum value of $y = -2 \sin(x)$?

Answer:

1. For negative amplitude, maximum equals the negative of the amplitude.

Flashcard 35: What is the effect of $d$ in $y = a \, \sin(bx + c) + d$?

Answer: Vertical shift. Parameter $d$ moves the graph up or down vertically.

Flashcard 36: What is the period formula for a sine function $y = \sin(bx)$?

Answer: $\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.

Flashcard 37: Identify the phase shift in $y = \cos(x + \frac{\pi}{3})$.

Answer: $\frac{\pi}{3}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/3}{1} = -\frac{\pi}{3}$ (left).

Flashcard 38: What is the standard form of the cosine function?

Answer: $y = a \, \cos(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

Flashcard 39: What is the period of $y = \sin(\frac{x}{2})$?

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

Flashcard 40: What is the standard form of the sine function?

Answer: $y = a \, \sin(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.

Flashcard 41: What is the frequency of the function $y = \sin(3x)$?

Answer:

1. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 3$.

Flashcard 42: What is the horizontal shift in $y = \cos(x - \frac{\pi}{2})$?

Answer: $\frac{\pi}{2}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/2)}{1} = \frac{\pi}{2}$ to the right.

Flashcard 43: State the phase shift formula for $y = \sin(bx + c)$.

Answer: $-\frac{c}{b}$. Standard formula for horizontal shift in transformed sine functions.

Flashcard 44: State the phase shift of $y = \sin(x + \frac{\pi}{6})$.

Answer: $\frac{\pi}{6}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/6}{1} = -\frac{\pi}{6}$ (left).

Flashcard 45: What is the maximum value of $y = 2 \, \cos(x)$?

Answer:

1. Maximum value equals the amplitude for positive coefficient.

Flashcard 46: What is the frequency of $y = \sin(5x)$?

Answer:

1. Frequency equals the coefficient $b$ of $x$.

Flashcard 47: What is the minimum value of $y = -6 \, \cos(x)$?

Answer: -6. For negative amplitude, minimum equals the amplitude value.

Flashcard 48: What is the effect of $a$ in $y = a \, \sin(bx + c) + d$?

Answer: Affects the amplitude. Parameter $a$ determines the vertical stretch or compression.

Flashcard 49: Identify the phase shift in $y = \sin(x - \frac{\pi}{4})$.

Answer: $\frac{\pi}{4}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/4)}{1} = \frac{\pi}{4}$ to the right.

Flashcard 50: What is the vertical shift in $y = \sin(x) + 6$?

Answer: 6 units up. The constant $+6$ shifts the entire graph 6 units upward.

Flashcard 51: State the period of $y = \cos(\frac{1}{3}x)$.

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

Flashcard 52: What is the effect of $a$ in $y = a \, \sin(bx + c) + d$?

Answer: Affects the amplitude. Parameter $a$ determines the vertical stretch or compression.

Flashcard 53: What is the vertical shift in $y = \sin(x) - 1$?

Answer: 1 unit down. The constant $-1$ shifts the entire graph 1 unit downward.

Flashcard 54: What is the range of the function $y = -3 \, \sin(x)$?

Answer: [-3, 3]. Range is $[-|a|, |a|] = [-3, 3]$ for amplitude 3.

Flashcard 55: Identify the period of $y = \sin(\frac{2}{3}x)$.

Answer: $3\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2/3} = 3\pi$.

Flashcard 56: What is the amplitude of $y = 7 \, \sin(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 57: What is the range of $y = -5 \, \sin(x) + 2$?

Answer: [-3, 7]. Range is $[2-5, 2+5] = [-3, 7]$ with amplitude 5 and vertical shift 2.

Flashcard 58: What is the frequency of the function $y = \sin(3x)$?

Answer:

1. Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 3$.

Flashcard 59: What is the maximum value of $y = 2 \, \cos(x)$?

Answer:

1. Maximum value equals the amplitude for positive coefficient.

Flashcard 60: State the phase shift formula for $y = \sin(bx + c)$.

Answer: $-\frac{c}{b}$. Standard formula for horizontal shift in transformed sine functions.

Flashcard 61: What is the frequency of $y = \sin(5x)$?

Answer: $5$. Frequency equals the coefficient $b$ of $x$.

Flashcard 62: What is the period of $y = \cos(2x)$?

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

Flashcard 63: What is the vertical shift in $y = \cos(x) - 2$?

Answer: 2 units down. The constant $-2$ shifts the entire graph 2 units downward.

Flashcard 64: What is the amplitude of $y = 3 \, \sin(x)$?

Answer:

1. The coefficient of sine gives the amplitude (distance from center to peak).

Flashcard 65: What is the horizontal shift in $y = \cos(x - \frac{\pi}{2})$?

Answer: $\frac{\pi}{2}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/2)}{1} = \frac{\pi}{2}$ to the right.

Flashcard 66: What is the phase shift in $y = \sin(2x + \pi)$?

Answer: $-\frac{\pi}{2}$. Phase shift is $-\frac{c}{b} = -\frac{\pi}{2} = -\frac{\pi}{2}$.

Flashcard 67: What is the vertical shift in $y = \sin(x) - 1$?

Answer: 1 unit down. The constant $-1$ shifts the entire graph 1 unit downward.

Flashcard 68: What is the effect of $d$ in $y = a \, \sin(bx + c) + d$?

Answer: Vertical shift. Parameter $d$ moves the graph up or down vertically.

Flashcard 69: What is the effect of $b$ in $y = a \, \cos(bx + c) + d$?

Answer: Affects the period. Parameter $b$ determines how many cycles occur in $2\pi$ units.

Flashcard 70: What is the range of the function $y = 2 \, \cos(x)$?

Answer: [-2, 2]. Range is $[-|a|, |a|] = [-2, 2]$ for amplitude 2.

Flashcard 71: What is the period formula for a sine function $y = \sin(bx)$?

Answer: $\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.

Flashcard 72: What is the phase shift in $y = \cos(3x - \pi)$?

Answer: $\frac{\pi}{3}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi)}{3} = \frac{\pi}{3}$ to the right.

Flashcard 73: What effect does a negative coefficient have on $y = -\sin(x)$?

Answer: Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.

Flashcard 74: What is the amplitude of the function $y = -4 \, \cos(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient, so $|-4| = 4$.

Flashcard 75: What is the range of the function $y = -3 \, \sin(x)$?

Answer: [-3, 3]. Range is $[-|a|, |a|] = [-3, 3]$ for amplitude 3.

Flashcard 76: What is the range of the function $y = 2 \, \cos(x)$?

Answer: [-2, 2]. Range is $[-|a|, |a|] = [-2, 2]$ for amplitude 2.

Flashcard 77: What is the maximum value of $y = 5 \, \sin(x)$?

Answer:

1. Maximum value equals the amplitude for positive coefficient.

Flashcard 78: What is the vertical shift in $y = \sin(x) + 6$?

Answer: 6 units up. The constant $+6$ shifts the entire graph 6 units upward.