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: 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 3: 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 4: Identify the amplitude of $y = 10 \, \cos(x)$.

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 5: 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 6: Identify the amplitude of $y = \frac{1}{2} \cos(x)$.

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

Flashcard 7: What is the period of $y = \cos(2x)$?

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

Flashcard 8: 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 9: What is the frequency of the function $y = \cos(4x)$?

Answer:

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

Flashcard 10: What is the minimum value of $y = -6 \, \cos(x)$?

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

Flashcard 11: What is the amplitude of $y = 7 \, \sin(x)$?

Answer:

1. Amplitude is the absolute value of the coefficient.

Flashcard 12: 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 13: 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 14: 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 15: 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 16: What is the maximum value of $y = -2 \sin(x)$?

Answer:

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

Flashcard 17: Identify the vertical shift in $y = \cos(x) + 4$.

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

Flashcard 18: 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 19: 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 20: 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 21: What is the frequency of the function $y = \sin(3x)$?

Answer:

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

Flashcard 22: State the phase shift formula for $y = \sin(bx + c)$.

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

Flashcard 23: 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 24: What is the maximum value of $y = 2 \, \cos(x)$?

Answer:

1. Maximum value equals the amplitude for positive coefficient.

Flashcard 25: What is the frequency of $y = \sin(5x)$?

Answer:

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

Flashcard 26: 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 27: 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 28: 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 29: 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 30: 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 31: 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 32: What is the amplitude of $y = 3 \, \sin(x)$?

Answer:

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

Flashcard 33: 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 34: 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 35: 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 36: 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 37: What is the range of the function $y = -3 \, \sin(x)$?

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

Flashcard 38: What is the range of the function $y = 2 \, \cos(x)$?

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

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

Answer:

1. Maximum value equals the amplitude for positive coefficient.