Sine and Cosine Function Graphs - AP Precalculus
Card 1 of 30
What is the amplitude of the function $y = -4 , \cos(x)$?
What is the amplitude of the function $y = -4 , \cos(x)$?
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- Amplitude is the absolute value of the coefficient, so $|-4| = 4$.
- Amplitude is the absolute value of the coefficient, so $|-4| = 4$.
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Identify the amplitude of $y = \frac{1}{2} \cos(x)$.
Identify the amplitude of $y = \frac{1}{2} \cos(x)$.
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$\frac{1}{2}$. Amplitude is the absolute value of the coefficient.
$\frac{1}{2}$. Amplitude is the absolute value of the coefficient.
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What is the period formula for a sine function $y = \sin(bx)$?
What is the period formula for a sine function $y = \sin(bx)$?
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$\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.
$\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.
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Identify the vertical shift in $y = \cos(x) + 4$.
Identify the vertical shift in $y = \cos(x) + 4$.
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4 units up. The constant $+4$ shifts the entire graph 4 units upward.
4 units up. The constant $+4$ shifts the entire graph 4 units upward.
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What is the standard form of the cosine function?
What is the standard form of the cosine function?
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$y = a , \cos(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.
$y = a , \cos(bx + c) + d$. General form where $a$ is amplitude, $b$ affects period, $c$ affects phase shift, and $d$ is vertical shift.
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What effect does a negative coefficient have on $y = -\sin(x)$?
What effect does a negative coefficient have on $y = -\sin(x)$?
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Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.
Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.
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What is the frequency of the function $y = \sin(3x)$?
What is the frequency of the function $y = \sin(3x)$?
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- Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 3$.
- Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 3$.
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What is the amplitude of $y = 3 , \sin(x)$?
What is the amplitude of $y = 3 , \sin(x)$?
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- The coefficient of sine gives the amplitude (distance from center to peak).
- The coefficient of sine gives the amplitude (distance from center to peak).
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What is the amplitude of $y = 3 , \sin(x)$?
What is the amplitude of $y = 3 , \sin(x)$?
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- The coefficient of sine gives the amplitude (distance from center to peak).
- The coefficient of sine gives the amplitude (distance from center to peak).
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What is the period formula for a sine function $y = \sin(bx)$?
What is the period formula for a sine function $y = \sin(bx)$?
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$\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.
$\frac{2\pi}{b}$. Period equals $\frac{2\pi}{b}$ where $b$ is the coefficient of $x$.
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What is the vertical shift in $y = \sin(x) + 6$?
What is the vertical shift in $y = \sin(x) + 6$?
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6 units up. The constant $+6$ shifts the entire graph 6 units upward.
6 units up. The constant $+6$ shifts the entire graph 6 units upward.
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What is the effect of $a$ in $y = a , \sin(bx + c) + d$?
What is the effect of $a$ in $y = a , \sin(bx + c) + d$?
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Affects the amplitude. Parameter $a$ determines the vertical stretch or compression.
Affects the amplitude. Parameter $a$ determines the vertical stretch or compression.
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Identify the phase shift in $y = \sin(x - \frac{\pi}{4})$.
Identify the phase shift in $y = \sin(x - \frac{\pi}{4})$.
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$\frac{\pi}{4}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/4)}{1} = \frac{\pi}{4}$ to the right.
$\frac{\pi}{4}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/4)}{1} = \frac{\pi}{4}$ to the right.
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What is the amplitude of the function $y = -4 , \cos(x)$?
What is the amplitude of the function $y = -4 , \cos(x)$?
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- Amplitude is the absolute value of the coefficient, so $|-4| = 4$.
- Amplitude is the absolute value of the coefficient, so $|-4| = 4$.
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What is the period of $y = \cos(2x)$?
What is the period of $y = \cos(2x)$?
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$\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.
$\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.
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What is the vertical shift in $y = \sin(x) + 3$?
What is the vertical shift in $y = \sin(x) + 3$?
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3 units up. The constant $+3$ shifts the entire graph 3 units upward.
3 units up. The constant $+3$ shifts the entire graph 3 units upward.
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What effect does a negative coefficient have on $y = -\sin(x)$?
What effect does a negative coefficient have on $y = -\sin(x)$?
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Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.
Reflects over the x-axis. Negative coefficient flips the graph vertically across the x-axis.
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Identify the phase shift in $y = \cos(x + \frac{\pi}{3})$.
Identify the phase shift in $y = \cos(x + \frac{\pi}{3})$.
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$\frac{\pi}{3}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/3}{1} = -\frac{\pi}{3}$ (left).
$\frac{\pi}{3}$ to the left. Phase shift is $-\frac{c}{b} = -\frac{\pi/3}{1} = -\frac{\pi}{3}$ (left).
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What is the maximum value of $y = 5 , \sin(x)$?
What is the maximum value of $y = 5 , \sin(x)$?
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- Maximum value equals the amplitude for positive coefficient.
- Maximum value equals the amplitude for positive coefficient.
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What is the minimum value of $y = -6 , \cos(x)$?
What is the minimum value of $y = -6 , \cos(x)$?
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-6. For negative amplitude, minimum equals the amplitude value.
-6. For negative amplitude, minimum equals the amplitude value.
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State the phase shift formula for $y = \sin(bx + c)$.
State the phase shift formula for $y = \sin(bx + c)$.
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$-\frac{c}{b}$. Standard formula for horizontal shift in transformed sine functions.
$-\frac{c}{b}$. Standard formula for horizontal shift in transformed sine functions.
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What is the horizontal shift in $y = \cos(x - \frac{\pi}{2})$?
What is the horizontal shift in $y = \cos(x - \frac{\pi}{2})$?
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$\frac{\pi}{2}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/2)}{1} = \frac{\pi}{2}$ to the right.
$\frac{\pi}{2}$ to the right. Phase shift is $-\frac{c}{b} = -\frac{(-\pi/2)}{1} = \frac{\pi}{2}$ to the right.
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What is the range of the function $y = -3 , \sin(x)$?
What is the range of the function $y = -3 , \sin(x)$?
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[-3, 3]. Range is $[-|a|, |a|] = [-3, 3]$ for amplitude 3.
[-3, 3]. Range is $[-|a|, |a|] = [-3, 3]$ for amplitude 3.
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What is the vertical shift in $y = \cos(x) - 2$?
What is the vertical shift in $y = \cos(x) - 2$?
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2 units down. The constant $-2$ shifts the entire graph 2 units downward.
2 units down. The constant $-2$ shifts the entire graph 2 units downward.
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What is the period of $y = \sin(\frac{x}{2})$?
What is the period of $y = \sin(\frac{x}{2})$?
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$4\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/2} = 4\pi$.
$4\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/2} = 4\pi$.
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What is the frequency of the function $y = \cos(4x)$?
What is the frequency of the function $y = \cos(4x)$?
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- Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 4$.
- Frequency equals $\frac{b}{2\pi} \cdot 2\pi = b = 4$.
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What is the maximum value of $y = 2 , \cos(x)$?
What is the maximum value of $y = 2 , \cos(x)$?
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- Maximum value equals the amplitude for positive coefficient.
- Maximum value equals the amplitude for positive coefficient.
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State the period of $y = \cos(\frac{1}{3}x)$.
State the period of $y = \cos(\frac{1}{3}x)$.
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$6\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/3} = 6\pi$.
$6\pi$. Period equals $\frac{2\pi}{b} = \frac{2\pi}{1/3} = 6\pi$.
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What is the phase shift in $y = \sin(2x + \pi)$?
What is the phase shift in $y = \sin(2x + \pi)$?
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$-\frac{\pi}{2}$. Phase shift is $-\frac{c}{b} = -\frac{\pi}{2} = -\frac{\pi}{2}$.
$-\frac{\pi}{2}$. Phase shift is $-\frac{c}{b} = -\frac{\pi}{2} = -\frac{\pi}{2}$.
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Identify the amplitude of $y = \frac{1}{2} \cos(x)$.
Identify the amplitude of $y = \frac{1}{2} \cos(x)$.
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$\frac{1}{2}$. Amplitude is the absolute value of the coefficient.
$\frac{1}{2}$. Amplitude is the absolute value of the coefficient.
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