Periodic Phenomena - AP Precalculus
Card 1 of 30
Find the range of $y = -4 , \text{cos}(x) + 2$.
Find the range of $y = -4 , \text{cos}(x) + 2$.
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$[-2, 6]$. Range is $[D - |A|, D + |A|]$ where $A = -4$ and $D = 2$.
$[-2, 6]$. Range is $[D - |A|, D + |A|]$ where $A = -4$ and $D = 2$.
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Find the period of $y = \text{sin}(\frac{1}{2}x)$.
Find the period of $y = \text{sin}(\frac{1}{2}x)$.
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$4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{2}$.
$4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{2}$.
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What is the period of the function $y = \text{sin}(3x)$?
What is the period of the function $y = \text{sin}(3x)$?
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$\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{B}$ where $B = 3$.
$\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{B}$ where $B = 3$.
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What is the frequency of $y = \text{cos}(3x)$?
What is the frequency of $y = \text{cos}(3x)$?
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- Frequency is the coefficient $B$ of $x$.
- Frequency is the coefficient $B$ of $x$.
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What is the vertical shift of $y = \text{cos}(x) - 5$?
What is the vertical shift of $y = \text{cos}(x) - 5$?
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Down 5. Vertical shift is the constant term added to the function.
Down 5. Vertical shift is the constant term added to the function.
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Determine the vertical shift in $y = \text{cos}(x) + 2$.
Determine the vertical shift in $y = \text{cos}(x) + 2$.
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Up 2. Vertical shift is the constant term added to the function.
Up 2. Vertical shift is the constant term added to the function.
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What is the frequency of $y = \text{cos}(8x)$?
What is the frequency of $y = \text{cos}(8x)$?
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- Frequency is the coefficient $B$ of $x$.
- Frequency is the coefficient $B$ of $x$.
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Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{2})$.
Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{2})$.
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$\frac{\pi}{2}$ to the right. Phase shift is $C$ in the form $(x - C)$.
$\frac{\pi}{2}$ to the right. Phase shift is $C$ in the form $(x - C)$.
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Determine the vertical shift in $y = \text{cos}(x) + 2$.
Determine the vertical shift in $y = \text{cos}(x) + 2$.
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Up 2. Vertical shift is the constant term added to the function.
Up 2. Vertical shift is the constant term added to the function.
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What is the general form of the sine function?
What is the general form of the sine function?
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$y = A , \text{sin}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.
$y = A , \text{sin}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.
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State the amplitude of $y = 4 , \text{cos}(x)$.
State the amplitude of $y = 4 , \text{cos}(x)$.
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- Amplitude is the absolute value of the coefficient of cosine.
- Amplitude is the absolute value of the coefficient of cosine.
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Find the phase shift of $y = \text{sin}(x - \frac{\pi}{4})$.
Find the phase shift of $y = \text{sin}(x - \frac{\pi}{4})$.
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$\frac{\pi}{4}$ to the right. Phase shift is $C$ in the form $(x - C)$.
$\frac{\pi}{4}$ to the right. Phase shift is $C$ in the form $(x - C)$.
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What is the range of the function $y = 2 , \text{sin}(x) + 1$?
What is the range of the function $y = 2 , \text{sin}(x) + 1$?
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$[-1, 3]$. Range is $[D - |A|, D + |A|]$ where $A = 2$ and $D = 1$.
$[-1, 3]$. Range is $[D - |A|, D + |A|]$ where $A = 2$ and $D = 1$.
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Determine the frequency of $y = \text{cos}(5x)$.
Determine the frequency of $y = \text{cos}(5x)$.
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- Frequency is the coefficient $B$ of $x$.
- Frequency is the coefficient $B$ of $x$.
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What is the period of the cosine function $y = \text{cos}(2x)$?
What is the period of the cosine function $y = \text{cos}(2x)$?
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$\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.
$\pi$. Period formula: $\frac{2\pi}{B}$ where $B = 2$.
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Identify the vertical shift in $y = \text{sin}(x) - 3$.
Identify the vertical shift in $y = \text{sin}(x) - 3$.
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Down 3. Vertical shift is the constant term added to the function.
Down 3. Vertical shift is the constant term added to the function.
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State the general form of the cosine function.
State the general form of the cosine function.
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$y = A , \text{cos}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.
$y = A , \text{cos}(B(x - C)) + D$. Standard form with amplitude $A$, frequency $B$, phase shift $C$, and vertical shift $D$.
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Find the period of $y = \text{sin}(\frac{1}{2}x)$.
Find the period of $y = \text{sin}(\frac{1}{2}x)$.
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$4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{2}$.
$4\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{2}$.
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What is the amplitude of $y = -3 , \text{sin}(x)$?
What is the amplitude of $y = -3 , \text{sin}(x)$?
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- Amplitude is the absolute value of the coefficient.
- Amplitude is the absolute value of the coefficient.
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What is the phase shift of $y = \text{sin}(x + \frac{\pi}{3})$?
What is the phase shift of $y = \text{sin}(x + \frac{\pi}{3})$?
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$\frac{\pi}{3}$ to the left. Phase shift is $\frac{\pi}{3}$ left for $(x + \frac{\pi}{3})$.
$\frac{\pi}{3}$ to the left. Phase shift is $\frac{\pi}{3}$ left for $(x + \frac{\pi}{3})$.
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What is the range of $y = -2 , \text{cos}(x)$?
What is the range of $y = -2 , \text{cos}(x)$?
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$[-2, 2]$. Range is $[-|A|, |A|]$ for standard cosine function.
$[-2, 2]$. Range is $[-|A|, |A|]$ for standard cosine function.
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What is the amplitude of $y = 5 , \text{cos}(x)$?
What is the amplitude of $y = 5 , \text{cos}(x)$?
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- Amplitude is the absolute value of the coefficient of cosine.
- Amplitude is the absolute value of the coefficient of cosine.
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Identify the period of $y = \cos(\frac{x}{3})$.
Identify the period of $y = \cos(\frac{x}{3})$.
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$6\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{3}$.
$6\pi$. Period formula: $\frac{2\pi}{B}$ where $B = \frac{1}{3}$.
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Find the frequency of $y = \sin(7x)$.
Find the frequency of $y = \sin(7x)$.
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- Frequency is the coefficient $B$ of $x$.
- Frequency is the coefficient $B$ of $x$.
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What is the phase shift of $y = \text{cos}(x + \frac{\pi}{2})$?
What is the phase shift of $y = \text{cos}(x + \frac{\pi}{2})$?
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$\frac{\pi}{2}$ to the left. Phase shift is $\frac{\pi}{2}$ left for $ (x + \frac{\pi}{2}) $.
$\frac{\pi}{2}$ to the left. Phase shift is $\frac{\pi}{2}$ left for $ (x + \frac{\pi}{2}) $.
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Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{6})$.
Determine the phase shift of $y = \text{cos}(x - \frac{\pi}{6})$.
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$\frac{\pi}{6}$ to the right. Phase shift is $C$ in the form $(x - C)$.
$\frac{\pi}{6}$ to the right. Phase shift is $C$ in the form $(x - C)$.
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What is the frequency of $y = \text{cos}(8x)$?
What is the frequency of $y = \text{cos}(8x)$?
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- Frequency is the coefficient $B$ of $x$.
- Frequency is the coefficient $B$ of $x$.
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Find the range of $y = 3 , \text{sin}(x) - 1$.
Find the range of $y = 3 , \text{sin}(x) - 1$.
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$[-4, 2]$. Range is $[D - |A|, D + |A|]$ where $A = 3$ and $D = -1$.
$[-4, 2]$. Range is $[D - |A|, D + |A|]$ where $A = 3$ and $D = -1$.
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What is the period of $y = \text{cos}(4x)$?
What is the period of $y = \text{cos}(4x)$?
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$\frac{\pi}{2}$. Period formula: $\frac{2\pi}{B}$ where $B = 4$.
$\frac{\pi}{2}$. Period formula: $\frac{2\pi}{B}$ where $B = 4$.
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Identify the amplitude of $y = 0.5 , \text{sin}(x)$.
Identify the amplitude of $y = 0.5 , \text{sin}(x)$.
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0.5. Amplitude is the absolute value of the coefficient.
0.5. Amplitude is the absolute value of the coefficient.
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