Sinusoidal Function Context and Data Modeling - AP Precalculus
Card 1 of 30
Identify the midline for $y = -3 \text{sin}(2x) + 5$.
Identify the midline for $y = -3 \text{sin}(2x) + 5$.
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Midline = $y = 5$. Midline equals the vertical shift D=5.
Midline = $y = 5$. Midline equals the vertical shift D=5.
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Find the amplitude of $y = 7 \text{sin}(x)$.
Find the amplitude of $y = 7 \text{sin}(x)$.
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Amplitude = 7. Coefficient A directly determines amplitude magnitude.
Amplitude = 7. Coefficient A directly determines amplitude magnitude.
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Convert $180^\text{o}$ to radians.
Convert $180^\text{o}$ to radians.
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$\text{π}$. $180° \times \frac{\pi}{180} = \pi$ radians.
$\text{π}$. $180° \times \frac{\pi}{180} = \pi$ radians.
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What is the period of $y = \text{cos}(x)$?
What is the period of $y = \text{cos}(x)$?
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Period = $2\text{π}$. Basic cosine has the same period as basic sine.
Period = $2\text{π}$. Basic cosine has the same period as basic sine.
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Calculate the phase shift for $y = \text{sin}(x + \frac{\text{π}}{3})$.
Calculate the phase shift for $y = \text{sin}(x + \frac{\text{π}}{3})$.
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Phase shift = $-\frac{\text{π}}{3}$. For $\sin(x+C)$, phase shift = $-C$.
Phase shift = $-\frac{\text{π}}{3}$. For $\sin(x+C)$, phase shift = $-C$.
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State the midline of $y = \text{cos}(x) + 2$.
State the midline of $y = \text{cos}(x) + 2$.
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Midline = $y = 2$. Midline is the horizontal line y=D.
Midline = $y = 2$. Midline is the horizontal line y=D.
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Identify the vertical shift in $y = 2 \text{sin}(x) + 6$.
Identify the vertical shift in $y = 2 \text{sin}(x) + 6$.
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Vertical shift = 6. D represents the upward shift from the x-axis.
Vertical shift = 6. D represents the upward shift from the x-axis.
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Determine the period of $y = 3 \text{sin}(5x) - 2$.
Determine the period of $y = 3 \text{sin}(5x) - 2$.
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Period = $\frac{2\text{π}}{5}$. Period = $\frac{2\pi}{|B|}$ where B=5.
Period = $\frac{2\text{π}}{5}$. Period = $\frac{2\pi}{|B|}$ where B=5.
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Find the amplitude of $y = -4 \text{cos}(3x)$.
Find the amplitude of $y = -4 \text{cos}(3x)$.
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Amplitude = 4. Amplitude equals $|A|$ regardless of sign.
Amplitude = 4. Amplitude equals $|A|$ regardless of sign.
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Convert $90^\text{o}$ to radians.
Convert $90^\text{o}$ to radians.
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$\frac{\text{π}}{2}$. $90° \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.
$\frac{\text{π}}{2}$. $90° \times \frac{\pi}{180} = \frac{\pi}{2}$ radians.
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What is the period of $y = \text{sin}(x)$?
What is the period of $y = \text{sin}(x)$?
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Period = $2\text{π}$. Standard sine and cosine functions complete one cycle in $2\pi$.
Period = $2\text{π}$. Standard sine and cosine functions complete one cycle in $2\pi$.
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Determine the vertical shift of $y = \text{sin}(x) - 4$.
Determine the vertical shift of $y = \text{sin}(x) - 4$.
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Vertical shift = $-4$. D value directly gives the vertical displacement.
Vertical shift = $-4$. D value directly gives the vertical displacement.
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Identify the vertical shift in $y = 2 \text{sin}(x) + 6$.
Identify the vertical shift in $y = 2 \text{sin}(x) + 6$.
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Vertical shift = 6. D represents the upward shift from the x-axis.
Vertical shift = 6. D represents the upward shift from the x-axis.
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Find the midline of $y = 2 \text{sin}(x) + 3$.
Find the midline of $y = 2 \text{sin}(x) + 3$.
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Midline = $y = 3$. Midline is determined by the vertical shift D.
Midline = $y = 3$. Midline is determined by the vertical shift D.
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Identify the phase shift of $y = 5 \text{cos}(x + \frac{\text{π}}{2})$.
Identify the phase shift of $y = 5 \text{cos}(x + \frac{\text{π}}{2})$.
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Phase shift = $-\frac{\text{π}}{2}$. Phase shift = $-\frac{C}{B}$ for the form $\cos(x+C)$.
Phase shift = $-\frac{\text{π}}{2}$. Phase shift = $-\frac{C}{B}$ for the form $\cos(x+C)$.
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What is the effect of parameter $C$ in $y = A \text{sin}(B(x - C)) + D$?
What is the effect of parameter $C$ in $y = A \text{sin}(B(x - C)) + D$?
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$C$ affects the phase shift. C determines horizontal shift left or right.
$C$ affects the phase shift. C determines horizontal shift left or right.
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What is the frequency of $y = \sin(3x)$?
What is the frequency of $y = \sin(3x)$?
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Frequency = $\frac{3}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=3.
Frequency = $\frac{3}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=3.
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Find the amplitude of $y = 7 \text{sin}(x)$.
Find the amplitude of $y = 7 \text{sin}(x)$.
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Amplitude = 7. Coefficient A directly determines amplitude magnitude.
Amplitude = 7. Coefficient A directly determines amplitude magnitude.
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Identify the phase shift of $y = 8 \text{cos}(x - \frac{\text{π}}{4})$.
Identify the phase shift of $y = 8 \text{cos}(x - \frac{\text{π}}{4})$.
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Phase shift = $\frac{\text{π}}{4}$. Phase shift = C when in the form $(x-C)$.
Phase shift = $\frac{\text{π}}{4}$. Phase shift = C when in the form $(x-C)$.
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Find the amplitude of $y = -2 \text{cos}(x)$.
Find the amplitude of $y = -2 \text{cos}(x)$.
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Amplitude = 2. Amplitude is absolute value of coefficient A.
Amplitude = 2. Amplitude is absolute value of coefficient A.
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Convert $45^\text{o}$ to radians.
Convert $45^\text{o}$ to radians.
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$\frac{\text{π}}{4}$. $45° \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.
$\frac{\text{π}}{4}$. $45° \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.
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What does the amplitude represent in a sinusoidal function?
What does the amplitude represent in a sinusoidal function?
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Amplitude is the peak vertical distance from the midline. Maximum deviation from the center line of oscillation.
Amplitude is the peak vertical distance from the midline. Maximum deviation from the center line of oscillation.
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Identify the vertical shift in $y = 3 \text{cos}(x) + 5$.
Identify the vertical shift in $y = 3 \text{cos}(x) + 5$.
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Vertical shift = 5. D value shows vertical displacement from x-axis.
Vertical shift = 5. D value shows vertical displacement from x-axis.
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Find the amplitude of $y = -2 \text{cos}(x)$.
Find the amplitude of $y = -2 \text{cos}(x)$.
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Amplitude = 2. Amplitude is absolute value of coefficient A.
Amplitude = 2. Amplitude is absolute value of coefficient A.
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State the phase shift for $y = \text{cos}(x + \text{π})$.
State the phase shift for $y = \text{cos}(x + \text{π})$.
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Phase shift = $-\text{π}$. For $\cos(x+\pi)$, phase shift = $-\pi$.
Phase shift = $-\text{π}$. For $\cos(x+\pi)$, phase shift = $-\pi$.
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What is the frequency of $y = \text{sin}(3x)$?
What is the frequency of $y = \text{sin}(3x)$?
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Frequency = $\frac{3}{2\text{π}}$. Frequency = $\frac{B}{2\pi}$ where B=3.
Frequency = $\frac{3}{2\text{π}}$. Frequency = $\frac{B}{2\pi}$ where B=3.
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Identify the midline for $y = 5 \text{sin}(x) - 3$.
Identify the midline for $y = 5 \text{sin}(x) - 3$.
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Midline = $y = -3$. Midline is y equals the vertical shift value.
Midline = $y = -3$. Midline is y equals the vertical shift value.
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What is the effect of parameter $D$ in $y = A \text{sin}(B(x - C)) + D$?
What is the effect of parameter $D$ in $y = A \text{sin}(B(x - C)) + D$?
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$D$ affects the vertical shift. D moves the entire graph up or down vertically.
$D$ affects the vertical shift. D moves the entire graph up or down vertically.
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Identify the phase shift of $y = 8 \text{cos}(x - \frac{\text{π}}{4})$.
Identify the phase shift of $y = 8 \text{cos}(x - \frac{\text{π}}{4})$.
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Phase shift = $\frac{\text{π}}{4}$. Phase shift = C when in the form $(x-C)$.
Phase shift = $\frac{\text{π}}{4}$. Phase shift = C when in the form $(x-C)$.
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What is the frequency of $y = 6 \cos(7x)$?
What is the frequency of $y = 6 \cos(7x)$?
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Frequency = $\frac{7}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=7.
Frequency = $\frac{7}{2\pi}$. Frequency = $\frac{B}{2\pi}$ where B=7.
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