Sinusoidal Function Transformations - AP Precalculus
Card 1 of 30
Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.
Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.
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Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).
Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).
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What is the effect of $B = 1$ in $y = A , \text{sin}(B(x - C)) + D$?
What is the effect of $B = 1$ in $y = A , \text{sin}(B(x - C)) + D$?
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Period is $2\text{π}$. When $B = 1$, the period remains the natural $2\pi$.
Period is $2\text{π}$. When $B = 1$, the period remains the natural $2\pi$.
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What is the phase shift if $C = -\frac{\text{π}}{3}$?
What is the phase shift if $C = -\frac{\text{π}}{3}$?
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Phase shift is $\frac{\text{π}}{3}$. Phase shift = $-C$ where $C = -\frac{\pi}{3}$.
Phase shift is $\frac{\text{π}}{3}$. Phase shift = $-C$ where $C = -\frac{\pi}{3}$.
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Find the amplitude of $y = -7 , \text{cos}(x)$.
Find the amplitude of $y = -7 , \text{cos}(x)$.
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Amplitude is 7. Amplitude is $|A| = |-7| = 7$ (absolute value of coefficient).
Amplitude is 7. Amplitude is $|A| = |-7| = 7$ (absolute value of coefficient).
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Find the period of $y = 5 , \text{cos}(4x)$.
Find the period of $y = 5 , \text{cos}(4x)$.
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Period is $\frac{\text{π}}{2}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{4} = \frac{\pi}{2}$.
Period is $\frac{\text{π}}{2}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{4} = \frac{\pi}{2}$.
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Identify the vertical shift in $y = \text{sin}(x) - 2$.
Identify the vertical shift in $y = \text{sin}(x) - 2$.
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Vertical shift is -2. Constant term $D = -2$ shifts the graph down 2 units.
Vertical shift is -2. Constant term $D = -2$ shifts the graph down 2 units.
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Determine the frequency of $y = 2 , \text{sin}(5x)$.
Determine the frequency of $y = 2 , \text{sin}(5x)$.
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Frequency is $\frac{5}{2\text{π}}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.
Frequency is $\frac{5}{2\text{π}}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.
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Calculate the period of $y = \sin\left(\frac{x}{2}\right)$.
Calculate the period of $y = \sin\left(\frac{x}{2}\right)$.
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Period is $4\pi$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi$.
Period is $4\pi$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi$.
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What does $A$ represent in the sinusoidal function $y = A , \text{sin}(B(x - C)) + D$?
What does $A$ represent in the sinusoidal function $y = A , \text{sin}(B(x - C)) + D$?
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$A$ represents the amplitude. The coefficient $A$ determines the vertical stretch and amplitude.
$A$ represents the amplitude. The coefficient $A$ determines the vertical stretch and amplitude.
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Calculate the period of $y = 2 , \text{sin}(x - \frac{\text{π}}{6})$.
Calculate the period of $y = 2 , \text{sin}(x - \frac{\text{π}}{6})$.
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Period is $2\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi$ since $B = 1$.
Period is $2\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1} = 2\pi$ since $B = 1$.
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What transformation occurs if $A = 1$ in $y = A , \text{sin}(B(x - C)) + D$?
What transformation occurs if $A = 1$ in $y = A , \text{sin}(B(x - C)) + D$?
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No change in amplitude. When $A = 1$, amplitude remains at standard value 1.
No change in amplitude. When $A = 1$, amplitude remains at standard value 1.
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Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.
Identify the phase shift of $y = \text{sin}(x - \frac{\text{π}}{2})$.
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Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).
Phase shift is $\frac{\text{π}}{2}$. Phase shift = $C = \frac{\pi}{2}$ (rightward shift).
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Determine the amplitude of $y = 6 , \text{sin}(x)$.
Determine the amplitude of $y = 6 , \text{sin}(x)$.
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Amplitude is 6. Amplitude is the coefficient $A = 6$.
Amplitude is 6. Amplitude is the coefficient $A = 6$.
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What is the effect of $C = \text{π}$ in $y = A , \text{cos}(B(x - C)) + D$?
What is the effect of $C = \text{π}$ in $y = A , \text{cos}(B(x - C)) + D$?
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Shift right $\text{π}$ units. Positive $C$ creates rightward horizontal shift.
Shift right $\text{π}$ units. Positive $C$ creates rightward horizontal shift.
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What does a phase shift of $-\text{π}$ indicate in $y = A , \text{cos}(B(x - C)) + D$?
What does a phase shift of $-\text{π}$ indicate in $y = A , \text{cos}(B(x - C)) + D$?
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Shift left $\text{π}$ units. Negative phase shift means leftward horizontal translation.
Shift left $\text{π}$ units. Negative phase shift means leftward horizontal translation.
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Determine the period of $y = 4 , \sin(3x) - 1$.
Determine the period of $y = 4 , \sin(3x) - 1$.
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Period is $\frac{2\pi}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ from coefficient of $x$.
Period is $\frac{2\pi}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ from coefficient of $x$.
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Calculate the period of $y = \text{sin}(\frac{x}{2})$.
Calculate the period of $y = \text{sin}(\frac{x}{2})$.
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Period is $4\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi$.
Period is $4\text{π}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{1/2} = 4\pi$.
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What transformation occurs if $D > 0$ in $y = A , \text{cos}(B(x - C)) + D$?
What transformation occurs if $D > 0$ in $y = A , \text{cos}(B(x - C)) + D$?
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Upward vertical shift. Positive $D$ moves the entire graph upward.
Upward vertical shift. Positive $D$ moves the entire graph upward.
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Determine the amplitude of $y = -2 , \text{sin}(x) + 3$.
Determine the amplitude of $y = -2 , \text{sin}(x) + 3$.
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Amplitude is 2. Amplitude is $|A| = |-2| = 2$ (absolute value).
Amplitude is 2. Amplitude is $|A| = |-2| = 2$ (absolute value).
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Determine the frequency of $y = 2 \sin(5x)$.
Determine the frequency of $y = 2 \sin(5x)$.
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Frequency is $\frac{5}{2\pi}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.
Frequency is $\frac{5}{2\pi}$. Frequency = $\frac{B}{2\pi} = \frac{5}{2\pi}$.
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Find the period of $y = 5 , \text{cos}(4x)$.
Find the period of $y = 5 , \text{cos}(4x)$.
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Period is $\frac{\pi}{2}$. Period = $\frac{2\pi}{B} = $\frac{2\pi}{4} = $\frac{\pi}{2}$
Period is $\frac{\pi}{2}$. Period = $\frac{2\pi}{B} = $\frac{2\pi}{4} = $\frac{\pi}{2}$
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Identify the period of $y = \text{cos}(3x) - 4$.
Identify the period of $y = \text{cos}(3x) - 4$.
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Period is $\frac{2\text{π}}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ regardless of other parameters.
Period is $\frac{2\text{π}}{3}$. Period = $\frac{2\pi}{B} = \frac{2\pi}{3}$ regardless of other parameters.
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Identify the maximum value of $y = 3 , \text{sin}(x) + 2$.
Identify the maximum value of $y = 3 , \text{sin}(x) + 2$.
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Maximum value is 5. Maximum = amplitude + vertical shift = $3 + 2 = 5$.
Maximum value is 5. Maximum = amplitude + vertical shift = $3 + 2 = 5$.
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What does $A$ represent in the sinusoidal function $y = A , \text{sin}(B(x - C)) + D$?
What does $A$ represent in the sinusoidal function $y = A , \text{sin}(B(x - C)) + D$?
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$A$ represents the amplitude. The coefficient $A$ determines the vertical stretch and amplitude.
$A$ represents the amplitude. The coefficient $A$ determines the vertical stretch and amplitude.
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Identify the amplitude in $y = 3 , \text{sin}(2x - \frac{\text{π}}{3}) + 4$.
Identify the amplitude in $y = 3 , \text{sin}(2x - \frac{\text{π}}{3}) + 4$.
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Amplitude is 3. Amplitude is the absolute value of coefficient $A = 3$.
Amplitude is 3. Amplitude is the absolute value of coefficient $A = 3$.
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What is the general form of a sinusoidal function?
What is the general form of a sinusoidal function?
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$y = A , \text{sin}(B(x - C)) + D$ or $y = A , \text{cos}(B(x - C)) + D$. Standard sinusoidal form with amplitude $A$, period factor $B$, phase shift $C$, and vertical shift $D$.
$y = A , \text{sin}(B(x - C)) + D$ or $y = A , \text{cos}(B(x - C)) + D$. Standard sinusoidal form with amplitude $A$, period factor $B$, phase shift $C$, and vertical shift $D$.
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Find the phase shift of $y = 4 , \text{cos}(2(x - \frac{\text{π}}{6}))$.
Find the phase shift of $y = 4 , \text{cos}(2(x - \frac{\text{π}}{6}))$.
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Phase shift is $\frac{\text{π}}{6}$. Phase shift equals $C = \frac{\pi}{6}$ (right shift).
Phase shift is $\frac{\text{π}}{6}$. Phase shift equals $C = \frac{\pi}{6}$ (right shift).
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What does $B$ represent in $y = A , \text{cos}(B(x - C)) + D$?
What does $B$ represent in $y = A , \text{cos}(B(x - C)) + D$?
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$B$ affects the period. Coefficient $B$ determines how many cycles fit in $2\pi$ units.
$B$ affects the period. Coefficient $B$ determines how many cycles fit in $2\pi$ units.
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What is the vertical shift in $y = A , \text{sin}(B(x - C)) + D$?
What is the vertical shift in $y = A , \text{sin}(B(x - C)) + D$?
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Vertical shift is $D$. The constant $D$ moves the midline up or down.
Vertical shift is $D$. The constant $D$ moves the midline up or down.
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Identify the vertical shift in $y = \text{sin}(x) - 2$.
Identify the vertical shift in $y = \text{sin}(x) - 2$.
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Vertical shift is -2. Constant term $D = -2$ shifts the graph down 2 units.
Vertical shift is -2. Constant term $D = -2$ shifts the graph down 2 units.
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