Modeling Periodic Phenomena with Trigonometric Functions - Pre-Calculus
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What is the vertical shift of $y=A\sin(Bx)+D$?
What is the vertical shift of $y=A\sin(Bx)+D$?
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$D$ units (midline $y=D$). The $+D$ term shifts the entire graph up by $D$ units.
$D$ units (midline $y=D$). The $+D$ term shifts the entire graph up by $D$ units.
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Identify the max and min values of $y=A\sin(Bx)+D$ using $A$ and $D$.
Identify the max and min values of $y=A\sin(Bx)+D$ using $A$ and $D$.
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max $=D+|A|$, min $=D-|A|$. Max occurs when sine equals 1, min when sine equals -1.
max $=D+|A|$, min $=D-|A|$. Max occurs when sine equals 1, min when sine equals -1.
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What is the horizontal phase shift of $y=A\cos(B(x-C))+D$?
What is the horizontal phase shift of $y=A\cos(B(x-C))+D$?
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$C$ units to the right. The $(x-C)$ form shifts the graph $C$ units right.
$C$ units to the right. The $(x-C)$ form shifts the graph $C$ units right.
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What is the frequency (cycles per $2\pi$ units) of $y=A\sin(Bx)+D$ in terms of $B$?
What is the frequency (cycles per $2\pi$ units) of $y=A\sin(Bx)+D$ in terms of $B$?
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$|B|$. Frequency counts how many complete cycles occur in $2\pi$ units.
$|B|$. Frequency counts how many complete cycles occur in $2\pi$ units.
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What is the period of $y=A\sin(B(x-C))+D$ in terms of $B$?
What is the period of $y=A\sin(B(x-C))+D$ in terms of $B$?
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$\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of the coefficient of $x$.
$\frac{2\pi}{|B|}$. Period formula divides $2\pi$ by the absolute value of the coefficient of $x$.
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What is the midline of $y=A\cos(B(x-C))+D$ in terms of $D$?
What is the midline of $y=A\cos(B(x-C))+D$ in terms of $D$?
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$y=D$. The midline is the horizontal line around which the function oscillates.
$y=D$. The midline is the horizontal line around which the function oscillates.
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What is the amplitude of $y=A\sin(B(x-C))+D$ in terms of $A$?
What is the amplitude of $y=A\sin(B(x-C))+D$ in terms of $A$?
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$|A|$. Amplitude is the absolute value of the coefficient of the trig function.
$|A|$. Amplitude is the absolute value of the coefficient of the trig function.
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What is the standard sinusoid form (using sine) that shows amplitude, midline, and period?
What is the standard sinusoid form (using sine) that shows amplitude, midline, and period?
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$y=A\sin(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.
$y=A\sin(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.
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What is the standard sinusoid form (using cosine) that shows amplitude, midline, and period?
What is the standard sinusoid form (using cosine) that shows amplitude, midline, and period?
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$y=A\cos(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.
$y=A\cos(B(x-C))+D$. Standard form where $A$ is amplitude, $B$ affects period, $C$ is phase shift, $D$ is midline.
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Choose $A$ and $D$ so the sinusoid has max $9$ and min $-3$ (no other changes).
Choose $A$ and $D$ so the sinusoid has max $9$ and min $-3$ (no other changes).
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$A=6$, $D=3$. Amp $= \frac{9-(-3)}{2} = 6$, midline $= \frac{9+(-3)}{2} = 3$.
$A=6$, $D=3$. Amp $= \frac{9-(-3)}{2} = 6$, midline $= \frac{9+(-3)}{2} = 3$.
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Identify a model for max $10$, min $2$, and period $12$ using cosine with no phase shift.
Identify a model for max $10$, min $2$, and period $12$ using cosine with no phase shift.
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$y=4\cos\left(\frac{\pi}{6}x\right)+6$. Amp $= \frac{10-2}{2} = 4$, midline $= \frac{10+2}{2} = 6$, $B = \frac{2\pi}{12} = \frac{\pi}{6}$.
$y=4\cos\left(\frac{\pi}{6}x\right)+6$. Amp $= \frac{10-2}{2} = 4$, midline $= \frac{10+2}{2} = 6$, $B = \frac{2\pi}{12} = \frac{\pi}{6}$.
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Find the amplitude, midline, and period of $y=-4\cos\left(\frac{\pi}{3}x\right)+7$.
Find the amplitude, midline, and period of $y=-4\cos\left(\frac{\pi}{3}x\right)+7$.
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amp $=4$, midline $y=7$, period $=6$. $|A| = 4$, $D = 7$, period $= \frac{2\pi}{\frac{\pi}{3}} = 6$.
amp $=4$, midline $y=7$, period $=6$. $|A| = 4$, $D = 7$, period $= \frac{2\pi}{\frac{\pi}{3}} = 6$.
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Find a cosine model with amplitude $5$, midline $y=-1$, and period $\pi$.
Find a cosine model with amplitude $5$, midline $y=-1$, and period $\pi$.
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$y=5\cos(2x)-1$. Use $B = \frac{2\pi}{\pi} = 2$ for period $\pi$.
$y=5\cos(2x)-1$. Use $B = \frac{2\pi}{\pi} = 2$ for period $\pi$.
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Find a sine model with amplitude $3$, midline $y=2$, and period $4\pi$.
Find a sine model with amplitude $3$, midline $y=2$, and period $4\pi$.
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$y=3\sin\left(\frac{1}{2}x\right)+2$. Use $B = \frac{2\pi}{4\pi} = \frac{1}{2}$ for period $4\pi$.
$y=3\sin\left(\frac{1}{2}x\right)+2$. Use $B = \frac{2\pi}{4\pi} = \frac{1}{2}$ for period $4\pi$.
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Which function starts at a maximum when $A>0$ and $C=0$: sine or cosine?
Which function starts at a maximum when $A>0$ and $C=0$: sine or cosine?
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cosine. $\cos(0) = 1$, so cosine starts at its maximum when $A > 0$.
cosine. $\cos(0) = 1$, so cosine starts at its maximum when $A > 0$.
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Which function starts at the midline and increases when $A>0$ and $C=0$: sine or cosine?
Which function starts at the midline and increases when $A>0$ and $C=0$: sine or cosine?
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sine. $\sin(0) = 0$ starts at midline; $\cos(0) = 1$ starts at maximum.
sine. $\sin(0) = 0$ starts at midline; $\cos(0) = 1$ starts at maximum.
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What is the period $T$ if a sinusoid completes $f$ cycles per unit on the $x$-axis?
What is the period $T$ if a sinusoid completes $f$ cycles per unit on the $x$-axis?
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$T=\frac{1}{f}$. Period is the reciprocal of frequency.
$T=\frac{1}{f}$. Period is the reciprocal of frequency.
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What is $B$ if a sinusoid has period $T$ (in radians) in $y=A\sin(Bx)+D$?
What is $B$ if a sinusoid has period $T$ (in radians) in $y=A\sin(Bx)+D$?
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$B=\frac{2\pi}{T}$. To get period $T$, set $B = \frac{2\pi}{T}$ in the standard form.
$B=\frac{2\pi}{T}$. To get period $T$, set $B = \frac{2\pi}{T}$ in the standard form.
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What is the midline value $D$ of a sinusoid with maximum $M$ and minimum $m$?
What is the midline value $D$ of a sinusoid with maximum $M$ and minimum $m$?
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$\frac{M+m}{2}$. Midline is the average of the maximum and minimum values.
$\frac{M+m}{2}$. Midline is the average of the maximum and minimum values.
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What is the amplitude of a sinusoid with maximum $M$ and minimum $m$?
What is the amplitude of a sinusoid with maximum $M$ and minimum $m$?
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$\frac{M-m}{2}$. Amplitude is half the distance between maximum and minimum values.
$\frac{M-m}{2}$. Amplitude is half the distance between maximum and minimum values.
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Write a sine model with amplitude $3$, period $10$, midline $-2$, and no phase shift.
Write a sine model with amplitude $3$, period $10$, midline $-2$, and no phase shift.
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$y=3\sin\left(\frac{\pi}{5}x\right)-2$. $B=\frac{2\pi}{10}=\frac{\pi}{5}$ for period 10; no phase shift means no $(x-C)$ term.
$y=3\sin\left(\frac{\pi}{5}x\right)-2$. $B=\frac{2\pi}{10}=\frac{\pi}{5}$ for period 10; no phase shift means no $(x-C)$ term.
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State the standard sinusoid model (using sine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.
State the standard sinusoid model (using sine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.
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$y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$. Standard form where $B=\frac{2\pi}{P}$ relates period to angular frequency.
$y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$. Standard form where $B=\frac{2\pi}{P}$ relates period to angular frequency.
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State the standard sinusoid model (using cosine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.
State the standard sinusoid model (using cosine) with amplitude $A$, period $P$, midline $D$, and phase shift $C$.
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$y=A\cos\left(\frac{2\pi}{P}(x-C)\right)+D$. Same as sine form but uses cosine; $B=\frac{2\pi}{P}$ converts period to angular frequency.
$y=A\cos\left(\frac{2\pi}{P}(x-C)\right)+D$. Same as sine form but uses cosine; $B=\frac{2\pi}{P}$ converts period to angular frequency.
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What is the midline of $y=A\sin(B(x-C))+D$ in terms of $D$?
What is the midline of $y=A\sin(B(x-C))+D$ in terms of $D$?
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$y=D$. The midline is the vertical shift, the constant $D$ added to the function.
$y=D$. The midline is the vertical shift, the constant $D$ added to the function.
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What is the period of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$ in terms of $P$?
What is the period of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$ in terms of $P$?
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$P$. When $B=\frac{2\pi}{P}$, the period is simply $P$.
$P$. When $B=\frac{2\pi}{P}$, the period is simply $P$.
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What is the frequency (cycles per unit) of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$?
What is the frequency (cycles per unit) of $y=A\sin\left(\frac{2\pi}{P}(x-C)\right)+D$?
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$\frac{1}{P}$. Frequency is reciprocal of period: $f=\frac{1}{P}$ cycles per unit.
$\frac{1}{P}$. Frequency is reciprocal of period: $f=\frac{1}{P}$ cycles per unit.
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Identify the angular frequency $\omega$ in $y=A\sin(\omega(x-C))+D$.
Identify the angular frequency $\omega$ in $y=A\sin(\omega(x-C))+D$.
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$\omega=|B|$. Angular frequency is the coefficient of $(x-C)$, which is $|B|$.
$\omega=|B|$. Angular frequency is the coefficient of $(x-C)$, which is $|B|$.
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Given max $M$ and min $m$ of a sinusoid, what is the amplitude?
Given max $M$ and min $m$ of a sinusoid, what is the amplitude?
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$\frac{M-m}{2}$. Amplitude is half the difference between max and min values.
$\frac{M-m}{2}$. Amplitude is half the difference between max and min values.
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Given max $M$ and min $m$ of a sinusoid, what is the midline value $D$?
Given max $M$ and min $m$ of a sinusoid, what is the midline value $D$?
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$\frac{M+m}{2}$. Midline is the average of max and min values.
$\frac{M+m}{2}$. Midline is the average of max and min values.
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Find the amplitude and midline of a sinusoid with maximum $10$ and minimum $2$.
Find the amplitude and midline of a sinusoid with maximum $10$ and minimum $2$.
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$A=4,\ D=6$. $A=\frac{10-2}{2}=4$, $D=\frac{10+2}{2}=6$ using the formulas.
$A=4,\ D=6$. $A=\frac{10-2}{2}=4$, $D=\frac{10+2}{2}=6$ using the formulas.
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