Modeling Periodic Phenomena with Trigonometric Functions - Geometry
Card 1 of 30
What is the period of $y = 5\sin(\frac{2\pi}{7}x)+1$?
What is the period of $y = 5\sin(\frac{2\pi}{7}x)+1$?
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$7$. Period = $\frac{2\pi}{\frac{2\pi}{7}} = 7$.
$7$. Period = $\frac{2\pi}{\frac{2\pi}{7}} = 7$.
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What is the amplitude of $y = A\sin(B(x-C))+D$ in terms of parameters?
What is the amplitude of $y = A\sin(B(x-C))+D$ in terms of parameters?
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$|A|$. Amplitude is the absolute value of coefficient $A$.
$|A|$. Amplitude is the absolute value of coefficient $A$.
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What is the midline of $y = A\sin(B(x-C))+D$ written as an equation?
What is the midline of $y = A\sin(B(x-C))+D$ written as an equation?
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$y = D$. The midline is the horizontal line $y = D$.
$y = D$. The midline is the horizontal line $y = D$.
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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: $\frac{2\pi}{|B|}$ for sine functions.
$\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for sine functions.
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What is the period of $y = A\cos(B(x-C))+D$ in terms of $B$?
What is the period of $y = A\cos(B(x-C))+D$ in terms of $B$?
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$\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for cosine functions.
$\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for cosine functions.
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Choose $\sin$ or $\cos$: a model that crosses the midline at $x=0$ increasing with $A>0$.
Choose $\sin$ or $\cos$: a model that crosses the midline at $x=0$ increasing with $A>0$.
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$\sin$. Sine crosses midline increasing when $A > 0$.
$\sin$. Sine crosses midline increasing when $A > 0$.
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What is the frequency (cycles per $2\pi$) of $y = A\sin(Bx)$ in terms of $B$?
What is the frequency (cycles per $2\pi$) of $y = A\sin(Bx)$ in terms of $B$?
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$|B|$. Frequency is $|B|$ cycles per $2\pi$ units.
$|B|$. Frequency is $|B|$ cycles per $2\pi$ units.
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What is the phase shift of $y = A\sin(B(x-C))+D$ in terms of $C$?
What is the phase shift of $y = A\sin(B(x-C))+D$ in terms of $C$?
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$C$ units right. Phase shift moves graph $C$ units horizontally.
$C$ units right. Phase shift moves graph $C$ units horizontally.
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What is the range of $y = A\sin(B(x-C))+D$ written with $A$ and $D$?
What is the range of $y = A\sin(B(x-C))+D$ written with $A$ and $D$?
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$[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.
$[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.
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What is the range of $y = A\cos(B(x-C))+D$ written with $A$ and $D$?
What is the range of $y = A\cos(B(x-C))+D$ written with $A$ and $D$?
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$[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.
$[D-|A|,\ D+|A|]$. Range extends $|A|$ units above and below midline $D$.
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What is the amplitude of $y = a\sin(x)+d$ if the max is $M$ and min is $m$?
What is the amplitude of $y = a\sin(x)+d$ if the max is $M$ and min is $m$?
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$\frac{M-m}{2}$. Amplitude is half the difference between max and min.
$\frac{M-m}{2}$. Amplitude is half the difference between max and min.
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What is the midline value $D$ if the maximum is $M$ and minimum is $m$?
What is the midline value $D$ if the maximum is $M$ and minimum is $m$?
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$\frac{M+m}{2}$. Midline value is the average of max and min.
$\frac{M+m}{2}$. Midline value is the average of max and min.
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What is the vertical shift $D$ in $y = A\sin(Bx)+D$ if the midline is $y=k$?
What is the vertical shift $D$ in $y = A\sin(Bx)+D$ if the midline is $y=k$?
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$D = k$. Vertical shift $D$ equals the midline value $k$.
$D = k$. Vertical shift $D$ equals the midline value $k$.
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Which function starts at a maximum when $A>0$ and $C=0$: $\sin$ or $\cos$?
Which function starts at a maximum when $A>0$ and $C=0$: $\sin$ or $\cos$?
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$\cos$. Cosine starts at maximum when no phase shift.
$\cos$. Cosine starts at maximum when no phase shift.
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Choose $\sin$ or $\cos$: a model that is at its maximum at $x=0$ with $A>0$ and no shift.
Choose $\sin$ or $\cos$: a model that is at its maximum at $x=0$ with $A>0$ and no shift.
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$\cos$. Cosine starts at maximum when $A > 0$.
$\cos$. Cosine starts at maximum when $A > 0$.
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What midline value $D$ is needed if a sinusoid has maximum $9$ and minimum $1$?
What midline value $D$ is needed if a sinusoid has maximum $9$ and minimum $1$?
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$D = 5$. Midline value = $\frac{9+1}{2} = 5$.
$D = 5$. Midline value = $\frac{9+1}{2} = 5$.
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What amplitude $A$ is needed if a sinusoid has maximum $9$ and minimum $1$?
What amplitude $A$ is needed if a sinusoid has maximum $9$ and minimum $1$?
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$A = 4$. Amplitude = $\frac{9-1}{2} = 4$.
$A = 4$. Amplitude = $\frac{9-1}{2} = 4$.
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What is the value of $B$ if $y = A\cos(Bx)+D$ has period $8\pi$?
What is the value of $B$ if $y = A\cos(Bx)+D$ has period $8\pi$?
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$B = \frac{1}{4}$. From period $8\pi = \frac{2\pi}{B}$, so $B = \frac{1}{4}$.
$B = \frac{1}{4}$. From period $8\pi = \frac{2\pi}{B}$, so $B = \frac{1}{4}$.
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What is the value of $B$ if $y = A\sin(Bx)+D$ has period $\pi$?
What is the value of $B$ if $y = A\sin(Bx)+D$ has period $\pi$?
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$B = 2$. From period $\pi = \frac{2\pi}{B}$, so $B = 2$.
$B = 2$. From period $\pi = \frac{2\pi}{B}$, so $B = 2$.
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What is the period of $y = \sin(\frac{\pi}{3}x)$?
What is the period of $y = \sin(\frac{\pi}{3}x)$?
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$6$. Period = $\frac{2\pi}{\frac{\pi}{3}} = 6$.
$6$. Period = $\frac{2\pi}{\frac{\pi}{3}} = 6$.
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Identify the range of $y = -2\cos(x)-6$ using interval notation.
Identify the range of $y = -2\cos(x)-6$ using interval notation.
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$[-8,\ -4]$. Range: $[-6-2, -6+2] = [-8, -4]$.
$[-8,\ -4]$. Range: $[-6-2, -6+2] = [-8, -4]$.
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Identify the range of $y = 4\sin(x)+1$ using interval notation.
Identify the range of $y = 4\sin(x)+1$ using interval notation.
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$[-3,\ 5]$. Range: $[1-4, 1+4] = [-3, 5]$.
$[-3,\ 5]$. Range: $[1-4, 1+4] = [-3, 5]$.
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Identify the amplitude of $y = \frac{1}{2}\sin(x)-3$.
Identify the amplitude of $y = \frac{1}{2}\sin(x)-3$.
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$\frac{1}{2}$. Amplitude is the coefficient $\frac{1}{2}$.
$\frac{1}{2}$. Amplitude is the coefficient $\frac{1}{2}$.
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What is the midline equation of $y = 2\cos(3x)-4$?
What is the midline equation of $y = 2\cos(3x)-4$?
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$y = -4$. Midline is $y = D = -4$.
$y = -4$. Midline is $y = D = -4$.
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Which function starts on the midline going upward when $A>0$ and $C=0$: $\sin$ or $\cos$?
Which function starts on the midline going upward when $A>0$ and $C=0$: $\sin$ or $\cos$?
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$\sin$. Sine starts at midline going upward when no phase shift.
$\sin$. Sine starts at midline going upward when no phase shift.
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What is the period of $y = A\sin(\frac{\pi}{6}x)+D$?
What is the period of $y = A\sin(\frac{\pi}{6}x)+D$?
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$12$. Period = $\frac{2\pi}{|\frac{\pi}{6}|} = \frac{2\pi}{\frac{\pi}{6}} = 12$.
$12$. Period = $\frac{2\pi}{|\frac{\pi}{6}|} = \frac{2\pi}{\frac{\pi}{6}} = 12$.
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What is the period of $y = 3\cos(4x)-2$?
What is the period of $y = 3\cos(4x)-2$?
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$\frac{\pi}{2}$. Period = $\frac{2\pi}{4} = \frac{\pi}{2}$.
$\frac{\pi}{2}$. Period = $\frac{2\pi}{4} = \frac{\pi}{2}$.
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What is the period of $y = \sin(2\pi x)$?
What is the period of $y = \sin(2\pi x)$?
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$1$. Period = $\frac{2\pi}{2\pi} = 1$.
$1$. Period = $\frac{2\pi}{2\pi} = 1$.
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Identify the error: period of $y=\cos(5x)$ was stated as $5$. What is correct period?
Identify the error: period of $y=\cos(5x)$ was stated as $5$. What is correct period?
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Period is $\frac{2\pi}{5}$. Period is $\frac{2\pi}{B}$, not $B$ itself.
Period is $\frac{2\pi}{5}$. Period is $\frac{2\pi}{B}$, not $B$ itself.
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Identify the error: amplitude of $y = 2\sin(3x)+5$ was stated as $5$. What is correct?
Identify the error: amplitude of $y = 2\sin(3x)+5$ was stated as $5$. What is correct?
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Amplitude is $2$. Amplitude is coefficient of trig function, not vertical shift.
Amplitude is $2$. Amplitude is coefficient of trig function, not vertical shift.
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