Transformations of Functions and Graphs - Algebra 2
Card 1 of 30
What is the effect of replacing $f(x)$ with $-f(x)$?
What is the effect of replacing $f(x)$ with $-f(x)$?
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Reflect the graph across the $x$-axis. Negative coefficient flips all $y$-values to opposite signs.
Reflect the graph across the $x$-axis. Negative coefficient flips all $y$-values to opposite signs.
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Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.
Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.
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Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.
Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.
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What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?
What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?
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Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.
Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.
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What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?
What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?
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Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.
Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.
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Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.
Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.
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$k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.
$k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.
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What is the effect of $g(x)=1\cdot f(x)$ on the graph of $f$?
What is the effect of $g(x)=1\cdot f(x)$ on the graph of $f$?
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No change. Multiplying by one produces an identical transformation.
No change. Multiplying by one produces an identical transformation.
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What is the effect of $g(x)=f(1\cdot x)$ on the graph of $f$?
What is the effect of $g(x)=f(1\cdot x)$ on the graph of $f$?
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No change. Multiplying input by one produces an identical transformation.
No change. Multiplying input by one produces an identical transformation.
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Identify the effect of $g(x)=2f(x)-3$ on the graph relative to $f(x)$.
Identify the effect of $g(x)=2f(x)-3$ on the graph relative to $f(x)$.
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Vertical stretch by $2$, then shift down $3$. First stretch vertically, then translate downward.
Vertical stretch by $2$, then shift down $3$. First stretch vertically, then translate downward.
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Identify the effect of $g(x)=f(x-4)+1$ on the graph relative to $f(x)$.
Identify the effect of $g(x)=f(x-4)+1$ on the graph relative to $f(x)$.
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Shift right $4$, then shift up $1$. First shift horizontally, then translate vertically.
Shift right $4$, then shift up $1$. First shift horizontally, then translate vertically.
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Identify the effect of $g(x)=f(-x)+5$ on the graph relative to $f(x)$.
Identify the effect of $g(x)=f(-x)+5$ on the graph relative to $f(x)$.
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Reflect across $y$-axis, then shift up $5$. First reflect horizontally, then translate vertically.
Reflect across $y$-axis, then shift up $5$. First reflect horizontally, then translate vertically.
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Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.
Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.
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Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.
Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.
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What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?
What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?
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Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.
Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.
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Identify the $y$-intercept of $g(x)=f(x+k)$ in terms of $f$.
Identify the $y$-intercept of $g(x)=f(x+k)$ in terms of $f$.
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$g(0)=f(k)$. The $y$-intercept becomes $f(k)$.
$g(0)=f(k)$. The $y$-intercept becomes $f(k)$.
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What is the point-mapping rule for $g(x)=k f(x)$ from a point $(x,y)$ on $f$?
What is the point-mapping rule for $g(x)=k f(x)$ from a point $(x,y)$ on $f$?
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$(x,y)\to(x,ky)$. Vertical scaling multiplies each $y$-coordinate by $k$.
$(x,y)\to(x,ky)$. Vertical scaling multiplies each $y$-coordinate by $k$.
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What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?
What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?
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Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.
Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.
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Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.
Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.
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$k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.
$k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.
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Find $k$ if $g(x)=k f(x)$ and a point $(-3,8)$ on $f$ becomes $(-3,2)$ on $g$.
Find $k$ if $g(x)=k f(x)$ and a point $(-3,8)$ on $f$ becomes $(-3,2)$ on $g$.
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$k=\frac{1}{4}$. Scaling factor is $\frac{2}{8}=\frac{1}{4}$.
$k=\frac{1}{4}$. Scaling factor is $\frac{2}{8}=\frac{1}{4}$.
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What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k>0$?
What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k>0$?
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Shift the graph up $k$ units. Adding $k$ to the function output translates all points vertically upward.
Shift the graph up $k$ units. Adding $k$ to the function output translates all points vertically upward.
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What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k<0$?
What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k<0$?
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Shift the graph down $|k|$ units. Adding negative $k$ translates all points downward by the magnitude.
Shift the graph down $|k|$ units. Adding negative $k$ translates all points downward by the magnitude.
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What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k>0$?
What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k>0$?
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Shift the graph right $k$ units. Subtracting $k$ from input shifts the graph rightward by $k$ units.
Shift the graph right $k$ units. Subtracting $k$ from input shifts the graph rightward by $k$ units.
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What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k<0$?
What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k<0$?
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Shift the graph left $|k|$ units. Subtracting negative $k$ from input shifts the graph leftward.
Shift the graph left $|k|$ units. Subtracting negative $k$ from input shifts the graph leftward.
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What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k>0$?
What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k>0$?
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Shift the graph left $k$ units. Adding $k$ to input shifts the graph leftward by $k$ units.
Shift the graph left $k$ units. Adding $k$ to input shifts the graph leftward by $k$ units.
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What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k<0$?
What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k<0$?
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Shift the graph right $|k|$ units. Adding negative $k$ to input shifts the graph rightward.
Shift the graph right $|k|$ units. Adding negative $k$ to input shifts the graph rightward.
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What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k>1$?
What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k>1$?
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Vertical stretch by factor $k$. Multiplying by $k>1$ stretches all $y$-values away from the $x$-axis.
Vertical stretch by factor $k$. Multiplying by $k>1$ stretches all $y$-values away from the $x$-axis.
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What is the graph effect of replacing $f(x)$ with $k f(x)$ for $0<k<1$?
What is the graph effect of replacing $f(x)$ with $k f(x)$ for $0<k<1$?
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Vertical compression by factor $k$. Multiplying by $0<k<1$ compresses all $y$-values toward the $x$-axis.
Vertical compression by factor $k$. Multiplying by $0<k<1$ compresses all $y$-values toward the $x$-axis.
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What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k<0$?
What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k<0$?
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Reflect across $x$-axis and scale by $|k|$. Negative coefficient flips across $x$-axis and scales by absolute value.
Reflect across $x$-axis and scale by $|k|$. Negative coefficient flips across $x$-axis and scales by absolute value.
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What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k>1$?
What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k>1$?
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Horizontal compression by factor $k$. Multiplying input by $k>1$ compresses the graph horizontally.
Horizontal compression by factor $k$. Multiplying input by $k>1$ compresses the graph horizontally.
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What is the graph effect of replacing $f(x)$ with $f(kx)$ for $0<k<1$?
What is the graph effect of replacing $f(x)$ with $f(kx)$ for $0<k<1$?
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Horizontal stretch by factor $ rac{1}{k}$. Multiplying input by $0<k<1$ stretches the graph horizontally.
Horizontal stretch by factor $ rac{1}{k}$. Multiplying input by $0<k<1$ stretches the graph horizontally.
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What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k<0$?
What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k<0$?
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Reflect across $y$-axis and scale horizontally by $|k|$. Negative input coefficient reflects across $y$-axis and scales horizontally.
Reflect across $y$-axis and scale horizontally by $|k|$. Negative input coefficient reflects across $y$-axis and scales horizontally.
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What is the effect of replacing $f(x)$ with $f(-x)$?
What is the effect of replacing $f(x)$ with $f(-x)$?
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Reflect the graph across the $y$-axis. Negative input creates a mirror image across the $y$-axis.
Reflect the graph across the $y$-axis. Negative input creates a mirror image across the $y$-axis.
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