Stretches of Graphs
Stretch Rule 1
For $y=pf(x)$, $p \gt 0$, the effect of $p$ is to vertically stretch the graph by a factor by $p$.
- If $p \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis.
- If $0 \lt p \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis.
Stretch Rule 2
For $y=f(kx)$, $k \gt 0$, the effect of $k$ is to horizontally compress the graph by a factor of $k$.
- If $k \gt 1$, it moves points of $y=f(x)$ further away from the $x$-axis.
- If $0 \lt k \lt 1$, it moves points of $y=f(x)$ closer to the $x$-axis.
Example 1
Given that the point $(-2,1)$ lies on $y=f(x)$, find the corresponding point on the image function $y=3f(2x)$.
$2x$ means that the graph is to horizontally compress by a factor of $2$.
Thus the $x$-value $-2$ transforms to $-1$.
$3f(2x)$ means that the graph is to vertically stretch by a factor of $3$.
Thus the $y$-value $1$ transforms to $3$.
Therefore the point $(-2,1)$ transforms $(-1,3)$.
Example 2
Find the point which is moved to the point $(-7,3)$ under the transformation $y=3f(2x)$.
$2x$ means that the graph is to horizontally compress by a factor of $2$.
Thus the $x$-value $-7$ was transformed from $-14$.
$3f(2x)$ means that the graph is to vertically stretch by a factor of $3$.
Thus the $y$-value $3$ was transformed from $1$.
Therefore the point $(-7,3)$ was transformed from $(-14,1)$.
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