Stretches of Graphs $y=pf(x)$ and $y=f(qx)$

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 $y=3f(2x)$.

Example 2

Find the point which is moved to the point $(-7,3)$ under the transformation $y=3f(2x)$.