# 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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