# How do I find the stretches of a transformed function?

## Do I look for the x and y intercepts and the invariant points?

Feb 9, 2018

Look at the variables of $a$ and $b$ to figure out what factor to stretch by.

#### Explanation:

Refer to: $y = a f \left(b \left(x - h\right)\right) + k$

A vertical stretch is the stretching of a function on the x-axis.
If $| a | > 1$, then the graph is stretched vertically by a factor of $a$ units.
If the values of $a$ are negative, this will result in the graph reflecting vertically across the x-axis.

A horizontal stretch is the stretching of a function on the y-axis.
If $| b | < 1$, then the graph is stretched horizontally by a factor of $b$ units.
If the values of $b$ are negative, this will result in the graph reflecting horizontally across the y-axis.

For example:
y=2f((1/2)x-h))+k
$a = 2$

$b = \frac{1}{2}$

To vertically stretch we use this formula:
${y}^{1} = a y$
${y}^{1} = 2 y$ So, the vertical stretch would be by a factor of 2.

To horizontally stretch we use this formula:
${x}^{1} = \frac{x}{b}$

${x}^{1} = \frac{x}{\frac{1}{2}}$

${x}^{1} = 2 x$ So, the horizontal stretch would be by a factor of 2 as well.

Extras:
If $| a | < 1$, this results in a vertical compression.
If $| b | > 1$, this results in a horizontal compression.