Graph stretches

Graph stretches involve expanding or compressing a graph either vertically or horizontally, changing its shape. Unlike translations, stretches alter the steepness or width of the graph without shifting its position.

Vertical Stretches

A vertical stretch changes the height of the graph by multiplying the function by a constant $a$. The function:

$$y=af(x)$$

stretches the graph of $y=f(x)$ vertically by a factor of $|a|$.

If $a>1$, the graph stretches, making it steeper.

If $0<a<1$1, the graph compresses, making it flatter.

If $a<0$, the graph is also reflected in the x-axis.

Horizontal Stretches

A horizontal stretch changes the width of the graph by dividing the input xxx by a constant $b$. The function:

$$y=f\left(\frac{x}{b}\right)$$

stretches the graph of $y=f(x)$ horizontally by a factor of $|b|$.

If $b>1$, the graph is compressed horizontally, making it narrower.

If $0<b<1$, the graph stretches horizontally, making it wider.

Reflections

A reflection flips the graph across an axis.

• Reflection in the x-axis: The function $y=-f(x)$ reflects the graph of $y=f(x)$ across the x-axis. $y=-f(x)$
• Reflection in the y-axis: The function $y=f(-x)$ reflects the graph across the y-axis. $y=f(-x)$

Combined Stretches

A graph can be stretched both vertically and horizontally by applying both transformations:

$$y=af\left(\frac{x}{b}\right)$$

This stretches the graph vertically by a factor of $|a|$ and horizontally by a factor of $|b|$.