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:

\[y = a f(x)\]

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:

\[y = a f\left(\frac{x}{b}\right)\]

This stretches the graph vertically by a factor of \(|a|\) and horizontally by a factor of \(|b|\).