Graph transformations

We can combine stretches and translations with functions. We can also use our knowledge of transformations to sketch transformed graphs.

Translations

A translation shifts the graph horizontally, vertically, or both.

• Horizontal Translation: The function $y=f(x+h)$ shifts the graph of $y=f(x)$ h units to the right if $h<0$, or h units to the left if $h>0$.
• Vertical Translation: The function $y=f(x)+k$ shifts the graph k units up if $k>0$, or k units down if $k<0$.

Stretches

A stretch either makes the graph narrower or wider, or taller or shorter, depending on whether it's horizontal or vertical.

• Vertical Stretch: The function $y=af(x)$ stretches the graph vertically by a factor of $|a|$. If $a>1$, the graph becomes steeper, and if $0<a<1$, it becomes flatter. $y=af(x)$
• Horizontal Stretch: The function $y=f\left(\frac{x}{b}\right)$ stretches the graph horizontally by a factor of $|b|$. If $b>1$, the graph becomes narrower, and if $0<b<1$, it becomes wider.$y=f\left(\frac{x}{b}\right)$

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. $y=f(-x)$