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 \(y = a f(x)\) stretches the graph vertically by a factor of \(|a|\). If \(a > 1\), the graph becomes steeper, and if \(0 < a < 1\), it becomes flatter. \(y = a f(x)\)
• 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. \(y = f(-x)\)