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

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 \(k < 0\).

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

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**. \(y = f(-x)\)

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