Vertical translations refer to movements of a graph of a function vertically along the y-axis by changing the y values. So, if y = f(x), then y = (x) + h results in a vertical shift. If h > 0, then the graph shifts h units up; while If h < 0, then the graph shifts h units down.

# Transformations of functions: Vertical translations

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Intros

__An Experiment to Study "Vertical Translations"__

Sketch and compare: $\left( y \right) = {x^2}$ VS. $\left( {y - 3} \right) = {x^2}$ VS. $\left( {y + 2} \right) = {x^2}$- Sketch all three quadratic functions on the same set of coordinate axes.
- Compared to the graph of $y = {x^2}$:

• the graph of $\left( {y - 3} \right) = {x^2}$ is translated "vertically" ________ units _____________.

• the graph of $\left( {y + 2} \right) = {x^2}$ is translated "vertically" ________ units _____________.

Examples

**Vertical Translations**

Given the graph of $y=f(x)$ as shown, sketch:- $y = f\left( x \right) - 8$
- $y = f\left( x \right) + 3$
- In conclusion:

• $\left( y \right) \to \left( {y + 8} \right)$: shift ________ units ______________ $\Rightarrow$ all $y$ coordinates _____________________________.

• $\left( y \right) \to \left( {y - 3} \right)$: shift ________ units ______________ $\Rightarrow$ all $y$ coordinates _____________________________.

- $y = f\left( x \right) - 8$