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- Lesson: 212:09

Horizontal translations refer to movements of a graph of a function horizontally along the x-axis by changing the x values. So, if y = f(x), then y = (x –h) results in a horizontal shift. If h > 0, then the graph shifts h units to the right; while If h < 0, then the graph shifts h units to the right.

Basic concepts: Completing the square, Converting from general to vertex form by completing the square, Shortcut: Vertex formula,

Related concepts: Graphing transformations of trigonometric functions, Determining trigonometric functions given their graphs,

Compared to $y=f(x)$:

$y=f(x-8)$: shift 8 units to the right

$y=f(x+3)$: shift 3 units to the left

$y=f(x-8)$: shift 8 units to the right

$y=f(x+3)$: shift 3 units to the left

- 1.a)Sketch the following functions on the same set of coordinate axes:

$y = {\left( x \right)^2}$ VS. $y = {\left( {x - 6} \right)^2}$ VS. $y = {\left( {x + 5} \right)^2}$b)Compared to the graph of $y = {x^2}$:

• the graph of $y = {\left( {x - 6} \right)^2}$ is translated "horizontally" ________ units to the ______________.

• the graph of $y = {\left( {x + 5} \right)^2}$ is translated "horizontally" ________ units to the ______________. - 2.
**Horizontal Translations**

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

• $\left( x \right) \to \left( {x-8} \right)$: shift __________ to the __________. All x coordinates $\Rightarrow$ ____________________

• $\left( x \right) \to \left( {x+3} \right)$: shift __________ to the __________. All x coordinates $\Rightarrow$ ____________________

2.

Transformations of Functions

2.1

Transformations of functions: Horizontal translations

2.2

Transformations of functions: Vertical translations

2.3

Reflection across the y-axis: $y = f(-x)$

2.4

Reflection across the x-axis: $y = -f(x)$

2.5

Transformations of functions: Horizontal stretches

2.6

Transformations of functions: Vertical stretches

2.7

Combining transformations of functions

2.8

Even and odd functions

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