Single Transformations of Functions

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Introduction

Vertical shift: (red) and (blue)
Horizontal shift: (red) and (blue)
Vertical reflection: (red) and (blue)
Horizontal reflection: (red) and (blue)

Translations

One kind of transformation involves shifting the entire graph of a function up, down, right, or left. The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function. In other words, we add the same constant to the output value of the function regardless of the input. For a function , the function is shifted vertically k units. For example, is the function shifted up by 4 units. is the function shifted down by 7.7 units.

Given a function f, a new function , where h is a constant, is a horizontal shift of the function f. If h is positive, the graph will shift h units to the right. If h is negative, the graph will shift h units to the left. For example, is the graph of shifted 3 units to the right. is the function shifted units to the left.

Reflections

Given a function , a new function is a vertical reflection of the function , sometimes called a reflection about (or over, or through) the x-axis. For example, is a vertical reflection of the function .

Given a function , a new function is a horizontal reflection of the function , sometimes called a reflection about the y-axis. For example, is a horizontal reflection of the function .

Even and Odd Functions

A function f is even if for all values of x, ; that is, a function is even if its horizontal reflection is identical to itself. For example, is an even function since .

A function f is odd if for all values of x, ; that is, a function is odd if a horizontal reflection and vertical reflection of the function results in the same function. For example, is an odd function since .

If a function satisfies neither of these conditions, then it is neither even nor odd. For example, is neither even nor odd because , which is not equal to , and , which is also not equal to .

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