1.2 Basic classes of functions  (Page 8/28)

Transformations of functions

We have seen several cases in which we have added, subtracted, or multiplied constants to form variations of simple functions. In the previous example, for instance, we subtracted 2 from the argument of the function $y=x^2$ to get the function $f\left(x\right)={\left(x-2\right)}^2.$ This subtraction represents a shift of the function $y=x^2$ two units to the right. A shift, horizontally or vertically, is a type of transformation of a function    . Other transformations include horizontal and vertical scalings, and reflections about the axes.

A vertical shift of a function occurs if we add or subtract the same constant to each output $y.$ For $c>0,$ the graph of $f\left(x\right)+c$ is a shift of the graph of $f(x)$ up $c$ units, whereas the graph of $f\left(x\right)-c$ is a shift of the graph of $f\left(x\right)$ down $c$ units. For example, the graph of the function $f(x)=x^3+4$ is the graph of $y=x^3$ shifted up $4$ units; the graph of the function $f(x)=x^3-4$ is the graph of $y=x^3$ shifted down $4$ units ( [link] ).

A horizontal shift of a function occurs if we add or subtract the same constant to each input $x.$ For $c>0,$ the graph of $f(x+c)$ is a shift of the graph of $f(x)$ to the left $c$ units; the graph of $f(x-c)$ is a shift of the graph of $f(x)$ to the right $c$ units. Why does the graph shift left when adding a constant and shift right when subtracting a constant? To answer this question, let’s look at an example.

Consider the function $f\left(x\right)=|x+3|$ and evaluate this function at $x-3.$ Since $f\left(x-3\right)=\left|x\right|$ and $x-3<x,$ the graph of $f\left(x\right)=|x+3|$ is the graph of $y=\left|x\right|$ shifted left 3 units. Similarly, the graph of $f\left(x\right)=|x-3|$ is the graph of $y=\left|x\right|$ shifted right $3$ units ( [link] ).