1.2 Basic classes of functions  (Page 9/28)

A vertical scaling of a graph occurs if we multiply all outputs $y$ of a function by the same positive constant. For $c>0,$ the graph of the function $cf(x)$ is the graph of $f(x)$ scaled vertically by a factor of $c.$ If $c>1,$ the values of the outputs for the function $cf(x)$ are larger than the values of the outputs for the function $f(x);$ therefore, the graph has been stretched vertically. If $0<c<1,$ then the outputs of the function $cf(x)$ are smaller, so the graph has been compressed. For example, the graph of the function $f\left(x\right)=3x^2$ is the graph of $y=x^2$ stretched vertically by a factor of 3, whereas the graph of $f\left(x\right)=x^2\text{/}3$ is the graph of $y=x^2$ compressed vertically by a factor of $3$ ( [link] ).

The horizontal scaling of a function occurs if we multiply the inputs $x$ by the same positive constant. For $c>0,$ the graph of the function $f(cx)$ is the graph of $f(x)$ scaled horizontally by a factor of $c.$ If $c>1,$ the graph of $f(cx)$ is the graph of $f(x)$ compressed horizontally. If $0<c<1,$ the graph of $f\left(cx\right)$ is the graph of $f\left(x\right)$ stretched horizontally. For example, consider the function $f\left(x\right)=\sqrt{2x}$ and evaluate $f$ at $x\text{/}2.$ Since $f(x\text{/}2)=\sqrt{x},$ the graph of $f\left(x\right)=\sqrt{2x}$ is the graph of $y=\sqrt{x}$ compressed horizontally. The graph of $y=\sqrt{x\text{/}2}$ is a horizontal stretch of the graph of $y=\sqrt{x}$ ( [link] ).

We have explored what happens to the graph of a function $f$ when we multiply $f$ by a constant $c>0$ to get a new function $cf(x).$ We have also discussed what happens to the graph of a function $f$ when we multiply the independent variable $x$ by $c>0$ to get a new function $f(cx).$ However, we have not addressed what happens to the graph of the function if the constant $c$ is negative. If we have a constant $c<0,$ we can write c as a positive number multiplied by $\mathrm{-1};$ but, what kind of transformation do we get when we multiply the function or its argument by $\mathrm{-1}?$ When we multiply all the outputs by $\mathrm{-1},$ we get a reflection about the $x$ -axis. When we multiply all inputs by $\mathrm{-1},$ we get a reflection about the $y$ -axis. For example, the graph of $f\left(x\right)=\text{−}(x^3+1)$ is the graph of $y=(x^3+1)$ reflected about the $x$ -axis. The graph of $f\left(x\right)={\left(\text{−}x\right)}^3+1$ is the graph of $y=x^3+1$ reflected about the $y$ -axis ( [link] ).

If the graph of a function consists of more than one transformation of another graph, it is important to transform the graph in the correct order. Given a function $f(x),$ the graph of the related function $y=cf\left(a\left(x+b\right)\right)+d$ can be obtained from the graph of $y=f(x)$ by performing the transformations in the following order.

1. Horizontal shift of the graph of $y=f(x).$ If $b>0,$ shift left. If $b<0,$ shift right.
2. Horizontal scaling of the graph of $y=f(x+b)$ by a factor of $\left|a\right|.$ If $a<0,$ reflect the graph about the $y$ -axis.
3. Vertical scaling of the graph of $y=f(a\left(x+b\right))$ by a factor of $\left|c\right|.$ If $c<0,$ reflect the graph about the $x$ -axis.
4. Vertical shift of the graph of $y=cf(a\left(x+b\right)).$ If $d>0,$ shift up. If $d<0,$ shift down.