New Functions from Old:
Scaled and Shifted Functions
miscellaneous on-line topics for
Calculus Applied to the Real World

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Let's start by looking at the graphs of some "well-known" functions.


$f(x) = x$
$f(x) = x^{2}$
$f(x) = x^3$

$f(x)=\frac{1}{x}$
$f(x) = \sqrt{x}$
$f(x) = \|x\|$

(To review these graphs, consult the section on functions and their graphs in Chapter 1 of Applied Calculus)

But what about more complicated functions? For example, what about $f(x) = (x-3)^2$ ? Notice that here we've taken $f(x) = x^2$ (one of the functions graphed above) and replaced $x$ by $(x-3)$ to get a new function. Well, here are some "shift" rules that tell you the effect of operations like this.

Shift Rules
RuleExample
Horizontal Shift

Let c be a fixed positive number.

  • Replacing $x$ by the quantity $(x-c)$ shifts the graph to the right $c$ units.
  • Replacing $x$ by the quantity $(x+c)$ shifts the graph to the left $c$ units.

 Here is a picture of the graph of $g(x) = \|x-4\|.$ It is obtained from the graph of $f(x) = \|x\|$ by shifting it to the right $4$ units.
Vertical Shift

Let c be a fixed positive number.

  • Replacing $f(x)$ by $f(x) + c$ shifts the graph up $c$ units.
  • Replacing $f(x)$ by $f(x) - c$ shifts the graph down $c$ units.
 Here is a picture of the graph of $g(x) = x^{2-1}.$ It is obtained from the graph of $f(x) = x^2$ by shifting it down $1$ unit.

Now here is one for you to do.

Example 1 Sketching a Shifted Function

Let $f(x) = \frac{1}{x+1}.$ Select the correct options and press "Check."

The graph of $f(x)$ is obtained from the graph of $\frac{1}{x}$ by shifting it unit$(s).$
       

Now click on the correct graph of the function $f.$



Here is one that is obtained by two successive translations.

Example 2 Multiple Shifts

Let $g(x) = \sqrt{x-2} + 1.$ Select the correct options and press "Check."

The graph of $g(x)$ is obtained from the graph of $\sqrt{x}$ by shifting it $2$ units, and
$1$ unit.
       

Now click on the correct graph of the function $f.$



Question: Why does the Horizontal Shift Rule work?
Answer

Question: Why does the Vertical Shift Rule work?
Answer

In addition to the shift rules, we also have the

Scaling Rules
RuleExample
Horizontal Scaling

If $g(x) = f(cx)$ with $c$ positive, then:
  • If $c>1,$ the graph of $g$ is the graph of $f,$ compressed in the $x$-direction by a factor of $c.$
  • If $0 < c < 1$, then the graph is stretched in the $x$-direction by a factor of $1/c$

 Here is a picture of the graph of $g(x) = (0.5x)^3$. Since $c = 0.5 < 1$, the graph is obtained from that of $f(x) = x^3$ by stretching it in the $x$-direction by a factor of $1/c = 2$.
Vertical Scaling

If $g(x) = cf(x)$ with $c$ positive, then:
  • If $c>1$, the graph of $g$ is the graph of $f$, stretched in the $y$-direction by a factor of $c$.
  • If $0 < c < 1$, then the graph is compressed in the $y$-direction by a factor of $1/c$.
 Here is a picture of the graph of $g(x) = 3\sqrt{x}.$ Since $c = 3>1,$ the graph is obtained from that of $f(x) = \sqrt{x}$ by stretching it in the $y$-direction by a factor of $c = 3.$

Now one for you.

Example 3 Sketching a Scaled Function

Let $g(x) = \frac{1}{3}\left(x+\frac{1}{x}\right).$ Select the correct options and press "Check."

The graph of $g(x)$ is obtained from the graph of $\left(x+\frac{1}{x}\right)$ by it by a factor of
in the direction.          

Now graph the function

on your graphing calculator (or here) and then click on the correct graph of the function $g.$



Here is one that is obtained by several successive operations.

Example 4 A Scaled and Shifted Function

Let $g(x) = \frac {(x-2) ^2}{3} +4.$ Select the correct options and press "Check."

Starting with $f(x) = x^2,$

Step 1: Replace $x$ by $(x-2),$ giving $(x-2)^2.$
  This

Step 2: Divide the last function by $3,$ giving $(x-2)^{2/3}.$
  This

Step 3: Now add $4$ to the last function, to obtain the given function $g.$
  This

Here are the graphs corresponding to these steps.

Original Function
$y = x^2$		 Step 1
$y = (x-2)^2$		 Step 2
$y = \frac{(x-2)^2}{3}$		 Step 3
$y = \frac{(x-2)^2}{3}+4$

Finally, we look at reflections.

Reflections
RuleExample
Horizontal Reflection
  • Replacing $x$ by the quantity $(-x)$ reflects the graph in $y$-axis

    • (In other words, it "flips it about the $y$-axis.")

Here is a picture of the graph of $g(x) = (-0.5x)^{3}+1.$ It is obtained from the graph of $f(x) = 0.5x^{3}+1$ by reflecting it in the $y$-axis.
Vertical Reflection
  • Replacing $f(x)$ by $-f(x)$ reflects the graph in the $x$-axis

    • (In other words, it "flips it about the $x$-axis.")
Here is a picture of the graph of $g(x) = -(x^{2} - 1).$ It is obtained from the graph of $f(x) = x^{2} - 1$ by reflecting it in the $x$-axis.

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Last Updated:January, 1998
Copyright © 1998 StefanWaner and Steven R. Costenoble