Tutorial: New functions from old: Scaled and shifted functions

Resources

Function evaluator and grapher

Excel grapher

Some basic functions

Here are the graphs of some common functions. Try to be able to instantly recognize each of them by its shape.

$f(x) = x$

$f(x) = x^2$

$f(x) = x^3$

$f(x) = \dfrac{1}{x}$

$f(x) = \sqrt{x}$

$f(x) = |x|$

(To see how these graphs are drawn, go to the %%functionstut and scroll down to "Graphing functions.")

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

Shift rules

Shift rules tell us which transformations of a function result in the graph being shifted left, right, up, or down:

Scaling rules

In addition to shifting the graph of a function, we can also stretch or compress it vertically and/or horizontally. The scaling rules tell us which transformations of a function are needed to do this:

Of course, we can combine combine shifts and scales in sequence:

Flipping rules

Finally, in addition to shifting and scaling the graph of a function, we can also flip it vertically and/or horizontally. The following rules tell us which transformations of a function are needed to do this: