Tutorial: New functions from old: Scaled and shifted functions

Game version

Adaptive game tutorial
This adaptive game tutorial will help you master this topic in a way that adapts to your ability as you practice the questions.
If this is your first time studying the material, you may find yourself requiring more help, and as a result your game scores could end up lower, or you could even "die." Do not be discouraged! Just press "New game" to play the game as many times as you need in order to require less help and improve your scores.
You can also try the non-game version (link on the side), which does not score you, is not randomized or adaptive and gives you all the answers, but will not give you additional practice at the level you might need.
To complete this game you must answer all the questions correctly; you will be allowed to have a number of tries for some questions.
The questions are randomized: Expect to see lots of differences differences every time you load the page.
Press "Scores" at any time to show the scores you currently have.
The game is automatically saved. Relaunching the game on the same computer with the same browser will bring up the saved version.
Press "New game" to discard the saved game and start a new game.
Caution: Clicking on the little pictures that appear from time to time on the left can have unexpected consequences! You might want to experiment with them; this is a game after all!

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: