Absolute Value Transformations can be tricky, since we have two different types of problems:
| Transformations of the Absolute Value Parent Function | Absolute Value Transformations of other Parent Functions |
Note: To review absolute value functions, see the Solving Absolute Value Equations and Inequalities section. For Parent Functions and general transformations, see the Parent Graphs and Transformations section.
Transformations of the Absolute Value Parent Function
Let’s first work with transformations on the absolute value parent function.
Since the vertex (the “point”) of an absolute value parent function $ y=\left| x \right|$ is $ \left( {0,\,0} \right)$, an absolute value equation with new vertex $ \left( {h,\,k} \right)$ is $ \displaystyle f\left( x \right)=a\left| {\frac{1}{b}\left( {x-h} \right)} \right|+k$, where $ a$ is the vertical stretch, $ b$ is the horizontal stretch, $ h$ is the horizontal shift to the right, and $ k$ is the vertical shift upwards. If $ a$ is negative, the graph points up instead of down. Here is an example with a t-chart:
Here’s an example of writing an absolute value function from a graph:
Absolute Value Transformations of other Parent Functions
Now let’s look at taking the absolute value of functions, both on the outside (affecting the $ y$’s) and the inside (affecting the $ x$’s). We’ll start out with a function of points.
Note that with the absolute value on the outside (affecting the $ \boldsymbol{y}$’s), we just take all negative $ \boldsymbol{y}$-values and make them positive, and with absolute value on the inside (affecting the $ \boldsymbol{x}$’s), we take all the 1st and 4th quadrant points and reflect them over the $ \boldsymbol{y}$-axis, so that the new graph is symmetric to the $ \boldsymbol{y}$-axis.
Here are examples of mixed absolute value transformations to show what happens when the inside absolute value is not just around the $ x$, versus just around the $ x$; you can see that this can get complicated.
Here are more absolute value examples with parent functions:
Note: These mixed transformations with absolute value are very tricky; it’s really difficult to know what order to use to perform them. The general rule of thumb is to perform the absolute value first for the absolute values on the inside, and the absolute value last for absolute values on the outside (work from the inside out). The best thing to do is to play around with them on your graphing calculator to see what’s going on.
For example, with something like $ y=\left| {{{2}^{x}}} \right|-3$, you perform the $ y$ absolute value function first (before the shift); with something like $ y=\left| {{{2}^{x}}-3} \right|$, you perform the $ y$ absolute value last (after the shift). (These two make sense, when you look at where the absolute value functions are.) But we saw that with $ y={{2}^{{\left| x \right|-3}}}$, we performed the $ x$ absolute value function last (after the shift). I also noticed that with $ y={{2}^{{\left| {x-3} \right|}}}$, you perform the $ x$ absolute value transformation first (before the shift). I don’t think you’ll get this detailed with your transformations, but you can see how complicated this can get!
Here’s an example where we’re using what we know about the absolute value transformation, but we’re using it on an absolute value parent function! Pretty crazy, huh?
More Absolute Value Transformations
What about $ \left| {f\left( {\left| x \right|} \right)} \right|$? Play around with this in your calculator with $ y=\left| {{{2}^{{\left| x \right|}}}-5} \right|$, for example. You’ll see that it shouldn’t matter which absolute value function you apply first, but it certainly doesn’t hurt to work from the inside out. And with $ -\left| {f\left( {\left| x \right|} \right)} \right|$, it’s a good idea to perform the inside absolute value first, then the outside, and then the flip across the $ x$-axis. So the rule of thumb with these absolute value functions and reflections is to move from the inside out.
Let’s do more complicated examples with absolute value and flipping – sorry that this stuff is so complicated! Just be careful about the order by trying real functions in your calculator to see what happens. These are for the more advanced Pre-Calculus classes!
Learn these rules, and practice, practice, practice!
On to Piecewise Functions – you are ready!