Graphing Ablsolute Value Functions and Piecewise Notation

The way absolute value functions work is that the points can never be negative, so the line never goes into the negative zone on the graph. It will reflect back up instead. The point where it reflects on the x axis is called the critical point or x-intercept. Our entire example will consist of one graph. This time we will focus on linear equations but I will include the picture of a quadratic to better understand the process. We will also find out what piecewise notation is. A linear absolute value function will look like y=|mx+b|

The absolute value function we will use is y=|3x+6|. We will graph y=3x+6 and then use the opposite signs to get the reflection, since the line can never go negative. You can think of the reflection as if the line continues downward but is flipped up.

Now we will get the reflected line

Now we will erase the lines below y=0, making it so y is greater or equal to 0. This is also the range. The domain is x is equal to all real numbers.

By this graph we can see that the critical point is (-2,0) and the y-intercept is (0,6)

We will now find the Piecewise notation of the graph. Piecewise notation is basically two equations with restrictions to outline which line is represented and where it is. We will do the same thing where we flipped the signs before, because they each represent the two parts of the reflected line.

If we are given y=|3x+6|, then it is split into the two equations we used before, y=3x+6 and y=-3x-6. The first equation is the original so we will use it first. It is anywhere on the x axis greater than or equal to -2. The second equation is anywhere on the x axis that is lower than -2, but not equal to. Both equations cannot be equal to the same number. We use all these same steps everytime. Our piecewise notation will look like so:

The first equation represents the right side of the line or the original and the bottom represents the left side or the reflection. The restriction represents where they are cut off.

I won’t go into detail about the quadratic version of this but instead of one point there may be 2 in the piecewise notation’s restrictions. The quadratic flips up where it goes negative just like the linear version. I will show a picture of the function . Because it is opening down the parts that go back up are the “arms”.

The x-intercepts are 0 and -2. We will use these points in the piecewise notation. The two equations shown on the graph because of the absolute value are, and The first one represents the middle and the last one represents the two arms or long parts going up. Because of this the Piecewise notation that represents this function is:

The original function is in between -2 and 0 and the reflected function is past or below 0 and -2, respectively.