## Shifting versus Point Plotting

Last week I taught my intermediate algebra students how to graph absolute value functions of the form $f(x)=a |x-h|+k$ using two methods.

Point Plotting

On Wednesday I used a point plotting approach. We started by finding the value of x that caused the expression inside the absolute value to equal 0. We then chose two values of x above this number and 2 values of x below this number. We then created a table of function values, plotted the points, and drew the V-shaped curve through them.

Some of students struggled with this approach, especially when it came to completing the table of function values. I know it’s just arithmetic at that point, but it is tedious. I was only able to get through a few examples, and while many students could duplicate the process I’m not sure how many understood what they were doing.

Shifting

As class ended on Wednesday I asked my students to hold off on their homework. (As I’m sure you would understand, they obliged!) I told them I had another technique to show them on Thursday. When class started the next day, I put up the graph from the last example of the previous class: $f(x)=|x-4|+6$. I asked them for the turning point of the graph, and they told me $(4,6)$. I then asked if there was a way to have found that directly from the function without the arithmetic, and they all pointed to the 4 and the 6 in the function.

We did talk a little about why the graph bottomed out at $(h,k)$, and then we worked out several examples. There was some discussion for how to determine whether the graph opened upward or downward, and we reasoned out that it depended on a. I also used the idea of slope and a to talk about how quickly the graph opened up (or down). At the end of class I knew that my students truly understood the graph of an absolute value function.

Graphing in this fashion also will help students to graph parabolas and some of the other functions we cover in this class. Overall I am very happy with the way this went. I prefer plotting points only to introduce a basic graph, but the quicker we can get to translations, rotations, and shifting, the better.

Do you have any graphing tips or stories that you’d like to share? Please share your experience and thoughts by leaving a comment, or reaching me through the contact page at my web site – georgewoodbury.com.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com.

• 1. Diane  |  September 5, 2010 at 12:49 pm

When I do the linear abs. value equations I like to point out that without the absolute value bars, it would be a line. And all the absolute value does is to make the negative parts positive, thereby ‘flipping’ the graph up from the negative to the positive, and that’s why the shape of the graph results in a a “V”. Of course you have to do one that has a turning point right on the x-axis to show this properly.

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