Function Operations

Using Multiple Representations on the TI-84+

Algebra 2 students are studying function operations and transformations of a parent function.  My student had learned about the graph of and how it gets shifted, flipped, and stretched by including parameters a, h, and k in the equation.

Now he was faced with this question: how to graph  the equation in #58:

It didn’t fit the model of    so it wasn’t a transformation of the absolute value parent function.  He knew how to graph each part individually, but didn’t know how to graph the combined equation.  The TI-84+ showed him the graph with an unusual shape—not the V-shape he expected.

TIP: use the   button to access the shortcut menus above the  , , and   keys. The absolute value template is used here.

“Why does the graph look like this?” he wanted to know. We decided to break up the equation into two parts, using ALPHA-TRACE to access the YVAR variable names.* The complete function is found by adding up the two partial functions.

Then we looked at a table of values, to get a numerical view of the situation.  I remind my students that if they are unsure how to graph a particular function, they can ALWAYS make a table of X-Y values as a backup plan—it isn’t the quickest method to graph, but is sure to work.  To get the Y-values of the combined function, add up the Y-values for the partial functions, since .

Initially, we “turned off” Y3 by pressing ENTER on the equals sign, so we could view the partial functions in the TABLE. I asked the student what he thought the values in the next column should be.

He mentally added them up, and then we verified his thinking by activating Y3 and viewing the table again.

To further illuminate the flat portion of the graph, we changed the table increment to 0.1 in order to “zoom in” on those values.

TIP: While in the table,  press  to change the increment , or press 2nd WINDOW to access the TBL SET screen.Success! The TI-84+ provided graphical and numerical representations that deepened our understanding of the algebraic equation. This task had challenged the student, because it didn’t fit the parent function model he had learned, but he built on his knowledge of function operations to solve his own problem and help some classmates as well.  One of our approaches to learning is to “use what you know.”**

NOTES & RESOURCES:

*You can use function notation on the home screen to perform calculations with any function from the Y= screen. Access the YVARs from ALPHA-TRACE.

**Much has been written about Classroom Norms. See Jo Boaler’s suggestions here and my messages to students here.

For more about transformations on parent functions, see this information about the Transformation Graphing App on the TI-84+ family of calculators.