### Graphing Absolute Value on the TI-83

|6x+8|+26
To do this we will create the expression
(abs(6X+8)+26)*3-1
which will have the value of -1 for all X where |6X+8|+26 is false, and the value 2 where that expression is true.

The final three figures on this page are meant to demonstrate the relation between what we say we are doing, and what we are really doing. The original problem was to solve

|6x+8|+26
Looking at this problem, here or in the text, we might notice that the variable Y does not appear in the problem. The original problem is an inequality in one variable, X. In the Figures above, we have worked to create a two-level graph of this problem, showing where the inequality is true and where it is false. We really created a two variable function, namely,
Y1=(abs(6X+8)+26)*3-1
Now we will look at the related problem
|6x+8|+2-60
where we have simply moved the 6 to the left side of the sign, giving us an expression on the left that is greater than or equal to 0. If we look at
Y2=abs(6X+8)+2-6
then we are really asking, where is Y2 greater than or equal to 0.

 Figure 14 In Figure 14 we have returned to the Y= screen and we have entered the new, unsimplified, equation. Figure 15 We return to the GRAPH screen and we can see not only the original two level function, but also the graph of our new function Y2=abs(6X+8)+2-6 which comes out as a "V". Notice that the "V" is at or above the X-axis for all X-values where the two level function is at the upper level, i.e., is true. And, the "V" is below the X-axis for those values of X where the two level function is at -1, i.e., is false. In order to better see this, Figure 15 has been augmented and reproduced as Figure 16. Figure 16 The change from Figure 15 to Figure 16 is that we have drawn two red and two blue vertical lines. The red lines are above and below the endpoints of the lower level segment of the two level function. That is, they mark the left and right ends of the region where the original expression was false. The blue lines are above and below the endpoints of the upper level segments of the two level function. That is, they mark the right and left ends of the region where the original expression was true. These red and blue lines were added to make it easier to see where the "V" graph of the second function is above the X-axis and where it is below the X-axis.