2.13: Using the General Absolute Value Equation and the Graphing Calculator
Mrs. Patel assigns the absolute value function \begin{align*}y = x + 3  2\end{align*}
"This is hard," George laments. "I'm going to need a calculator."
"No, it's not," Sarai counters. "I can tell you what the vertex is without even graphing by hand."
Who is right and what is the vertex?
Guidance
In the problem set of the previous concept, we were introduced to the general equation of an absolute value function. Let’s formally define it here.
General Form of an Absolute Value Function: For any absolute value function, the general form is \begin{align*}y = axh+k\end{align*}
You probably made these connections during the problem set from the previous concept. Now, we will put it all to use together.
Example A
Graph \begin{align*}y = x\end{align*}
Solution: You can make a table for all three of these functions. However, now that we have a better understanding of absolute value functions, let’s use some patterns. First, look at the vertex. Nothing is being added or subtracted, so the vertex for all three will be (0, 0). Second, look at “\begin{align*}a\end{align*}
\begin{align*}y = \begin{cases}x;x \ge 0\\ x;x < 0\end{cases} \ \text{(blue)}, \quad y = \begin{cases} \frac{1}{2}x;x \ge 0\\  \frac{1}{2}x;x < 0\end{cases} \text{(red), and} \ y = \begin{cases} 2x;x \ge 0\\ 2x;x < 0\end{cases} \ \text{(green)}\end{align*}
Comparing the three, we see that if the slope is between 1 and 0, the opening is wider than the parent graph. If the slope, or \begin{align*}a\end{align*}
Now, in addition to drawing a table, we can use the general form of an absolute value equation and the value of \begin{align*}a\end{align*}
Example B
Without making a table, sketch the graph of \begin{align*}y = x62\end{align*}
Solution: First, determine the vertex. From the general form, we know that it will be (6, 2). Notice that the \begin{align*}x\end{align*}
Lastly, we can use a graphing calculator to help us graph absolute value equations. The directions given here pertain to the TI83/84 series; however every graphing calculator should be able to graph absolute value functions.
Example C
Use a graphing calculator to graph \begin{align*}y = 4x+12\end{align*}
Solution: For the TI83/84
1. Press the \begin{align*}Y=\end{align*}
2. Clear any previous functions (press CLEAR) and turn off any previous plots (arrow up to Plot 1 and press ENTER).
3. Press the MATH button, arrow over to NUM and highlight 1:abs(. Press ENTER.
4. Type in the remaining portion of the function. The screen:
5. Press GRAPH. If your screen is off, press ZOOM, scroll down to 6:ZStandard, and press ENTER.
The graph looks like:
As you can see from the graph, the vertex is not (1, 2). The \begin{align*}y\end{align*}
\begin{align*}4x+1 &= 0\\
4x &= 1\\
x &= \frac{1}{4}\end{align*}
The vertex is \begin{align*}\left( \frac{1}{4}, 2\right)\end{align*}
Intro Problem Revisit Sarai is right. The absolute value function is written in general form, so a calculator is not necessary. The vertex is (–3, –2).
Vocabulary
 General Form of an Absolute Value Function

For any absolute value function, the general form is \begin{align*}y = axh+k\end{align*}
y=ax−h+k , where \begin{align*}a\end{align*}a controls the width of the “\begin{align*}V\end{align*}V ” and \begin{align*}(h, k)\end{align*}(h,k) is the vertex.
 Breadth
 The wideness or narrowness of a function with two symmetric sides.
Guided Practice
1. Graph \begin{align*}y = 3x+45\end{align*}
2. Graph \begin{align*}y = 2x5+1\end{align*}
Answers
1. First, use the general form to find the vertex, (4, 5). Then, use \begin{align*}a\end{align*}
The domain is all real numbers and the range is all reals greater than and including 5.
Domain: \begin{align*}x \in \mathbb{R}\end{align*}
Range: \begin{align*}y \in [5, \infty)\end{align*}
2. Using the steps from Example C, the function looks like:
Practice
 Graph \begin{align*}y = 3x\end{align*}, \begin{align*}y = 3x\end{align*}, and \begin{align*}y = 3x\end{align*} on the same set of axes. Compare the graphs.
 Graph \begin{align*}y = \frac{1}{4}x+1\end{align*}, and \begin{align*}y = \frac{1}{4}x+1\end{align*} on the same set of axes. Compare the graphs.
 Without graphing, do you think that \begin{align*}y = 2x,y = 2x,\end{align*} and \begin{align*}y = 2x\end{align*} will all produce the same graph? Why or why not?
 We know that the domain of all absolute value functions is all real numbers. What would be a general rule for the range?
Use the general form and pattern recognition to graph the following functions. Determine the vertex, domain, and range. No graphing calculators!
 \begin{align*}y = x2+5\end{align*}
 \begin{align*}y = 2x+3\end{align*}
 \begin{align*}y = \frac{1}{3}x+4\end{align*}
 \begin{align*}y = 2x+12\end{align*}
 \begin{align*}y =  \frac{1}{2}x7\end{align*}
 \begin{align*}y = x8+6\end{align*}
Use a graphing calculator to graph the following functions. Sketch a copy of the graph on your paper. Identify the vertex, domain, and range.
 \begin{align*}y = 42x+1\end{align*}
 \begin{align*}y = \frac{2}{3}x4+ \frac{1}{2}\end{align*}
 \begin{align*}y = \frac{4}{3}2x3 \frac{7}{2}\end{align*}
Graphing Calculator Extension Use the graphing calculator to answer questions 1416.
 Graph \begin{align*}y = x^2 4\end{align*} on your calculator. Sketch the graph and determine the domain and range.
 Graph \begin{align*}y = x^2 4\end{align*} on your calculator. Sketch graph and determine the domain and range.
 How do the two graphs compare? How are they different? What could you do to the first graph to get the second?
Breadth
Breadth refers to the distance between the halves of a function with two symmetric sides (such as an absolute value function).General Form of an Absolute Value Function
The general equation for an absolute value function is , where controls the width of the “” and is the vertex.Image Attributions
Here you'll learn how to graph more complicated absolute value functions and use the graphing calculator.
Breadth
Breadth refers to the distance between the halves of a function with two symmetric sides (such as an absolute value function).General Form of an Absolute Value Function
The general equation for an absolute value function is , where controls the width of the “” and is the vertex.We need you!
At the moment, we do not have exercises for Using the General Absolute Value Equation and the Graphing Calculator.