Name: ____________________ Pre- Calc. 11 Date: _____________
Chapter 7 – Absolute Value and Reciprocal Functions Section 7.2B – Graphing Absolute Value Functions of the Form continue
Notice the ‘ ’ value is inside the absolute value. If this is the case, the absolute value
should be graphed using a completely different method. Example 1: Sketch the graph of 3 1y x
Step 1 : Sketch the graph of
y-intercept: slope: Step 2 : Calculate the -intercept by setting in the
absolute value equation. The -intercept
can also be located graphically from step 1. The -intercept of is also the
-intercept of . This is
called an invariant point.
Step 3 : Reflect in the -axis the part of the graph of
that is below the -axis. The V- shaped graph that results is the graph of
.
Step 4 : State the domain and range:
Since the right branch is from the equation and the left branch is from the equation
_________________________________ , the graph can be described as a piecewise function:
Invariant points are points that remain unchanged when a transformation is applied. What other points on the graph are invariant points?
Example 2: Sketch the graph of . State the domain and range, and express as a
Piecewise function.
Quadratic absolute functions
Graphing an Absolute Value Function of the Form
Example 3: Sketch the graph of by first sketching the graph of
. Then state the domain and range of the absolute value graph only.
1) Factor the corresponding quadratic equation to find the roots (x-intercepts). 2) Complete the square on the function to find the vertex. Graph the parabola. 3) Reflect in the x-axis the part of the graph that lies below the x-axis in order to build the absolute value graph. The negative y values in the original parabola will have the absolute value applied to them, thereby making them positive.
The absolute value graph above is actually a combination of two parabolas. What are the quadratic functions of the two parabolas?
We can define the absolute value function as a piecewise function of the two quadratic functions:
Example 3: Sketch the graph of and
a) State the domain and range of the absolute value graph. b) Express the absolute value as a piecewise function.
Assignment: Pg. 376-379 Q. 7, 8, 10, 11cd, 13, 22