Absolute Value Lesson 2
Comparing the graphs of functions and related absolute value functions using reflection
Use the Lesson 2 Graphing Absolute Value Functions AS to have students complete questions 1 -3. Once the majority of students have completed the first three questions, move to the next phase of the lesson.
Whole Class Discussion
Discuss with students what they noticed and what they wondered about the graphs of the function and their absolute values.
The following discussion should take place to help students complete question 3 on the Graphing Absolute Value Functions student activity sheet. The first example directly relates to problem 1 on the activity sheet.
f(x) = x and g(x)=|x |
Using the method of your choice [SMP 5]:
when x is positive, g(x)=|x |is the same as f(x) = x.
when x is negative, g(x)=|x |is the opposite of f(x) = x>
Let's look at the behavior of the function g(x) as a piecewise function.
g(x)=x when x > 0
=〖-x〗^ when x<0
Now graph the functions f(x) = x -3 and g(x)=|x-3| on the same coordinate plane.
Let's look at the behavior of g(x) as a piecewise function.
g(x)=x-3 when x ≥ 3
= -(x^ -3) when x<3
After this example, allow students time to write g(x) from problem 2 on the activity sheet as a piecewise function.
After students complete problem 2 and you have a short discussion about the answer, ask the following question to look at non-linear functions.
Is there similar behavior when graphing other types of parent functions? For example: quadratics, cubics, quartics, logarithms, rational functions, etc. [SMP 7]
Have students graph f(x)=x^2-4 and g(x)=|x^2-4| on the same coordinate plane, using the method of your choice, on the activity sheet [SMP 5]. Encourage students to use a table of values if they are having a difficult time thinking about what the absolute value of a quadratic function looks like.
Ask students how we could write g(x) as a piecewise function. One key idea is use the zeros of the function. You could direct students to do this on their own or complete this question as a class activity.
Where are the zeros? [(2, -2).]
Let's look at g(x) as a piecewise function.
g(x)=x^2-4, when x<-2
= -(x^2-4) when -2 ≤ x ≤ 2
=x^2-4 when x>2
What is the relationship between the parent function and the absolute value function? Write a general rule for graphing an absolute value function and how it relates to a reflection. Where is the parent function reflected?
After this discussion, work the following example:
Students graph f(x)=x^2+2x-3 and
using their preferred method of your choice. Ask them to describe the behavior of g(x), and write g(x)as a piecewise function.
g(x)=x^2+2x-3 when x<-3
= -(x^2+2x-3) when -3 ≤ x ≤ 1
=x^2+2x-3 when x>1
If time allows consider using the following examples.
(Choose a function your class is familiar with and might need to review!)
Leave your thoughts in the comments below.
I like the deeper understanding students will have of absolute value after working these two lessons.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS