Absolute Value Lesson 2

  • Absolute Value in the Coordinate Plane

    Lesson 2 of 3
    HS Algebra

    40–60 minutes

    Description

    Comparing the graphs of functions and related absolute value functions using reflection

    Materials

    Introduce

    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.

    Explore

    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]:

    • What do you notice about the graphs where x > 0?
      [The graph of the absolute value function g(x)=|x | and the linear graph f(x) = x are the same.]
    • What do you notice about the graphs where x < 0?
      [The graph of the absolute value function g(x)=|x| is the reflection of the linear function f(x) = x, over the x-axis.]

    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.

    graph reflected on the x axis
    • Where is the graph of the absolute value function the same as the graph of the linear function?
      [ when x > 3, then g(x)=|x-3| is the same as f(x)=x-3.]
    • Where is the graph reflected over the x-axis?
      [when x < 3, then g(x)=|x-3| is the reflection of f(x)=x-3 over the x-axis.]

    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

    Synthesize

    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 g(x)=|x^2+2x-3|
    using their preferred method of your choice. Ask them to describe the behavior of g(x), and write g(x)as a piecewise function.

    Answer:

    g(x)=x^2+2x-3 when x<-3
    = -(x^2+2x-3) when -3 ≤ x ≤ 1
    =x^2+2x-3 when x>1

    Teacher note:

    If time allows consider using the following examples. (Choose a function your class is familiar with and might need to review!)

    [y=|log(x)|.]
    [y=|1/x|.]
    [y=|sin(x)|.]

    Teacher Reflection

    • How did students make connections between the graphs of the absolute value functions and their related piecewise function definitions?
    • In what ways did students use the parent graphs to give meaning to the graphs of the absolute value functions?

    Leave your thoughts in the comments below.

    Related Material

    Available in hardcopy and eBook format. By Nathalie Sinclair, David Pimm, Melanie Skelin, Rose Mary Zbiek

    Other Lessons in This Activity

    Lesson 1 of 3
    Use dynagraphs to visualize and represent absolute value relationships.
    Lesson 3 of 3
    Solving absolute value equations graphically and algebraically.
  • Comments

  • 1 Comments

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      I like the deeper understanding students will have of absolute value after working these two lessons.

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  • Essential Question(s)

    • How does reflection help us understand the graph of linear and non-linear absolute value functions?

    Standards

    CCSS, Content Standards to specific grade/standard

    • HSF.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
    • HSF.IF.C.7.B Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    • HSF.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

    CCSS, Standards for Mathematical Practices

    • SMP 5 Use appropriate tools strategically.
    • SMP 7 Look for and make use of structure.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Use and connect mathematical representations.
    • Implement tasks that promote reasoning and problem solving.