Lesson Plan Lecture Version Absolute Value of Linear Functions Objectives: Students will: Discover how absolute value affects linear functions. Prerequisite Knowledge Students are able to: Graph linear functions. Reflect linear functions in a line of symmetry. Evaluate the absolute value of a constant. Resources This lesson assumes that your classroom has only one computer, from which you can lecture. For classrooms with enough computers for all your students either working individually or in small groups, see the lab version of this lesson. Rulers, pencils, and paper, handout 1, handout 2, and teacher resource1 Access to http://www.explorelearning.com/ Copies of the worksheet for each students (optional) Lesson Preparation Before conducting this lesson, be sure to read through it thoroughly, and familiarize yourself with the Absolute value of a linear function activity at ExploreLearning.com. You may want to bookmark the activity page for your students. If you like, make copies of the worksheet for each student. Lesson Motivation Provide the students with handout 1 and a ruler. Explain to students that the three figures represent holes on a miniature golf course. Have the students ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 1 of 10
determine a path for the golf ball (the white circle) that will result in a hole-in-one for each hole. After several minutes, have the students explain their shots. Try to have the students explain their shots in terms of equal angles, equal distances, and reflections. Teacher resource 1 provides possible hole in one paths. After the discussion, ask the students the following questions: What class of functions could be used to model the paths of the golf balls? Are the paths straight sections or curves? Could polynomials be used to model the paths? After a few minutes provide the students with handout 2. Tell them that they are going to collect data about the paths. As a class, construct a hole-in-one graph for each hole. As they construct the graph, have them keep track of the coordinates where the ball starts, where the ball hits a wall, and where the ball enters the hole. The axes are marked in units of 2. These coordinates are listed below. Hole 1 Hole 2 x y x y 10 8 10 8 2 0 0-2 -6 8-10 8 Hole 3 x y 10 8 2 0 0 2-2 0-10 8 Since the data does not fit any function they currently know of, they need to explore a new class of functions. The Absolute value of a linear function activity To explore functions that are similar to the path of the golf balls, go to the Absolute value of a linear function activity at ExploreLearning.com. ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 2 of 10
y = f(x) When the activity loads up the following graph should be showing. Ask the student the following questions about the graph: On which intervals are the values of y positive? Negative? How does the y values of y = f(x) compare to the y values of y = f(x)? Have the students make an x, y chart for each function, then compare charts. How could the graph of y = f(x) be changed to match the graph of y = f(x)? What line of symmetry would y = f(x) have? Select the black box beside y = f(x) and have the students check their answers The black graph represents y = f(x). ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 3 of 10
Repeat this exercise with y = 2x 4. Have them make conjectures relating y = f(x) to y = f(x). y = f( x ) Graph y = x 6 using the activity; then have the students make a table like the one below for y = f(x). ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 4 of 10
x y 2-4 4-2 -2-8 -4-10 0-6 Now have them make a table for y = x - 6 for the same x values. The table should like the one below. x y 2-4 4-2 -2-4 -4-2 0-6 Ask the students the following questions: How does (x, y) compare to (-x, y) for y = x - 6? Does y = x - 6 have a line of symmetry? What would be the line of symmetry? How could the graph of y = x 6 be used to aid in the graphing of y = x - 6? They should explain that the graph of y = x - 6 could be obtained by reflecting the graph of y = x 6 (for x> 0) in the y-axis. To check their answer using the activity, select the box next to y = f( x ). The blue graph represents y = f( x ). ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 5 of 10
Graph y = -x + 4 using the activity. Using the graph of y = -x + 4 as an aid, ask the students to describe the features of the graph of y = - x + 4. They should use the activity to see if their descriptions were correct. Have the students make conjectures comparing the graph of y = f(x) to the graph of y = f( x ). y = f( x ) Using the activity, graph y = 2x 6 and y = 2 x - 6 ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 6 of 10
Ask the students the following questions: On what intervals is y = 2 x -6 negative? How would the graph of y = (2 x - 6) be different than the graph of y = 2 x -6 on that interval? On what intervals is y = 2 x -6 positive? Would the graph of y = (2 x - 6) be different than the graph of y = 2 x -6 on those intervals? Check the red box so the students can see if their answers were correct. Graph y = -x + 2 using the activity. Next select the blue box so students can see ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 7 of 10
y = - x + 2 on the same axis. Using y = - x + 2 as an aid, ask them to describe the graph of y = (- x + 2). They should use the activity to check their answers. Have the students make conjectures comparing the graph of y = f(x) to the graph of y = f( x ). Back to the golf course Have the students look at the graph of hole 1 on handout 2. The graph looks like the one below. Have the students look at the segment of the graph from the tee to the wall. Using the coordinates of the tee and the coordinates of the point where the ball hits the wall, have them find the equation of the line that contains this segment of the graph. They should have found the equation of the line to be y = x 2. ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 8 of 10
Now experiment with the activity to find the function that converts y = x 2 to the function that represent the path of the golf ball. They should have found that y = f(x) = x 2 models the path of the golf ball. Now have the students look at hole 2 on handout 2. The segment from the tee to the wall is also contained in the line y = x 2, so experiment with the activity to find the function that converts y = x 2 to the function that represent the path of the golf ball. They should have found that y = f( x ) = x 2 models the path of the golf ball. ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 9 of 10
Now have the students look at hole 3 on handout 2. The segment from the tee to the first impact with the wall is also contained in the line y = x 2, so experiment with the activity to find the function that converts y = x 2 to the function that represent the path of the golf ball. They should have found that y = f( x ) = ( x 2) models the path of the golf ball. Conclusion The graphs of linear functions involving absolute value are generally a combination of several line segments. Reflecting certain intervals of y = f(x) can yield the graphs of absolute value functions. The graphs of absolute value functions can be used to solve absolute value equations graphically. Certain applications can be modeled using the absolute value functions. ExploreLearning.com Lesson Plan >> Absolute value of linear functions (Lecture Version)>> Page 10 of 10