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2.7 - Absolute Value Functions and Transformations

Learn how to graph absolute value functions and apply transformations using translations, reflections, stretches, and shrinks. Also, explore the relationship between different functions within the absolute value family.

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2.7 - Absolute Value Functions and Transformations

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  1. 2.7 - Absolute Value Functions and Transformations Objectives: • To graph an absolute value function by performing transformations on the parent • To apply transformations to graphing any function

  2. Vocabulary As a group, define each of these without your book. Give an example of each word and leave a bit of space for additions and revisions.

  3. The Absolute Value Function The absolute value function is defined by f (x) = |x|. This is the absolute value parent function.

  4. Parent Function • V-shape • It is symmetric about the y-axis • The vertexis the minimum point on the graph

  5. Investigation 1 In this Investigation, we are going to discover how to perform transformations on the absolute value function. Step 1: Press Y= and enter the following equations.

  6. Investigation 1 Step 2:Graph the equations using the ZOOMSTD setting. Step 3:Describe how the family of graphs y = |x| + k is related to y = |x|.

  7. Investigation 1 Step 4: Press Y= and enter the following equations. Step 5: Graph the equations using the ZOOMSTD setting.

  8. Investigation 1 Step 6: Describe how the family of graphs y = |x – h | is related to y = |x|.

  9. Translation A translation is a transformation that shifts a graph horizontally or vertically, but doesn’t change the overall shape or orientation.

  10. Translation The graph of y = |x – h| + k is the graph of y = |x| translated h horizontal units and y vertical units. • The new vertex is at (h, k)

  11. Investigation 2 In this Investigation, we will continue to expand our knowledge of transformations. Step 1: Press Y= and enter the following equations.

  12. Investigation 2 Step 2:Graph the equations using the ZOOMSTD setting. Step 3:Describe how the family of graphs y = a|x| is related to y = |x|.

  13. Stretching and Shrinking The graph of y = a|x| is graph of y = |x| vertically stretched or shrunk depending on the |a|. The value of a acts like the slope.

  14. Exercise 1 Use your graphing calculator to graph the following: • y = |x| • y = -|x| Describe how the graph of y = a|x| is related to y = |x| when a < 0.

  15. Reflection The graph of y = a|x| is graph of y = |x| reflected across the x-axis when a < 0.

  16. Multiple Transformations In general, the graph of an absolute value function of the form y = a|x – h| + kcan involve translations, reflections, stretches or shrinks. To graph an absolute value function, start by identifying the vertex.

  17. Graphing Absolute Value Functions Graphing y = a|x – h| + kthese things is easy: • Plot the vertex (h, k). • Use the a value as slope to plot one point. • Use symmetry to find a corresponding point. • Connect the dots in a V-shape.

  18. Exercise 2 Without a graphing calculator, graph the following functions. How do they compare to the parent function? • y = |x – 2| + 5 • y = (1/2)|x| • y = 2|x + 1| – 3 • f (x) = -3|x + 1| – 2

  19. Transformations in General You can perform transformations on the graph of any function in manner similar to transformations on the absolute value function.

  20. Exercise 3 The graph of y = f (x) is shown. Sketch the graph of the given function. • y = 2  f (x) • y = -f (x + 2) + 1 • y = 2  f (x – 2) – 5

  21. Assignment • P. 127-129: 1, 2-32 even, 33, 34, 39, 42-44 • P. 131: 1-10 • Working with Other Functions Worksheet: Evens

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