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Understanding Absolute Value Functions and Transformations in Graphs

This section explores the properties and transformations of absolute value functions, including vertical stretches, shrinks, and reflections. Using examples such as translating the graph of ( y = |x| ) to create ( y = |x - 1| + 3 ), we learn how to manipulate graphs through horizontal and vertical shifts. Several comparisons illustrate the effects of stretching and shrinking the function. For practical application, we will also draft equations for geometric designs based on graph properties. Review assignments are included for reinforcement.

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Understanding Absolute Value Functions and Transformations in Graphs

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  1. Section 2.7 Use Absolute Value Functions and Transformations Read all of p. 123.

  2. Example 1 • Graph y = | x – 1 | + 3. Compare the graph with the graph of y = | x |. • Comparison • y = | x – 1 | + 3 translate y = | x | one unit to the right and three units up.

  3. Read about stretches, shrinks, and reflections on p. 124.

  4. Example 2 • Compare each graph with the graph y = | x |. • Comparison • (a) y = 1/3|x | is the graph of y = | x | vertically shrunk by a factor of 1/3 • (b) y = -2|x| is the graph of y = | x | vertically stretched by a factor of 2 and reflected over the x-axis.

  5. Read 1st paragraph on p. 125.

  6. Example 3 • Compare the graph with the graph of y = | x |. • Comparison • The graph of y = ¼| x + 3 | − 2 is the graph of • y = | x |first vertically shrunk by a factor of ¼ then translated 3 units to the left and 2 units down.

  7. Example 4 • A landscaper sketches the design for a triangular shrub protector on graph paper. Write an equation for the shrub protector.

  8. The vertex is (5, 6). A point on the graph is either (0, 0) or (10, 0). Now solve for a. Read 1st paragraph and key concept on p. 126.

  9. Example 5 • The graph of a • function y = f (x) is • shown. Sketch the • graph of the given • function.

  10. HW: • Thursday’s HW: Sections 2.5 and 2.6 Quiz Review • Monday’s HW: pp. 127-129 (4-26 even, 32, 36, 38, 40)

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