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Absolute value functions

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  • Absolute-Value Functions

    Warm upLessonHomework

  • Warm UpEvaluate each expression for f(4) and f(-3).

    1. f(x) = |x + 1|

    2. f(x) = 2|x| 1

    3. f(x) = |x + 1| + 2

    5; 2

    7; 5

    7; 4

    Let g(x) be the indicated transformation of f(x). Write the rule for g(x).

    4. f(x) = 2x + 5; vertical translation 6 units downg(x) = 2x 1

    g(x) = 2x + 85. f(x) = x + 2; vertical stretch by a factor of 4

  • Graph and transform absolute-value functions.

    Your Goal Today is

  • absolute-value function

    Vocabulary

  • An absolute-value function is a function whose rule contains an absolute-value expression. The graph of the parent absolute-value function f(x) = |x| has a V shape with a minimum point or vertex at (0, 0).

  • The absolute-value parent function is composed of two linear pieces, one with a slope of 1 and one with a slope of 1. In Lesson 2-6, you transformed linear functions. You can also transform absolute-value functions.

  • The general forms for translations are

    Vertical:

    g(x) = f(x) + k

    Horizontal:

    g(x) = f(x h)

    Remember!

  • Example 1A: Translating Absolute-Value Functions

    Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

    5 units down

    Substitute.

    The graph of g(x) = |x| 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, 5).

    f(x) = |x|

    g(x) = f(x) + k

    g(x) = |x| 5

  • Example 1A Continued

    The graph of g(x) = |x| 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, 5).

    f(x)

    g(x)

  • Example 1B: Translating Absolute-Value Functions

    Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

    1 unit left

    Substitute.

    f(x) = |x|

    g(x) = f(x h )

    g(x) = |x (1)| = |x + 1|

  • Example 1B Continued

    f(x)

    g(x)

    The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (1, 0).

  • 4 units down

    Substitute.

    f(x) = |x|

    g(x) = f(x) + k

    g(x) = |x| 4

    Write In Your Notes! Example 1a

    Let g(x) be the indicated transformation of f(x) = |x|. Write the rule for g(x) and graph the function.

  • f(x)

    g(x)

    Check It Out! Example 1a Continued

    The graph of g(x) = |x| 4 is the graph of f(x) = |x| after a vertical shift of 4 units down. The vertex of g(x) is (0, 4).

  • Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

    2 units right

    Substitute.

    f(x) = |x|

    g(x) = f(x h)

    g(x) = |x 2| = |x 2|

    Write In Your Notes! Example 1b

  • f(x)

    g(x)

    Check It Out! Example 1b Continued

    The graph of g(x) = |x 2| is the graph of f(x) = |x| after a horizontal shift of 2 units right. The vertex of g(x) is (2, 0).

  • Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.

  • Example 2: Translations of an Absolute-Value Function

    Translate f(x) = |x| so that the vertex is at (1, 3). Then graph.

    g(x) = |x h| + k

    g(x) = |x (1)| + (3) Substitute.

    g(x) = |x + 1| 3

  • Example 2 Continued

    The graph confirms that the vertex is (1, 3).

    f(x)

    The graph of g(x) = |x + 1| 3 is the graph of f(x) = |x| after a vertical shift down 3 units and a horizontal shift left 1 unit.

    g(x)

  • Write In Your Notes! Example 2

    Translate f(x) = |x| so that the vertex is at (4, 2). Then graph.

    g(x) = |x h| + k

    g(x) = |x 4| + (2) Substitute.

    g(x) = |x 4| 2

  • The graph confirms that the vertex is (4, 2).

    Check It Out! Example 2 Continued

    g(x)

    The graph of g(x) = |x 4| 2 is the graph of f(x) = |x| after a vertical down shift 2 units and a horizontal shift right 4 units.

    f(x)

  • Reflection across x-axis: g(x) = f(x)

    Reflection across y-axis: g(x) = f(x)

    Remember!

    Absolute-value functions can also be stretched, compressed, and reflected.

    Vertical stretch and compression : g(x) = af(x)

    Horizontal stretch and compression: g(x) = f

    Remember!

  • Example 3A: Transforming Absolute-Value Functions

    Perform the transformation. Then graph.

    g(x) = f(x)

    g(x) = |(x) 2| + 3

    Take the opposite of the input value.

    Reflect the graph. f(x) =|x 2| + 3 across the y-axis.

  • g f

    Example 3A Continued

    The vertex of the graph g(x) = |x 2| + 3 is (2, 3).

  • g(x) = af(x)

    g(x) = 2(|x| 1) Multiply the entire function by 2.

    Example 3B: Transforming Absolute-Value Functions

    Stretch the graph. f(x) = |x| 1 vertically by a factor of 2.

    g(x) = 2|x| 2

  • Example 3B Continued

    The graph of g(x) = 2|x| 2 is the graph of f(x) = |x| 1 after a vertical stretch by a factor of 2. The vertex of g is at (0, 2).

    f(x) g(x)

  • Example 3C: Transforming Absolute-Value Functions

    Compress the graph of f(x) = |x + 2| 1 horizontally by a factor of .

    g(x) = |2x + 2| 1 Simplify.

    Substitute for b.

  • f

    The graph of g(x) = |2x + 2| 1 is the graph of

    f(x) = |x + 2| 1 after a horizontal compression by

    a factor of . The vertex of g is at (1, 1).

    Example 3C Continued

    g

  • Perform the transformation. Then graph.

    g(x) = f(x)

    g(x) = |x 4| + 3

    Take the opposite of the input value.

    Reflect the graph. f(x) = |x 4| + 3 across the y-axis.

    Write In Your Notes! Example 3a

    g(x) = |(x) 4| + 3

  • The vertex of the graph g(x) = |x 4| + 3 is (4, 3).

    Check It Out! Example 3a Continued

    fg

  • Compress the graph of f(x) = |x| + 1 vertically

    by a factor of .

    Simplify.

    Write In Your Notes! Example 3b

    g(x) = a(|x| + 1)

    g(x) = (|x| + 1)

    g(x) = (|x| + )

    Multiply the entire function by .

  • Check It Out! Example 3b Continued

    f(x)

    g(x)

    The graph of g(x) = |x| + is the graph of g(x) = |x| + 1 after a vertical compression by a factor of . The vertex of g is at ( 0, ).

  • Substitute 2 for b.

    Stretch the graph. f(x) = |4x| 3 horizontally by a factor of 2.

    g(x) = |2x| 3

    Write In Your Notes! Example 3c

    Simplify.

    g(x) = f( x)

    g(x) = | (4x)| 3

  • Check It Out! Example 3c Continued

    g

    The graph of g(x) = |2x| 3 the graph of f(x) = |4x| 3 after a horizontal stretch by a factor of 2. The vertex of g is at (0, 3).

    f