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Chapter 2 Functions and Graphs Section 2 Elementary Functions: Graphs and Transformations
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2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 2.2 The student will become familiar with a beginning library of elementary functions. The student will be able to transform functions using vertical and horizontal shifts. The student will be able to transform functions using reflections, stretches, and shrinks. The student will be able to graph piecewise-defined functions. Elementary Functions; Graphs and Transformations
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3 Barnett/Ziegler/Byleen Business Calculus 12e Identity Function Domain: All reals (- , ) Range: All reals (- , )
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4 Barnett/Ziegler/Byleen Business Calculus 12e Square Function Domain: All reals (- , ) Range: [0, ∞)
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5 Barnett/Ziegler/Byleen Business Calculus 12e Cube Function Domain: All reals (- , ) Range: All reals (- , )
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6 Barnett/Ziegler/Byleen Business Calculus 12e Square Root Function Domain: [0, ∞) Range: [0, ∞)
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7 Barnett/Ziegler/Byleen Business Calculus 12e Cube Root Function Domain: All reals (- , ) Range: All reals (- , )
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8 Barnett/Ziegler/Byleen Business Calculus 12e Absolute Value Function Domain: All reals (- , ) Range: [0, ∞)
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9 Transformations Types of transformations performed on graphs: Vertical shift (translation) Horizontal shift (translation) Vertical stretch/shrink (dilation) Horizontal stretch/shrink (dilation) Reflection Each one can be determined by examining the equation of the graph. Barnett/Ziegler/Byleen Business Calculus 12e
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10 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Shift The graph of y = f(x) + h Shifts the graph of y = f(x) up h units The graph of y = f(x) - h Shifts the graph of y = f(x) down h units Graph y = |x|, y = |x| + 4, and y = |x| – 5.
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11 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Shift State the domain and range of each function.
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12 Domain & Range y = |x| D: (- , )R: [0, ) y = |x| + 4 D: (- , )R: [4, ) y = |x| – 5 D: (- , )R: [-5, ) Barnett/Ziegler/Byleen Business Calculus 12e
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13 Barnett/Ziegler/Byleen Business Calculus 12e Horizontal Shift The graph of y = f(x + h) Shifts the graph of y = f(x) left h units The graph of y = f(x - h) Shifts the graph of y = f(x) right h units Graph y = |x|, y = |x + 4|, and y = |x – 5|.
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14 Barnett/Ziegler/Byleen Business Calculus 12e Horizontal Shift State the domain and range of each function.
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15 Domain & Range y = |x| D: (- , )R: [0, ) y = |x+4| D: (- , )R: [0, ) y = |x-5| D: (- , )R: [0, ) Barnett/Ziegler/Byleen Business Calculus 12e
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16 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Stretching/Shrinking The graph of y = Af(x) can be obtained from the graph of y = f(x) by multiplying each y-coordinate of f(x) by A. If A > 1, the result is a vertical stretch by a factor of A. If 0 < A < 1, the result is a vertical shrink by a factor of A. Graph y = |x|, y = 2|x|, and y = 0.5|x|
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17 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Stretching/Shrinking Vertical shrink Vertical stretch State the domain and range of each function.
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18 Domain & Range y = |x| D: (- , )R: [0, ) y = 2|x| D: (- , )R: [0, ) y = 0.5|x| D: (- , )R: [0, ) Barnett/Ziegler/Byleen Business Calculus 12e
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19 Barnett/Ziegler/Byleen Business Calculus 12e Horizontal Stretching/Shrinking
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20 Horizontal Stretching/Shrinking Barnett/Ziegler/Byleen Business Calculus 12e x y Horizontal shrink Horizontal stretch State the domain and range of each function.
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21 Domain & Range Barnett/Ziegler/Byleen Business Calculus 12e
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22 Reflections Barnett/Ziegler/Byleen Business Calculus 12e
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23 Reflections Barnett/Ziegler/Byleen Business Calculus 12e x y Reflected over x-axis Reflected over y-axis State the domain and range of each function.
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24 Domain & Range Barnett/Ziegler/Byleen Business Calculus 12e
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25 Multiple Transformations It is common for a graph to have multiple transformations. It’s important to know what the parent looks like so you can perform each transformation on it. Barnett/Ziegler/Byleen Business Calculus 12e
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26 Example 1 Describe the transformations for the function: y = -|x + 3| y = |x| shifted left 3, reflected over x-axis Barnett/Ziegler/Byleen Business Calculus 12e x y x y
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27 Example 2 Describe the transformations for : y = (x – 5) 2 + 4 Barnett/Ziegler/Byleen Business Calculus 12e x y x y
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28 Example 3 Barnett/Ziegler/Byleen Business Calculus 12e x y x y
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29 Writing Equations of Functions Barnett/Ziegler/Byleen Business Calculus 12e
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30 Barnett/Ziegler/Byleen Business Calculus 12e Piecewise-Defined Functions Functions whose definitions involve more than one rule for different parts of its domain are called piecewise- defined functions. Graphing one of these functions involves graphing each rule over the appropriate portion of the domain.
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31 Barnett/Ziegler/Byleen Business Calculus 12e Example of a Piecewise-Defined Function Graph the function Notice that the point (2,0) is included but the point (2, –2) is not.
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32 Piecewise Practice Barnett/Ziegler/Byleen Business Calculus 12e
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33 Barnett/Ziegler/Byleen Business Calculus 12e x y
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