Chapter 2 Functions and Graphs Section 2 Elementary Functions: Graphs and Transformations

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

3 Barnett/Ziegler/Byleen Business Calculus 12e Identity Function Domain: All reals (- ,  ) Range: All reals (- ,  )

4 Barnett/Ziegler/Byleen Business Calculus 12e Square Function Domain: All reals (- ,  ) Range: [0, ∞)

5 Barnett/Ziegler/Byleen Business Calculus 12e Cube Function Domain: All reals (- ,  ) Range: All reals (- ,  )

6 Barnett/Ziegler/Byleen Business Calculus 12e Square Root Function Domain: [0, ∞) Range: [0, ∞)

7 Barnett/Ziegler/Byleen Business Calculus 12e Cube Root Function Domain: All reals (- ,  ) Range: All reals (- ,  )

8 Barnett/Ziegler/Byleen Business Calculus 12e Absolute Value Function Domain: All reals (- ,  ) Range: [0, ∞)

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

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.

11 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Shift State the domain and range of each function.

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

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|.

14 Barnett/Ziegler/Byleen Business Calculus 12e Horizontal Shift State the domain and range of each function.

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

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|

17 Barnett/Ziegler/Byleen Business Calculus 12e Vertical Stretching/Shrinking Vertical shrink Vertical stretch State the domain and range of each function.

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

19 Barnett/Ziegler/Byleen Business Calculus 12e Horizontal Stretching/Shrinking

20 Horizontal Stretching/Shrinking Barnett/Ziegler/Byleen Business Calculus 12e x y Horizontal shrink Horizontal stretch State the domain and range of each function.

21 Domain & Range Barnett/Ziegler/Byleen Business Calculus 12e

22 Reflections Barnett/Ziegler/Byleen Business Calculus 12e

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.

24 Domain & Range Barnett/Ziegler/Byleen Business Calculus 12e

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

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

27 Example 2  Describe the transformations for : y = (x – 5) Barnett/Ziegler/Byleen Business Calculus 12e x y x y

28 Example 3 Barnett/Ziegler/Byleen Business Calculus 12e x y x y

29 Writing Equations of Functions Barnett/Ziegler/Byleen Business Calculus 12e

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.

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.

32 Piecewise Practice Barnett/Ziegler/Byleen Business Calculus 12e

33 Barnett/Ziegler/Byleen Business Calculus 12e x y

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