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College Algebra Chapter 2 Functions and Graphs

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1 College Algebra Chapter 2 Functions and Graphs
Section 2.7 Analyzing Graphs of Functions and Piecewise-Defined Functions

2 1. Test for Symmetry 2. Identify Even and Odd Functions 3. Graph Piecewise-Defined Functions 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function 5. Determine Relative Minima and Maxima of a Function

3 Test for Symmetry Consider an equation in the variables x and y. Symmetric with respect to the y-axis: Substituting –x for x results in equivalent equation. Symmetric with respect to the x-axis: Substituting –y for y results in equivalent equation. Symmetric with respect to the origin: Substituting –x for x and –y for y results in equivalent equation.

4 Example 1: Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

5 Example 2: Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

6 Example 3: Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

7 Example 4: Determine whether the graph of the equation is symmetric to the x-axis, y-axis, origin, or none of these.

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10 1. Test for Symmetry 2. Identify Even and Odd Functions 3. Graph Piecewise-Defined Functions 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function 5. Determine Relative Minima and Maxima of a Function

11 Identify Even and Odd Functions
Even function: f(–x) = f(x) for all x in the domain of f. (Symmetric with respect to the y-axis) Odd function: f(–x) = –f(x) for all x in the domain of f. (Symmetric with respect to the origin)

12 Example 5: Determine if the function is even, odd, or neither.

13 Example 6: Determine if the function is even, odd, or neither.

14 Example 7: Determine if the function is even, odd, or neither.

15 Example 8: Determine if the function is even, odd, or neither.

16 Example 9: Determine if the function is even, odd, or neither.

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19 1. Test for Symmetry 2. Identify Even and Odd Functions 3. Graph Piecewise-Defined Functions 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function 5. Determine Relative Minima and Maxima of a Function

20 Example 10: Evaluate the function for the given values of x.

21 Example 11: Evaluate the function for the given values of x.

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23 Example 12: Graph the function.

24 Example 13: Graph the function.

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27 Graph Piecewise-Defined Functions
Greatest integer function: is the greatest integer less than or equal to x.

28 Example 14: Evaluate.

29 Example 15: Graph.

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31 Example 16: A new job offer in sales promises a base salary of $3000 a month. Once the sales person reaches $50,000 in total sales, he earns his base salary plus a 4.3% commission on all sales of $50,000 or more. Write a piecewise-defined function (in dollars) to model the expected total monthly salary as a function of the amount of sales, x.

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33 1. Test for Symmetry 2. Identify Even and Odd Functions 3. Graph Piecewise-Defined Functions 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function 5. Determine Relative Minima and Maxima of a Function

34 Investigate Increasing, Decreasing, and Constant Behavior of a Function

35 Example 17: Use interval notation to write the interval(s) over which is increasing, decreasing, and constant. Increasing: _____________________ Decreasing: Constant:

36 Example 18: Use interval notation to write the interval(s) over which is increasing, decreasing, and constant. Increasing: _____________________ Decreasing: Constant:

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38 1. Test for Symmetry 2. Identify Even and Odd Functions 3. Graph Piecewise-Defined Functions 4. Investigate Increasing, Decreasing, and Constant Behavior of a Function 5. Determine Relative Minima and Maxima of a Function

39 Determine Relative Minima and Maxima of a Function

40 Example 19: Identify the location and value of any relative maxima or minima of the function. The point ________ is the lowest point in a small interval surrounding x = ____. At x = ____ the function has a relative minimum of _____.

41 Example 19 continued: The point ________ is the highest point in a small interval surrounding x = ____. At x = ____ the function has a relative maximum of _____.

42 Example 20: Identify the location and value of any relative maxima or minima of the function. At x = ____ the function has a relative minimum of _____. At x = ____ the function has a relative maximum of _____.

43 Example 21: Identify the location and value of any relative maxima or minima of the function.

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