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College Algebra Chapter 2 Functions and Graphs

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

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College Algebra Chapter 2 Functions and Graphs

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  1. College AlgebraChapter 2Functions 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 xresults in equivalent equation. Symmetric with respect to the x-axis: Substituting –yfor yresults in equivalent equation. Symmetric with respect to the origin: Substituting –x for xand –yfor yresults 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.

  8. 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

  9. 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)

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

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

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

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

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

  15. 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

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

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

  18. Example 12: Graph the function.

  19. Example 13: Graph the function.

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

  21. Example 14: Evaluate.

  22. Example 15: Graph.

  23. 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.

  24. 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

  25. Investigate Increasing, Decreasing, and Constant Behavior of a Function Increasing Decreasing Constant

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

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

  28. 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

  29. Determine Relative Minima and Maxima of a Function

  30. 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 _____.

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

  32. 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 minimum of _____. At x = ____ the function has a relative maximum of _____.

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

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