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Chapter 3 – The Nature of Graphs

Chapter 3 – The Nature of Graphs. 3.1 Symmetry. Point symmetry Line symmetry. Identify the point of symmetry for both of these functions:. A function f(x) has a graph that is symmetric to the origin if and only if f(-x) = -f(x).

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Chapter 3 – The Nature of Graphs

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  1. Chapter 3 – The Nature of Graphs

  2. 3.1 Symmetry • Point symmetry • Line symmetry

  3. Identify the point of symmetry for both of these functions:

  4. A function f(x) has a graph that is symmetric to the origin if and only if f(-x) = -f(x).

  5. Example: Determine whether the graph of f(x) = x5 is symmetric with respect to the origin. Identify the following line symmetry:

  6. Identify the following line symmetry:

  7. Functions whose graphs are symmetric with respect to the y-axis are even functions. Functions whose graphs are symmetric with respect to the origin are odd functions. Even functions Odd functions

  8. 3.2 Families of Graphs Def: A parent graph is an anchor graph from which other graphs in the family are derived.

  9. What is this function and how does it act?

  10. Ex: Graph: and Ex: Graph:

  11. Graph: Graph:

  12. Graph: Graph:

  13. Types of transformations and how we use them:

  14. 3.3 Inverse Functions and Relations Def: Two relations are inverse relations if and only if one relation contains the element (a,b), whenever the other relation contains the element (b,a). Example: Graph of f(x) = x3 and its inverse.

  15. Example: Find the inverse of f(x) = x2 + 2. Then graph f(x) and its inverse. You can use the horizontal line test to determine if the graph of the inverse of a function is also a function.

  16. Ex: Sketch the graphs of the following: a. b.

  17. 3.4 Rational Functions and Asymptotes Vertical asymptote: The line x = a is a vertical asymptote for a function f(x) if or as from either the left or the right. Horizontal asymptote: The line y = b is a horizontal asymptote for a function f(x) if as or as . Slant asymptote: The oblique line l is a slant asymptote for a function f(x) if the graph of the f(x) approaches l as or as .

  18. Example: Determine the asymptotes for the graph of the following: a. b.

  19. Use the parent graph to graph the following a. b. c. d.

  20. Ex: Determine the slant asymptote for Whenever the denominator and numerator of a rational function contain a common factor, a hole may appear in the graph of the function. Ex: Graph

  21. Ex: Graph

  22. 3.5 Graphs of Inequalities Graph: a. b.

  23. Solve the following problems: a. b.

  24. Maximum: Minimum: Point of inflection: Continuous:

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