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. x>0 x<0 f(1) = 1 f(-1) = -1 f(2) = ½ f(-2) = -½ f(3) = 1/3
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
Example: Find the inverse of f(x) = x 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
a b.

23. Solve the following problems:
a b.

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