1. 4.8 Symmetry, IVT and Number line sign studies for composite trig functions

2. Recall the definitions of even/odd functions:
If f is an even function, then it’s graph is symmetric with respect to the y-axis and f(-x)=f(x).
If f is an odd function, then it’s graph is symmetric with respect to the origin and f(-x)= -f(x).

3. Evaluate f(-x) and determine if each function is even, odd or neither.

4. Recall: The Intermediate Value Theorem (IVT) p.206 in Pre-Calc Text

5. Making Sense of the IVT
Think of the Intermediate Value Theorem as “crossing a river.” In the picture below, if you are walking on a continuous path from f(a) to f(b), and there is a river across your path at the horizontal line y=y0 , then you would have to cross the river to reach your destination.

6. Use the Intermediate value Theorem to determine if a zero must exist on the interval:
Note: the fact that the IVT does not guarantee a zero does not mean that one does not exist in the interval. For instance, check f(π/2) in number 2.

7. Example 1: Answer the following questions about on [0, 2π].
What are the zeros of f ?
Describe the symmetry of f.
Do a number line sign study for f and use interval notation to identify where f > 0.

8. Example 3: Answer the following questions about on [0, 4π].
What are the zeros of f ?
Do a number line sign study for f .
Identify the intervals for which f < 0.

9. Assignment A4.8, Sections I, II and III to be completed by Monday
Test #11 will be at the end of this week and includes Polar Equations and Complex Numbers.
See you Tmrrw!!