4.8 Symmetry, IVT and Number line sign studies for composite trig functions
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).
Evaluate f(-x) and determine if each function is even, odd or neither.
Recall: The Intermediate Value Theorem (IVT) p.206 in Pre-Calc Text
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.
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.
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.
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.
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 Tmrrw!!