Do Now

1. Do Now • Determine the end behavior of the following functions.

2. Objectives • 24.September.2012 • Scholars will be able to… • Determine intervals on which functions are increasing, constant, or decreasing, and determine maxima and minima of functions. • Determine the average rate of change of a function.

3. Increasing and Decreasing • On an open interval for a function : • If for every and such that , then , is increasing. • If for every and such that , then , is decreasing. • If for every and such that , then , is constant.

4. Example • Graph it what does it look like? • Use a table. • Decreasing on the interval • This refers to the -axis

5. Example • Graph it what does it look like? • Use a table. • Increasing on the interval • Decreasing on the interval • Increasing on the interval • What is the function doing at and ?

6. Czech for Understanding

7. In Pears • Answer the question: • How do we estimate the location of the real -intercepts (zeroes) of a continuous function? • Why is this true?

8. Extrema • Extremaare the critical points of a function at which a function change its increasing or decreasing behavior. • Extrema is the plural of extremum (Latin for extreme.) • 1 maximum. • 2 maxima. • 1 minimum. • 2 minima.

9. Relative Extrema • For a function on an open interval . • is a relative maximum of if • is a relative minimum of if

10. Examples

11. Average Rate of Change • The average rate of change of a function on the closed interval is the slope of the secant line between those two points. • Positive average rates of change mean the graph increases on average over that interval. • Negative… I bet you can figure that one out.

12. Example • Find the average rate of change between of

13. Exit Ticket • Given the function , which of the following intervals has the smallest average rate of change?