Properties of Functions Section 3.3 1. 2 3 Even, Odd or Neither? 4

1

Properties of FunctionsSection 3.3

2

Even Functions: Notice here that .

3

Odd Functions: Notice that .

4

Even, Odd or Neither?

5

Identify Even/Odd/Neither From Equation1. Exercise 35 Page 238 2. Exercise 37 Page 238 3. Exercise 42 Page 238

6

Recall Interval NotationThe following interval contains 3 parts:

• Values less than or equal to -6 : • Values greater than -2 but less than or equal to 2: • Values greater than 4: Put them all together :

7

Increasing and Decreasing

8

Where is the graph increasing/decreasing?

9

Where is the function inc/dec?

10

Where is the function inc/dec?

11

Local Maxima and MinimaThese are the values where the graph reaches a high/low point.Endpoints such as A and E can never be local max/min.B and D are local minima.C is a local maximum.

12

Absolute Maxima and MinimaOver the entire domain of the graph, these are values that correspond to the largest or smallest value.A is the absolute maximum.D is the absolute minimum.

13

Example• There is a local minimum at The local minimum value is .• There is NOT a local maximum at Why?• There is an absolute minimum at . The absolute minimum value is .• There is no absolute maximum. Why?

14

Example• There is an absolute minimum at .• There is no absolute maximum.• There is not a local minimum.• There is not a local maximum.

15

Exercise 48 – Page 2381. Local max?2. Local min?3. Absolute max?4. Absolute min?

16

Exercise 49 – Page 2381. Local max?2. Local min?3. Absolute max?4. Absolute min?

17

Exercise 54 – Page 2385 Use the calculator to find any local max or min between and 3.

18

Exercise 57 – Page 238 Use the calculator to find any local max or min between and 4.

19

Average Rate of ChangeTwo points on the graph: and .The average rate of change is simply the slope of the line between these 2 points:

20

Secant LineThe line between 2 points on the graph is called the secant line.The equation is

The slope is the average rate of change.

21

Exercise 67 – Page 239 a) Find the average rate of change from to 1.b) Find the equation of the secant line from to .

22

Exercise 70 – Page 239 a) Find the average rate of change from to 3.b) Find the equation of the secant line from to .