1. SECTION 1.3 PROPERTIES OF FUNCTIONS PROPERTIES OF FUNCTIONS

2. EVEN AND ODD FUNCTIONS Identifying Graphically: g(x) = x 3 - 6x See the graph. Odd functions are symmetric with respect to the origin.

3. EVEN AND ODD FUNCTIONS Identifying Graphically: h(x) = 0.1x 4 - 2x 2 + 5 See the graph. Even functions are symmetric with respect to the y-axis.

4. EVEN AND ODD FUNCTIONS Identifying Algebraically: Odd function: f (- x) = - f (x) g(x) = x 3 - 6x g(- x) = (- x) 3 - 6(- x) = - x 3 + 6x = - g(x) = - x 3 + 6x = - g(x)

5. EVEN AND ODD FUNCTIONS Identifying Algebraically: Even function: f (- x) = f (x) h(x) = 0.1x 4 - 2x 2 + 5 h(- x) = 0.1(- x) 4 - 2(- x) 2 + 5 = 0.1x 4 - 2x 2 + 5 = 0.1x 4 - 2x 2 + 5 = h(x) = h(x)

6. EXAMPLE: Identify the following functions as even, odd, or neither. g(x) = x 3  x  h(x) = x 5 + 1 k(x) =  x 5  Use both graphical and algebraic methods of identification.

7. INCREASING AND DECREASING FUNCTIONS INCREASING FUNCTION A function is increasing when as x increases, y also increases. DECREASING FUNCTION A function is decreasing when as x increases, y decreases.

8. Determine the intervals on which the function is decreasing. Determine the intervals on which the function is decreasing. k(x) = x 3 - 6x 2 - 13x k(x) = x 3 - 6x 2 - 13x -.89 < x < 4.89 -.89 < x < 4.89 EXAMPLE:

9. When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then at c the value of f is called a local maximum of the function. When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c, then at c the value of f is called a local maximum of the function. LOCAL MAXIMUM

10. When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then at c the value of f is called a local minimum of the function. When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c, then at c the value of f is called a local minimum of the function. LOCAL MINIMUM

11. Use a graphing calculator to determine where f has a local maximum and a local minimum: Use a graphing calculator to determine where f has a local maximum and a local minimum: f(x) = 6x 3 - 12x + 5 for - 2 < x < 2 f(x) = 6x 3 - 12x + 5 for - 2 < x < 2 max: (- 0.81, 11.53) min: (0.81, -1.53) EXAMPLE:

12. AVERAGE RATE OF CHANGE The average rate of change for a function is a measure of the change in the y-coordinate as the corresponding x-coordinate changes. The average rate of change for a function is a measure of the change in the y-coordinate as the corresponding x-coordinate changes.

13. AVERAGE RATE OF CHANGE OF f BETWEEN c and x If c is in the domain of a function y = f(x), the average rate of change of f between c and x is defined as This expression is also called the difference quotient of f at c.

14. EXAMPLE: Suppose you drop a ball from a cliff 1000 feet high. You measure the distance s the ball has fallen after time t using a motion detector and obtain the data in Table 3. Suppose you drop a ball from a cliff 1000 feet high. You measure the distance s the ball has fallen after time t using a motion detector and obtain the data in Table 3.

15. SecondsFeet 0 0 1 16 2 64 3144 4256 5400 6576 7784

16. EXAMPLE Draw a scatter diagram of the data treating time as the independent variable. Draw a line from (0,0) to (2,64). Find the average rate of change between these two points.

17. EXAMPLE Rate of Change =

18. EXAMPLE Interpret the average rate of change found between the points. The ball is dropping at an average speed of 32 ft/sec between 0 and 2 seconds.

19. EXAMPLE Now draw a line from (5,400) to (7,784). Find the average rate of change between these two points.

20. EXAMPLE Rate of Change =

21. EXAMPLE Interpret the average rate of change found between the points. The ball is dropping at an average speed of 192 ft/sec between 5 and 7 seconds.

22. EXAMPLE What is happening to the average rate of change or the speed as time passes? It is increasing as time passes since the ball is accelerating due to the effect of gravity.

23. FINDING AVERAGE RATES OF CHANGE Do Examples 6 and 8

24. CONCLUSION OF SECTION 1.3 CONCLUSION OF SECTION 1.3