Chapter 3 Section 3: Properties of Functions

1. Chapter 3 Section 3: Properties of Functions • In this section, we will… • Determine if a given function is even, odd or neither • Determine where a function is increasing, decreasing or constant • Find local maxima and minima

2. What symmetry is this? If you replace x with –x in the equation and an equivalent equation results, the function is even. 3.3 Determine if a Function is Even, Odd or Neither

3. What symmetry is this? If you replace x with –x in the equation and an equation which has one side negated results, the function is odd. 3.3 Determine if a Function is Even, Odd or Neither

4. Example: Determine graphically whether each function given is an even function, an odd function, or a function that is neither even nor odd. 3.3 Determine if a Function is Even, Odd or Neither

5. Example: Determine algebraically whether each function given is an even function, an odd function, or a function that is neither even nor odd. 3.3 Determine if a Function is Even, Odd or Neither

6. Example: Determine algebraically whether each function given is an even function, an odd function, or a function that is neither even nor odd. 3.3 Determine if a Function is Even, Odd or Neither

7. We talk about a function increasing over an open interval of x-values. Where is the function increasing? 3.3 Determine where a Function is Increasing, Decreasing or Neither

8. We talk about a function decreasing over an open interval of x-values. Where is the function decreasing? 3.3 Determine where a Function is Increasing, Decreasing or Neither

9. We talk about a function being constant over an open interval of x-values. Where is the function constant? 3.3 Determine where a Function is Increasing, Decreasing or Neither

10. Example: Find the intervals over which the function below is increasing, decreasing or constant. increasing: decreasing: constant: 3.3 Determine where a Function is Increasing, Decreasing or Neither

11. Example: Find the intervals over which the function is increasing, decreasing or constant; consider the function over (-2, 5). increasing: decreasing: constant: What if we considered the function over its entire domain? 3.3 Determine where a Function is Increasing, Decreasing or Neither

12. What are the local maxima? At what values do the local maxima occur? 3.3 Finding Local Maxima and Minima

13. What are the local minima? At what values do the local minima occur? 3.3 Finding Local Maxima and Minima

14. Example: Find the local maxima and local minima of the function; determine where the function is increasing, where it is decreasing and where it is constant: Round answers to the nearest hundredth. local minimum: increasing: local maximum: decreasing: constant: 3.3 Properties of Functions

15. Example: The height h of a ball (in feet) thrown with an initial velocity of 80 feet per second from an initial height of 6 feet is given as a function of the time t (in seconds) by • Graph h • Determine the time at which the height is at its maximum. • What is the maximum height? • State the intervals over which the function is increasing and decreasing. • Describe what this means in the context of the problem. • increasing: • decreasing: • interpretation: 3.3 Properties of Functions

16. Independent Practice You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect. Read pages 231-238 Homework: pp. 238-241#21-49 odds, 65 3.3 Properties of Functions