Understanding Function Properties: Even, Odd, Increasing, Decreasing

1. Sullivan Algebra & Trigonometry: Section 3.3Properties of Functions • Objectives • Determine Even and Odd Functions from a Graph • Identify Even and Odd Functions from the Equation • Determine Where a Function is Increasing, Decreasing, or is Constant • Locate Maxima and Minima • Find the Average Rate of Change of a Function

2. A function f is even if for every number x in its domain the number -x is also in the domain and f(x) = f(-x). A function is even if and only if its graph is symmetric with respect to the y-axis. A function f is odd if for every number x in its domain the number -x is also in the domain and -f(x) = f(-x). A function is odd if and only if its graph is symmetric with respect to the origin.

3. y x (0,0) y x Example of an Even Function. It is symmetric about the y-axis Example of an Odd Function. It is symmetric about the origin (0,0)

4. Determine whether each of the following functions is even, odd, or neither. Then determine whether the graph is symmetric with respect to the y-axis or with respect to the origin. a.) Even function, graph symmetric with respect to the y-axis.

5. b.) Not an even function. Odd function, and the graph is symmetric with respect to the origin.

6. A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2). A function f is constant on an open interval I if, for any choice of x in I, the values of f(x) are equal.

7. y (2, 3) 4 (4, 0) 0 (1, 0) x (0, -3) (10, -3) (7, -3) -4 Determine where the following graph is increasing, decreasing and constant. Increasing on (0,2) Decreasing on (2,7) Constant on (7,10)

8. A function f has a local maximum at c if there is an interval I containing c so that, for all x in I, f(x) < f(c). We call f(c) a local maximum of f. A function f has a local minimum at c if there is an interval I containing c so that, for all x in I, f(x) > f(c). We call f(c) a local minimum of f.

9. y (2, 3) 4 (4, 0) 0 (1, 0) x (0, -3) (10, -3) (7, -3) -4 Referring to the previous example, find all local maximums and minimums of the function:

10. If c is in the domain of a function y = f(x), the average rate of change of fbetween c and x is defined as This expression is also called the difference quotient of f at c.

11. The average rate of change of a function can be thought of as the average “slope” of the function, the change is y (rise) over the change in x (run). y = f(x) Secant Line (x, f(x)) f(x) - f(c) (c, f(c)) x - c

12. Example: The function gives the height (in feet) of a ball thrown straight up as a function of time, t (in seconds). a. Find the average rate of change of the height of the ball between 1 and t seconds.

14. b. Using the result found in part a, find the average rate of change of the height of the ball between 1 and 2 seconds. Average Rate of Change between 1 second and t seconds is: -4(4t - 21) If t = 2, the average rate of change between 1 second and 2 seconds is: -4(4(2) - 21) = 52 ft/second.