Section 2.2 The Graph of a Function

The graph of f(x) is given below (0, -3) (2, 3) (4, 0) (10, 0) (1, 0) x y

What is the domain and range of f ? Find f(0), f(4), and f(12) Does the graph represent a function? For which x is f(x)=0? For what numbers is f(x) < 0? For what value of x does f(x) = 3? What are the intercepts (zeros)? How often does the line x = 1 intersect the graph?

Section 2.3 Properties of Functions

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

A function f is increasing on an open interval I if, for any choice of x 1 and x 2 in I, with x 1 < x 2, we have f(x 1 ) < f(x 2 ). A function f is decreasing on an open interval I if, for any choice of x 1 and x 2 in I, with x 1 f(x 2 ). 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.

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

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.

Referring to the previous example, find all local maximums and minimums of the function: (hint: must be over an interval) (0, -3) (2, 3) (4, 0) (10, -3) (1, 0) x y (7, -3)

Extreme Value Theorem: If a function f whose domain is a closed interval [a, b], then f has an absolute maximum and an absolute minimum on [a, b]. Note: An absolute max/min may also be a local max/min.

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

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.

x - c f(x) - f(c) (x, f(x)) (c, f(c)) Secant Line y = f(x) 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).

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.

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. If t = 2, the average rate of change between 1 second and 2 seconds is: -4(4(2) - 21) = 52 ft/second. Average Rate of Change between 1 second and t seconds is: -4(4t - 21)

Section 2.4 Library of Functions; Piecewise-Defined Functions

When functions are defined by more than one equation, they are called piecewise- defined functions. Example: The function f is defined as: a.) Find f (1)= 3 Find f (-1)= (-1) + 3 = 2 Find f (4)= - (4) + 3 = -1

b.) Determine the domain of f Domain: in interval notation or in set builder notation c.) Graph f x y

d.) Find the range of f from the graph found in part c. Range: in interval notation or in set builder notation x y