Title: 1.5 Analyzing Graphs of a Function
11.5 Analyzing Graphs of a Function
2Objective
- Use the Vertical Line Test for functions.
- Find the zeros of functions.
- Determine intervals on which functions are
increasing or decreasing and determine relative
maximum and relative minimum values of functions. - Determine the average rate of change of a
function. - Identify even and odd functions
3The Graph of a Function
- The graph of a function f is the collection of
ordered (x, f(x)) such that x is the domain of f. - x the directed distance from the y-axis
- y f(x) the directed distance from the x-axis
4Example 1 Finding the Domain and Range of a
Function
- Use the graph of the function f to find (a) the
domain of f, (b) the function values f(-1) and
f(2) and (c) the range of f.
5Vertical Line Test for Functions
- A set of points in a coordinate plane is the
graph of y as a function of x if and only if no
vertical line intersects the graph at more than
one point.
6Example 2 Use the Vertical Line Test to decide
whether the graphs represent y as a function of
x. One y for every x.
7Two ys for every x.
8One y for every x. This is a piecewise function.
There are two pieces of functions.
9Zeros of a Function
- The zeros of a function f of x are the x-values
for which f(x) 0.
10Example 3 Finding the zeros of a Function
11(No Transcript)
12 13(No Transcript)
14(No Transcript)
15Increasing and Decreasing Functions
- A function f is increasing on an interval if, for
any x1 and x2 in the interval, x1 lt x2 implies
f(x1) lt f(x2). - A function f is decreasing on an interval if, for
any x1 and x2 in the interval, x1 lt x2 implies
f(x1) gt f(x2). - A function f is constant on an interval if, for
any x1 and x2 in the interval, f(x1) f(x2).
16Example 4 Increasing and Decreasing Functions
17(No Transcript)
18(No Transcript)
19Relative Minimum
- A function value f(a) is called a relative
minimum of f if there exists an interval (x1,x2)
that contains a such that
20Relative Maximum
- A function value f(a) is called a relative
maximum of f if there exists an interval (x1,x2)
that contains a such that
21Finding Local Maxima and Local Minima from a Graph
- At what numbers, if any, does f have a local
maximum? x -2.5 - What are the local maxima? (-2.5, 5)
- At what numbers, if any, does f have a local
minimum? X 2.5 - What are the local minima?
- (2.5,4)
22Using the calculator to find local maxima and
local minima
- Find the local maxima and minima for
2nd Trace
Select maximum, enter
23Place cursor to left side of maximum enter
Place cursor to right side of maximum enter
Shows maximum at (-2.1, 4.06)
Guess, Enter
24- To find the minimum, do the same thing but select
minimum instead of maximum.
25- Find the local maxima and minima for
26Average Rate of Change
- For a nonlinear graph whose slope changes at each
point, the average rate of change between any two
points is the slope of the line through the two
points. - The line through the two points is called the
secant line.
27- Average rate of change of f from
28- Example 8 Find the average rates of change of
- a) from
29From
30Even and Odd Functions
- A function is said to be even if its graph is
symmetric with respect to the y-axis and a
function is said to be odd if its graph is
symmetric with respect to the origin.
31Tests for Even and Odd Functions.
- A function is even if, for each x in the domain
of f, f(-x) f(x). - A function is odd if, for each x in the domain
of f, f(-x) -f(x).
32Example 9 Even and Odd Functions
- Determine whether the following functions are
even, odd, or neither.
No y-axis or origin symmetry
33Origin symmmetry