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1.5 Analyzing Graphs of a Function

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1.5 Analyzing Graphs of a Function Objective Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which functions are ... – PowerPoint PPT presentation

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Title: 1.5 Analyzing Graphs of a Function


1
1.5 Analyzing Graphs of a Function
2
Objective
  • 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

3
The 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

4
Example 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.

5
Vertical 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.

6
Example 2 Use the Vertical Line Test to decide
whether the graphs represent y as a function of
x. One y for every x.
7
Two ys for every x.
8
One y for every x. This is a piecewise function.
There are two pieces of functions.
9
Zeros of a Function
  • The zeros of a function f of x are the x-values
    for which f(x) 0.

10
Example 3 Finding the zeros of a Function
  • Examples on next slide.

11
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12
  • Graph

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15
Increasing 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).

16
Example 4 Increasing and Decreasing Functions
17
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19
Relative 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

20
Relative 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

21
Finding 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)

22
Using the calculator to find local maxima and
local minima
  • Find the local maxima and minima for

2nd Trace
Select maximum, enter
23
Place 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

26
Average 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

29
From
30
Even 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.

31
Tests 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).

32
Example 9 Even and Odd Functions
  • Determine whether the following functions are
    even, odd, or neither.

No y-axis or origin symmetry
33
Origin symmmetry
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