Transcript 1.5 Analyzing Graphs of a Function

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
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
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 xaxis
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.
Domain  {x : x }
Range = { y : y  4}
f (1)  3
f (2)  4
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.
Example 2
Use the Vertical Line Test to
decide whether the graphs represent y as a
function of x. One y for every x.
Two y’s for every x.
One y for every x. This is a piecewise function.
There are two pieces of functions.
Zeros of a Function
• The zeros of a function f of x are the xvalues for which f(x) = 0.
Example 3
Finding the zeros of a Function
Examples on next slide.
f ( x)  2 x  13x  24
2
Set y = 0 and solve.
0  (2 x  3)( x  8) Factor
0  2 x  3 and 0  x  8
3  2x
8  x
3
x
2
• Graph
g ( x)  8  x 2
0  8  x2
0  8  x 2 Square both sides
8  x2
 8x
2.8  x
domain {x : 2.8  x  2.8} or [2.8, 2.8]
range { y : 0  y  2.8} or [0, 2.8]
2t  4
t 1
0  2t  4
h(t ) 
4  2t
2t
y  intercept
2(0)  4
0 1
y  4
domain {t : t  1}
y
range { y : y  2}
Increasing and Decreasing
Functions
• A function f is increasing on an interval if,
for any x1 and x2 in the interval, x1 < x2
implies f(x1) < f(x2).
• A function f is decreasing on an interval if,
for any x1 and x2 in the interval, x1 < x2
implies f(x1) > f(x2).
• A function f is constant on an interval if, for
any x1 and x2 in the interval, f(x1) = f(x2).
Example 4 Increasing and
Decreasing Functions
decreasing on the interval (-,3)
increasing on the interval (3,)
increasing on the interval (,-1)
decreasing on the interval (-1,1)
increasing on the interval (1,)
increasing on the interval (-,0)
decreasing on the interval (0, )
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
x1  x  x2 implies f (a )  f ( x).
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
x1  x  x2 implies f (a )  f ( x).
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)
Using the calculator to find local
maxima and local minima
• Find the local maxima and minima for
f ( x)  x  2 x  5 x  6
3
2
2nd Trace
Select maximum, enter
Place cursor to left
side of maximum
enter
Guess, Enter
Place cursor to right
side of maximum
enter
Shows maximum at (-2.1,
4.06)
• To find the minimum, do the same thing
but select minimum instead of maximum.
• Find the local maxima and minima for
4
2
f ( x)  x  2 x  1
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.
• Average rate of change of f from
f ( x2 )  f ( x1 )
x1 to x2 
x2  x1
changein y

changein x
 msec
• Example 8 Find the average rates of
change of
2
f ( x)  x  2 x
• a) from
x1  1 and x2  3
From
x1  0 and x2  1
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.
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).
Example 9 Even and Odd
Functions
• Determine whether the following functions
are even, odd, or neither.
f ( x)  2 x  1
3
No y-axis or origin symmetry
g ( x)  x  x
3
Origin symmmetry