September 08, 2014
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September 08, 2014
Section 1.3
Graphs of Functions
Objective: Analyze the graphs of functions.
Important Vocabulary
Graph of a function
The collection of ordered pairs ( x, f(x)) such that x is in the domain of f.
Graph same as equations in two variables since y=f(x).
Greatest integer function
The greatest integer function is denoted by [|x|] and is defined as the greatest integer less
than or equal to x.
Use "int(" to graph
Step function
A function whose graph resembles a set of stair steps.
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Exa
Even function
A function y = f(x) is even if, for each x in the domain of f, f(– x) = f(x).
Odd function
A function y = f(x) is odd if, for each x in the domain of f, f(– x) = – f(x).
I. The Graph of a Function
Explain the use of open or closed dots in the graphs of functions.
The use of dots, open or closed, at the extreme left or right points of a
graph indicates that the graph does not extend beyond these points.
If no such dots are shown, assume that the graph extends beyond these
points.
To find the domain of a function from its graph, . . .
examine the graph to see for which x-values the graph is plotted.
To find the range of a function from its graph, . . .
examine the graph to see for which y-values the graph is plotted.
The Vertical Line Test for functions states . . .
that 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.
September 08, 2014
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II. Increasing and Decreasing Functions
Graph f(x)=x3+3x2
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).
Given a graph of a function, to find an interval on which the function is increasing . . .
search for an interval over which the graph rises from left to right.
Given a graph of a function, to find an interval on which the function is decreasing. . .
search for an interval over which the graph falls from left to right.
Given a graph of a function, to find an interval on which the function is constant . . .
search for an interval over which the graph is level.
September 08, 2014
III. Relative Minimum and Maximum Values
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).
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).
The point at which a function changes from increasing to decreasing is a relative
maximum .
The point at which a function changes from decreasing to increasing is a relative
minimum .
To approximate the relative minimum or maximum of a function using a graphing utility, . . .
use the zoom and trace features to approximate the lowest or highest point on the
graph, OR use a built-in program or feature for finding minimum or maximum values.
Ex: Suppose a function C represents the annual number of cases (in millions) of
chicken pox reported for the year x in the United States from 1960 through 2000.
Interpret the meaning of the function’s minimum at (1998, 3).
The lowest number of cases of chicken pox reported during this period was 3
million cases in 1998.
Ex: The profit for a new company can be modeled by
P=0.225x3 - 17.21x2 + 315x + 132.1
Graph the function and use the maximum command to find the relative max.
Should be approximately $1,822,680.
September 08, 2014
IV. Graphing Step Functions and Piecewise-Defined Functions
The Greatest Integer Function
The graph of the greatest integer function jumps vertically one unit at each integer and is
constant (a horizontal line segment) between each pair of consecutive integers.
Ex:
Let f(x)= [ x ], the greatest integer function. Find f(3.74).
Graph MATH: NUM 5: int(
Note: change MODE from Connected to Dot and ZOOM in
Answer: 3
Piecewise-Defined Function
sketch the graph of each equation of the piecewise-defined function over the appropriate
portion of the domain.
Ex: Sketch the graph of f(x)=
x+1, if x < 2
f(x)= 1- x, if x > 2
x+1, if x < 2
1- x, if x > 2
September 08, 2014
V. Even and Odd Functions
A graph is symmetric with respect to the y-axis if, whenever (x, y) is on the graph,
(- x, y) is also on the graph.
A graph is symmetric with respect to the x-axis if, whenever (x, y) is on the graph,
(x, - y) is also on the graph.
A graph is symmetric with respect to the origin if, whenever (x, y) is on the graph,
(- x, - y) is also on the graph.
even
odd
A function whose graph is symmetric with respect to the y-axis is a(n)
A function whose graph is symmetric with respect to the origin is a(n)
function.
function.
The graph of a (nonzero) function cannot be symmetric with respect to the x-axis .
So...
even ---> f(-x)=f(x)
odd ---> f(-x)=-f(x)
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September 08, 2014
Assignment:
p. 38 (2-26e, 32, 33, 41-43, 48, 50, 54, 66,
68, 82, 87, 89, 90)