3.2 Graphs of Functions

MATH 1324 – Business College Algebra
3.2 Graphs of Functions
Chapter 3 Functions and Graphs

Concepts and Objectives
The objectives for this section are
• Connect functions to their graphs
• Graph piecewise-defined functions
• Graph absolute value functions
• Graph greatest-integer functions
• Interpret graphs
• Use the vertical line test to determine a function

Graphs of Functions
• The graph of a function f is defined to be the graph of the
equation y = f(x).
• It consists of all points (x, f(x)) – that is, every point whose
first coordinate is an input number from the domain of f
and whose second coordinate is the corresponding output
number from the function.

Graphs of Functions (cont.)
• For the most part, we use f (x) and y interchangeably to
denote a function of x, but there are some subtle
differences.
• y is the output variable, while f (x) is the rule that
produces the output variable.
• An equation with two variables, x and y, may not be a
function at all.
Example: is a circle, but not a function
2 2
4
x y
+ =

Graphs of Functions (cont.)
• Example: The graph of the function g(x) = .5x – 3 is the
graph of the equation y = .5x – 3.
This equation is in slope-intercept
form, which means we can plot
the y-intercept (0, –3), and then
count the slope (up 1, over 2) to
find the second point (2, –2).
We could also have found the
x-intercept at (6, 0).

Graphs of Piecewise Functions
• Example: Graph the following function:
The graph of f consists of:
the part of the line y = x +1 where x  2 and
the part of the line y = –2x + 7 where x > 2.
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• Example: Graph the following:
• For the first equation, plot the y-
intercept and count the slope to
find the second point.
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• For the first equation, plot the y-
intercept and count the slope to
find the second point.
• Now, draw a line through the two
points, extending it until you
reach x = 2. End your line with a
closed circle.
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• If we evaluate the second
equation at x = 2, we get (2, 3),
which is the same point on our
first equation (this doesn’t
always happen).
• Use the slope (–2) to find the
second point.
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• Draw the line through the two
points to complete the graph.
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• We can also use technology to graph the function. I
prefer using Desmos. You control the interval being
graphed by using braces:
( )
1 if 2
2 7 if 2
x x
f x
x x
+ 

= 
− + 


• Using Desmos, we list the two parts.
( )
2 if 3
8 if 3
x x
f x
x x
− 

= 
− + 


• Unlike the last problem, the two lines do not intersect.
This means we have to indicate which line actually
includes x = 3. To do this, we use open and closed
circles.
• For the first equation, 3 – 2 = 1, so (3, 1) is closed.
• For the second, –(3) + 8 = 5, so (3, 5) is open.
( )
2 if 3
8 if 3
x x
f x
x x
− 

= 
− + 


( )
2 if 3
8 if 3
x x
f x
x x
− 

= 
− + 

Use the setup wheel to
change the color and
type of circle.

( )
2 if 3
8 if 3
x x
f x
x x
− 

= 
− + 


• Click “Done” to finish
( )
2 if 3
8 if 3
x x
f x
x x
− 

= 
− + 


Absolute-Value Function
• The absolute-value function is an example of a piecewise
function. It’s defined as
and looks like this:
( )
if 0
if 0
x x
f x x
x x


= = 
− 


Writing Piecewise Functions
• Example: The Birmingham Water Works charges its
residential customers with a ⅝-inch water line a basic
service charge of $25 per month. The first 15 hundred
cubic feet (CCF) of water cost $2 per CCF. To encourage
water conservation, excessive water usages is charged at
a higher rate of $4 per CCF above 15.
a) How much would a customer who used 10 CCF of
water pay?

a) How much would a customer who used 10 CCF of
water pay?
25 + 2(10) = $45

b) How much would a customer who used 20 CCF of
water pay?

b) How much would a customer who used 20 CCF of
water pay?
25 + 2(15) + 4(5) = $75

c) Find a rule for the function f(x) that gives the amount
of the water bill when x CCF of water are used.

For the higher rate, notice that 25 + 2(15) = 55.

( )
( )
25 2 if 0 15
55 4 15 if 15
x x
f x
x x
+  


= 
+ − 



d) Graph the function.

Greatest-Integer Function
• The greatest-integer function, usually written
is defined by saying that [x] denotes the largest integer
that is less than or equal to x.
• Be careful with negatives! Notice that –3 is less than –2.6!
( ) ,
f x x
=  
 
[8] 8
[7.45] 7
[] 3
[-1] -1
[-2.6] -3

Greatest-Integer Function
• Here’s another way to look at it:
• Here’s how the graph looks:
x –2  x  –1 –1  x  0 0  x  1 1  x  2
[x] –2 –1 0 1
The graph should
demonstrate why
this is an example of
a step function.

Step Functions
• Example: To mail a letter to Canada in 2016, the U.S. Post
Office charged $1.20 for the first two ounces, an
additional $0.47 for an additional ounce, and then an
additional $0.49 up to a maximum weight of 3.5 ounces.
If L(x) represents the cost of sending a letter weighing x
ounces, graph L(x) for x in the interval (0, 3.5].

Step Functions
• Example: To mail a letter to Canada in 2016, the U.S. Post
Office charged $1.20 for the first ounce, an additional
$0.47 for an additional two ounces, and then an additional
$0.49 up to a maximum weight of 3.5 ounces. If L(x)
represents the cost of sending a letter weighing x ounces,
graph L(x) for x in the interval (0, 3.5].
• Let’s figure out our steps:
0 1: $1.20
1 3: $1.20 $0.47 $1.67
3 3.5: $1.67 $0.49 $2.16
x
x
x
 
  + =
  + =

Step Functions
• This would give us a function that looked like:
• And a graph of
( )
1.2 if 0 1
1.67 if 1 3
2.16 if 3 3.5
x
L x x
x
 


=  

  


Reading and Interpreting Graphs
• Graphs are often used in business and the social sciences
to present data (recall the NASDAQ graph from last class).
It is just as important to know how to read such graphs as
it is to construct them.

• Example: The graph below shows the median sales prices
(in thousands of dollars) of new privately owned, one-
family houses by region of the United States (Northeast,
Midwest, South, and West) for the years 2008 to 2015.
a) How do prices in the
West compare with
those in the Midwest?

a) How do prices in the West compare with those in the
Midwest?
The houses in the West started out about $100,000 higher than
those in the Midwest. In both regions, prices declined initially,
but they rebounded. By
2012, the gap between the
two regions had shrunk to
less than $50,000, but
then widened again to
about $75,000 by 2015.

b) Which region typically had the highest prices?

a) Which region typically had the highest prices?
The Northeast had the highest median sales price for all years
shown.

• Example: The average price f(x) of a barrel of crude oil (in
dollars) in year x is shown for the years 2005 to 2015, with
projected prices shown through the year 2025.

a) Estimate the price of crude in 2015 and 2025.

a) Estimate the price of crude in 2015 and 2025.
2015: $50/barrel 2025: $80/barrel

b) During what period did oil prices drop below $40 per
barrel?

b) During what period did oil prices drop below $40 per
barrel?
2016

Vertical Line Test
• No vertical line intersects the graph of a function y = f(x) at
more than one point.
• Notice the difference:
Not a function Function

By Next Class
• 3.2 Graphs of Functions in MyMathLab
• Quiz 3.2 in Canvas
• Notes on Section 3.3

3.2 Graphs of Functions

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