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Graphing a Function Rule
Section 4-4
Goals
Goal
• To graph equations that
represent functions.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Continuous graph
• Discrete graph
Definition
• Some graphs are connected lines or curves called
continuous graphs. Some graphs are only distinct
points. They are called discrete graphs.
• Example discrete graphs:
– The graph on theme park attendance is an example
of a discrete graph. It consists of distinct points
because each year is distinct and people are
counted in whole numbers only. The values
between whole numbers are not included, since
they have no meaning for the situation.
• Example continuous graphs:
– The graph of a car approaching a traffic light is an
example of a continuous graph. It consists of
continuous line and all the points on the line,
because any point on the line has meaning.
Car Approaching Traffic Light
Example:
Henry begins to drain a water tank by opening a valve. Tell whether
the graph is continuous or discrete.
Water Level
Water tank
The graph is continuous.
Time
Example:
Jamie is taking an 8-week keyboarding class. At the end of each
week, she takes a test to find the number of words she can type
per minute. Tell whether the graph is continuous or discrete.
The graph is discrete.
Your Turn:
A small bookstore sold between 5 and 8 books each
day for 7 days. Tell whether the graph is continuous or
discrete.
The graph is discrete.
Your Turn:
A truck driver enters a street, drives at a constant speed, stops at a
light, and then continues. Tell whether the graph is continuous or
discrete.
y
Speed
The graph is continuous.
Time
x
Graphing Linear Functions
Many stretches on the German
autobahn have a speed limit of 120
km/h. If a car travels continuously
at this speed, y = 120x gives the
number of kilometers y that the car
would travel in x hours.
Notice that the graph is a straight line. An equation whose graph forms
a straight line is a linear equation. Also notice that this is a function. A
function represented by a linear equation is a linear function.
For any two points, there is exactly one line that contains them both.
This means you need only two ordered pairs to graph a line.
However, graphing three points is a good way to check that your line
is correct.
Graphing
Procedure: Graphing Functions
Step 1
Use the function to generate ordered pairs by
choosing several values for x.
Step 2
Plot enough points to see a pattern for the graph.
Step 3
Connect the points with a line or smooth curve.
Example: Graphing Linear
Functions
Graph y = 2x + 1.
Step 1 Choose three values of x and generate
ordered pairs.
x
1
y = 2x + 1
y = 2(1) + 1 = 3

(x, y)

(1, 3)
0
y = 2(0) + 1 = 1
(0, 1)
–1
y = 2(–1) + 1 = –1
(–1, –1)
Step 2 Plot the points and connect them with a straight line.

Example: Graphing Linear
Functions
Graph 15x + 3y = 9.
Step 1 Choose three values of x and generate
ordered pairs
x
y = –5x + 3
(x, y)
1
y = –5(1) + 3 = –2
(1, –2)
0
y = –5(0) + 3 = 3
(0, 3)
–1
y = –5(–1) + 3 = 8
(–1, 8)
Step 2 Plot the points and connect them with a straight line.



Your Turn:
Graph y = 4x.
Step 1 Choose three values of x and generate
ordered pairs
x
y = 4x

(x, y)
1
y = 4(1) = 4
(1, 4)
0
y = 4(0) = 0
(0, 0)
–1
y = 4(–1) = –4
(–1, –4)
Step 2 Plot the points and connect them with a straight line.


Your Turn:
Graph y + x = 7.
Step 2 Choose three values of x and generate
ordered pairs
x
y = –x + 7
(x, y)
1
y = –(1) + 7 = 6
(1, 6)
0
y = –(0) + 7 = 7
(0, 7)
y = –(–1) + 7 = 8
(–1, 8)
–1
Step 3 Plot the points and connect them with a straight line.



Graphing Real-World
Functions
• In many real-world situations, the x and y values must be
restricted. For example, quantities such as time, distance,
and number of people can be represented using only
nonnegative values.
• Sometimes x and y values are restricted even further to a
set of points. For example, a quantity such as number of
people can only be whole numbers. When this happens, the
graph is not actually connected because every point on the
line is not a solution. However, you may see these graphs
shown connected to indicate that the linear pattern, or
trend, continues.
Example: Application
The relationship between human years and dog years is given by
the function y = 7x, where x is the number of human years.
Graph this function.
Choose several values of x and make a
table of ordered pairs.
x
y = 7x
(x, y)
2
y = 7(2) = 14
(2, 14)
4
y = 7(4) = 28
(4, 28)
6
y = 7(6) = 42
(6, 42)
The ages are continuous
starting with 0.
Example: Continued
Graph the ordered pairs.
Human Years vs. Dog Years

(6, 42)

(4, 28)

(2, 14)
Any point on the line is
a solution in this
situation. The arrow
shows that the trend
continues.
Your Turn:
At a salon, Sue can rent a station for $10.00 per day
plus $3.00 per manicure. The amount she would pay
each day is given by f(x) = 3x + 10, where x is the
number of manicures. Graph this function.
Your Turn: Solution
Choose several values of x and make a table of
ordered pairs.
x f(x) = 3x + 10
0 f(0) = 3(0) + 10 = 10
1 f(1) = 3(1) + 10 = 13
2 f(2) = 3(2) + 10 = 16
3 f(3) = 3(3) + 10 = 19
4 f(4) = 3(4) + 10 = 22
5 f(5) = 3(5) + 10 = 25
The number of manicures
must be a whole number.
Your Turn: Solution
Graph the ordered pairs.
The individual
points are solutions
in this situation.
The line shows that
the trend continues.
Example: Graphing Nonlinear
Functions
Graph the function g(x) = |x| + 2.
Step 1 Choose several values of x and generate ordered pairs.
x
g(x) = |x| + 2
(x, g(x))
–2
g(x) = |–2| + 2= 4
(–2, 4)
–1
g(x) = |–1| + 2= 3
(–1, 3)
0
g(x) = |0| + 2= 2
(0, 2)
1
g(x) = |1| + 2= 3
(1, 3)
2
g(x) = |2| + 2= 4
(2, 4)
3
g(x) = |3| + 2= 5
(3, 5)
Example: Continued
Graph the function g(x) = |x| + 2.
Step 2 Plot enough points to see a pattern.
Example: Continued
Graph the function g(x) = |x| + 2.
Step 3 The ordered pairs appear to
form a v-shape. Draw lines through
all the points to show all the ordered
pairs that satisfy the function. Draw
arrowheads on the “ends” of the “v”.
Your Turn:
Graph the function y = |x – 1|.
Step 1 Choose several values of x and generate ordered pairs.
x
y = |x – 1|
(x, y)
–2
y = |–2 – 1| = 3
(–2, 3)
–1
y = |–1 – 1| = 2
(–1, 2)
0
y = |0 – 1| = 1
(0, 1)
1
y = |1 – 1| = 0
(1, 0)
2
y = |2 – 1| = 1
(2, 1)
Continued
Graph the function y = |x – 1|.
Step 2 Plot enough points to see a pattern.
Continued
Graph the function y = |x – 1|.
Step 3 The ordered pairs
appear to form a V-shape.
Draw a line through the
points to show all the
ordered pairs that satisfy the
function. Draw arrowheads
on both “ends” of the “V”..
Your Turn:
Graph the function y = x2 + 1.
Solution:
y = x2 + 1
x
–2
–1
0
1
2
x2 + 1
(–2)2 + 1
y
5
(–1)2 + 1
(0)2 + 1
(1)2 + 1
2
1
2
(2)2 + 1
5
Plot the
points
and
connect
them with
a smooth
curve.
Joke Time
• What do you call a man who lives in an envelope?
• Bill.
• What dog can't bark?
• A hot dog.
• What do you call a monkey on a mine field?
• A Ba-Boom.
Assignment
• 4-4 Exercises Pg. 274 – 276: #10 – 40 even