Graphing Functions
Additional Example 1: Finding Solutions of
Equations with Two Variables
Use the given x-values to write solutions
of the equation y = 4x + 2 as ordered
pairs. x =1, 2, 3, 4.
Make a function table by
Write these
using the given values for x
solutions as
to find values for y.
ordered pairs.
x
1
4x + 2
4(1) + 2
y
6
(x, y)
2
4(2) + 2
10
3
4
4(3) + 2
4(4) + 2
14
18
(2, 10)
(3, 14)
(1, 6)
(4, 18)
12-2 Course 1: Graphing Functions
Learn to represent linear functions using
ordered pairs and graphs.
Try This: Example 1
Use the given x-values to write solutions
of the equation y = 3x + 2 as ordered
pairs. x = 2, 3, 4, 5.
Make a function table by
using the given values for x
to find values for y.
x
2
3
3x + 2
3(2) + 2
3(3) + 2
y
8
11
4
5
3(4) + 2
3(5) + 2
14
17
Write these
solutions as
ordered pairs.
(x, y)
(2, 8)
(3, 11)
(4, 14)
(5, 17)
Insert Lesson Title Here
Additional Example 2: Checking Solutions of
Equations with Two Variables
Check if an ordered pair is a
solution of an equation by putting
the x and y values into the
equation to see if they make it a
true statement.
Determine whether the ordered pair is a
solution to the given equation.
(3, 21); y = 7x
y = 7x
?
21 = 7(3)
Write the equation.
Substitute 3 for x and 21 for y.
?
21 = 21 
So (3, 21) is a solution to y = 7x.
1
Insert Lesson Title Here
Try This: Example 2
Determine whether the ordered pair is a
solution to the given equation.
(4, 20); y = 5x
y = 5x
Write the equation.
?
20 = 5(4)
?
20 = 20
Substitute 4 for x and 20 for y.

You can also graph the solutions of
an equation on a coordinate plane.
When you graph the ordered pairs
of some functions, they form a
straight line. The equations that
express these functions are called
linear equations.
So (4, 20) is a solution to y = 5x.
Insert Lesson Title Here
Additional Example 3: Reading Solutions
on Graphs
Use the graph of the linear function to find
the value of y for the given value of x.
x=4
Start at the origin and
move 4 units right.
y
4
2
x
-4
-2
0
2
4
-2
Move up until you
reach the graph. Move
left to find the y-value
on the y-axis.
When x = 4, y = 2.
The ordered pair is (4, 2).
-4
Insert Lesson Title Here
Try This: Example 3
Use the graph of the linear function to find
the value of y for the given value of x.
x=2
Start at the origin and
move 2 units right.
y
4
2
x
-4
-2
0
2
4
-2
Move up until you
reach the graph. Move
left to find the y-value
on the y-axis.
When x = 2, y = 4.
The ordered pair is (2, 4).
-4
Insert Lesson Title Here
Additional Example 4: Graphing Linear
Functions
Graph the function described by the
equation. y = –x – 2
Make a function table.
Write these solutions as
ordered pairs.
(x, y)
(–1, –1)
x
–x – 2
y
–1
–(–1) – 2
–1
0
–(0) – 2
–2
(0, –2)
1
–(1) – 2
–3
(1, –3)
Additional Example 4 Continued
Graph the ordered pairs on a coordinate plane.
y
5
4
3
2
1
-5 -4 -3 -2 -1 0 1 2
-1
-2
-3
-4
-5
x
3 4 5
Draw a line through the
points to represent all the
values of x you could have
chosen and the
corresponding values of y.
2
Insert Lesson Title Here
Try This: Example 4
Try This: Example 4
Graph the function described by the
equation. y = –x – 4
Graph the ordered pairs on a coordinate plane.
y
x
–x – 4
y
–1
–(– 1) – 4
–3
0
–(0) – 4
–4
1
–(1) – 4
–5
Write these solutions as
ordered pairs.
(x, y)
(–1, –3)
(0, –4)
(1, –5)
12-4 Course 2: Linear Functions
5
4
3
2
1
x
-5 -4 -3 -2 -1 0 1 2
-1
-2
-3
-4
-5
3 4 5
Draw a line through the
points to represent all the
values of x you could have
chosen and the
corresponding values of y.
The graph at right shows how far
an inner tube travels down a
river if the current flows 2 miles
per hour. The graph is linear
because all the points fall on a
line. It is part of the graph of a
linear equation.
y
6
Miles
Make a function table.
4
2
x
0
0
2
4
Hours
A linear equation is an equation whose graph is
a line. The solutions of a linear equation are the
points that make up its graph. Linear equations
and linear graphs can be different representations
of linear functions. A linear function is a function
whose graph is a non vertical line.
Additional Example 1A: Graphing Linear Functions
You need to know only two points to
draw the graph of a linear function.
However, graphing a third point
serves as a check. You can use a
function table to find each ordered
pair.
Graph the linear function.
A. y = 4x – 1
Input
Rule
x
4x – 1
0
1
–1
Output Ordered Pair
y
(x, y)
4(0) – 1
–1
(0, –1)
4(1) – 1
3
(1, 3)
4(–1) – 1
–5
(–1, –5)
3
Additional Example 1A Continued
Graph the linear function.
Graph each linear function.
A. y = 4x – 1
B. y = –1
The equation y = –1 is the same equation
as y = 0x – 1.
y
4
(1, 3)
2
x
–4 –2
–2
0
Additional Example 1B: Graphing Linear Functions
2 4
(0, –1)
Place each ordered pair
on the coordinate grid and
then connect the points
with a line.
–4
(–1, –5)
Rule
x
0x – 1
0
0(0) – 1
–1
(0, –1)
3
0(3) – 1
–1
(3, –1)
–2
0(–2) – 1
–1
(–2, –1)
Additional Example 1B Continued
Graph the linear function.
B. y = –1
2
x
0
(–2, –1) (0, –1)(3, –1)
–2
Place each ordered
pair on the coordinate
grid and then connect
the points with a line.
–4
Try This: Example 1A
Input
Graph the linear function.
A. y = 3x + 1
Output Ordered Pair
x
3x + 1
3(0) + 1
y
(x, y)
1
(0, 1)
1
3(1) + 1
4
(1, 4)
–1
3(–1) + 1
–2
(–1, –2)
Try This: Example 1B
Graph each linear function.
B. y = 1
The equation y = 1 is the same equation as
y = 0x + 1.
y
(1, 4)
4
–4
Rule
0
Try This: Example 1A Continued
0
2 4
–4 –2
–2 (–1, –2)
(x, y)
A. y = 3x + 1
y
(0, 1)
y
Graph the linear function.
4
2
Output Ordered Pair
Input
x
Place each ordered
pair on the coordinate
grid and then connect
the points with a line.
Output Ordered Pair
Input
Rule
x
0x + 1
y
0
0(0) + 1
1
(0, 1)
3
0(3) + 1
1
(3, 1)
–2
0(–2) + 1
1
(–2, 1)
(x, y)
4
Try This: Example 1B Continued
Additional Example 2: Earth Science Application
Graph the linear function.
B. y = 1
y
4
(–2, 1) 2(0, 1) (3, 1)
x
0
Place each ordered
pair on the coordinate
grid and then connect
the points with a line.
The fastest-moving tectonic plates on Earth
move apart at a rate of 15 centimeters per
year. Scientists began studying two parts of
these plates when they were 30 centimeters
apart. How far apart will the two parts be
after 4 years?
The function y = 15x + 30, where x is the
number of years and y is the spread in
centimeters.
–2
–4
Insert Lesson Title Here
Try This: Example 2
Additional Example 2 Continued
Dogs are considered to age 7 years for each
human year. If a dog is 3 years old today, how
old in human years will it be in 4 more years?
Write a linear equation which would show this
relationship. Then make a graph to show how
the dog will age in human years over the next
4 years.
y = 15x + 30
y
100
Input
Rule
Output
x
15(x) + 30
y
0
15(0) + 30
30
2
15(2) + 30
60
4
15(4) + 30
90
80
60
40
The function y = 7x + 21, would describe
this situation where x is the number of
years, 21 is the current age and y would be
the future age.
20
x
0
2
4
8
10
12
Insert Lesson Title Here
12-2 Graphing Functions
Try This: Example 2
Warm Up
Write an equation for each function. Tell what
each variable you use represents.
y = 7x + 21
y
Input
Rule
Output
x
7(x) + 21
y
0
7(0) + 21
21
2
7(2) + 21
35
4
7(4) + 21
49
80
60
40
20
0
x
2
4
8
10
1. The length of a wall is 4 ft more than
three times the height.
l = 3h + 4, where l is length and h is height.
2. The number of trading cards is 3 less
than the number of buttons.
c = b – 3, where c is the number of cards
and b is the number of buttons.
Course 1
5
12-2 Graphing Functions
Insert Lesson Title Here
Lesson Quiz
Problem of the Day
Steve saved $1.50 each week. How
many weeks did it take him to save
enough to buy a $45 skateboard?
30
1. Use the given x-values to write solutions as
ordered pairs to the equation y = –3x + 1 for x =
0, 1, 2, and 3. (0, 1), (1, –2), (2, –5), (3, –8)
2. Determine whether (4, –2) is a solution to the
equation y = –5x + 3. No, –2 ≠ –5(4) + 3
3. Graph the function described by the equation
y = –x + 3.
Course 1
6