Chapter 4: Graphing
Linear Equations and
Functions
Y=mx+b
X-axis
Y-axis
The Coordinate Plane
4
3
2
The x-axis is the
horizontal axis on
a coordinate plane
-5 -4
The origin occurs where
the x-axis and the
y-axis intersect at (0, 0)
1
-3
-2
1
2
3
4
-1 -1
-2
-3
-4
The y-axis is the vertical axis
on a coordinate plane
5
Quadrants of the Coordinate Plane
Coordinate planes are divided up into four quadrants:
4
3
2
Quadrant II
(--, +)
Quadrant I
(+, +)
1
1
X-axis
-5 -4
-3
Quadrant III
(--, --)
-2
2
3
4
5
-1 -1
-2
-3
-4
Y-axis
Quadrant IV
(+, --)
Ordered Pairs
An ordered pair consists of an x-coordinate and a y-coordinate,
usually surrounded by parenthesis.
The x-coordinate
is always the first
number.
(x, y)
The y-coordinate
is always the second
number.
Example: Plot the point (3, -2)
x
.
y
(3, -2)
Starting at the origin,
move 3 units right
and then
2 units down.
Linear Equations
A linear equation is written in the form Ax+By=C, where neither
A nor B are zero.
A solution of an equation is an ordered pair (x, y) that makes
the equation true.
Example: Determine whether (4, -2) is a solution of 2x+3y=2
Step 1: plug in 4 for x and -2 for y.
(Remember that x is the first number
in an ordered pair and y is the second.)
2(4)+3(-2)=2
Step 2: multiply and simplify.
(Be sure to pay attention to any negative signs.)
8+(-6)=2
2=2
Since both sides are equal, (4, -2) is a solution of 2x+3y=2
Finding Solutions of Linear
Equations
The first step when finding solutions is to REWRITE the equation.
Write original equation:
4x-2y=10
Move x to the other side of the equation:
-2y=-4x+10
Divide by -2 to get your rewritten equation: y=2x-5
The next step is to CHOOSE x-values for your solutions and make a
table.
x
-3
-2
-1
0
1
2
3
y
-11
-9 -7
-5
-3
-1
1
Then PLUG IN the values for x in the new equation to get the y-values
All of the ordered pairs on the table are solutions for the given equation.
Graphing Linear Equations Using
Tables
By finding solutions for an equation, you are essentially finding points
on the graph of that equation.
Example: Graph y-2x=-1 by using a table of values.
y=2x-1
Rewrite the equation:
Choose x-values, make a table,
and plug the x-values into the
equation to find the y-values:
x
-3
-2
-1
0
1
2
3
y
-7
-5
-3 -1
1
3
5
Plot the points on the
Coordinate plane and
connect the dots
.
.
.
.
.
.
.
Graphing Horizontal and Vertical Lines
Graphing an equation such as x=2 or y=-1 is actually easier than it seems.
When an equation is x=2, it simply means
Similarly, when an equation is y=-1,
that for every y-value, the x-value is 2,
it means that for every x-value, the
making it a VERTICAL line.
y-value is -1, making it a HORIZONTAL line
.
.
.
..
x
y
. ... .
y
x
X- and Y-Intercepts
An x-intercept is the x-coordinate of a point where a graph crosses the x-axis.
..
A y-intercept is the y-coordinate of a point where a graph crosses the y-axis.
A line that is neither horizontal nor vertical has exactly one x-intercept and one y-intercep
Finding Intercepts
An x-intercept occurs where y=0, so to find the x-intercept, plug in 0 for
y in the given equation.
Example: Find the x-intercept of 7y-3x=21
7(0)-3x=21
-3x=21
x=-7
The x-intercept for this equation is -7.
A y-intercept occurs where x=0, so you can plug in 0 for x in the
equation to find the y-intercept.
Example: Find the y-intercept of 7y-3x=21
7y-3(0)=21
7y=21
y=3
The y-intercept for this equation is 3.
Graphing Using Intercepts
Once you find the intercepts of an equation, you can graph that equation
by plotting the intercepts and connecting the two points.
Example: Plot the graph of the equation that has the
intercept points (3,0) and (0,-2).
.
.
Slope of a Line
To find the slope of a line, it’s necessary to divide the rise by the run.
Vertical rise=2
Horizontal run=4
Slope =
Vertical rise
Horizontal run
=
2
4
=
1
2
Finding Slope
.
(x2 , y2)
.
y2 -y1
(x1 , y1)
x2 -x1
The slope m of a line that passes through the points above is:
m=
rise
run
=
change in y
Change in x
=
y2 -y1
x2 -x1
Positive and Negative Slope
If a line moves down and to the right, then it has a negative slope.
Example: Let two points on a line be (1, 4) and (7, 3). Find the slope.
m=
3-4 =
7-1
-1
6
If a line moves up and to the right, then it has a positive slope.
Example: Let two points on a line be (6, -2) and (9, 7). Find the slope.
7-(-2)
m=
9-6
=
9
3
=3
Zero and Undefined Slope
The slope of any horizontal line is always zero.
Example: Let two points on a line be (5, -2) and (3, -2). Find the slope.
m=
-2-(-2)=
0
3-5
-2
Zero divided by any number is 0. Slope is 0.
The slope of any vertical line is always undefined.
Example: Let two points on a line be (3, -1) and (3, 4). Find the slope.
m=
4-(-1)
3-3
=
5
0
Cannot divide by zero, so slope is undefined.
Direct Variation
When two quantities y and x have a constant ratio k, they are said to have
direct variation.
If y/x = k, then y=kx.
Example: If variables x and y vary directly and one pair of values is
y=24 and x=3, write an equation that relates x and y.
y=kx
24=k(3)
8=k
Write the model for direct variation.
Plug in the x and y values.
Divide to find the value of k.
By plugging the value of k back into the original model, an equation
that relates x and y is y=8x.
Slope-Intercept Form
The linear equation y=mx+b is written in slope-intercept form,
where m is the slope and b is the y-intercept.
Example: Graph the equation y=2x-4.
Step 1: Recognize that the slope is 2 and the y-intercept is -4.
Step 2: Plot the point (0,b) where b is -4.
Step 3: Use the slope to locate a second point on the line.
m=
2
1
=
rise
run
Move 2 units down, 1 unit right.
Parallel Lines
Parallel lines are different lines in the same plane that never intersect.
Two nonvertical lines are parallel if they have the same slope and
different y-intercepts. Any two vertical lines are parallel.
The red lines are parallel to each other
because they have the same slope.
The blue lines are parallel to each
other because they are vertical.
Functions and Relations
A function is a rule that establishes a relationship between two quantities,
called the input and the output, where for each input,
there is exactly one output.
A relation is any set of ordered pairs.
A relation is a function if for every input there is exactly one output.
Example: Determine whether the relation is a function.
Input
1
2
3
4
Output
3
5
7
The relation is not a function because the input 1 has two outputs.
Vertical Line Test for Functions
A graph is a function if no vertical line intersects the graph at more than
one point.
Function
Not a function