Digital Lesson
Graphs of Equations
The graph of an equation in two variables x and y
is the set of all points (x, y) whose coordinates
satisfy the equation.
For instance, the point (–1, 3) is on the graph of
2y – x = 7 because the equation is satisfied when
–1 is substituted for x and 3 is substituted for y.
That is,
2y – x = 7
2(3) – (–1) = 7
7=7
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Original Equation
Substitute for x and y.
Equation is satisfied.
2
To sketch the graph of an equation,
1. Find several solution points of the equation
by substituting various values for x and
solving the equation for y.
2. Plot the points in the coordinate plane.
3. Connect the points using straight lines or
smooth curves.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
3
Example: Sketch the graph of y = –2x + 3.
1. Find several solution points of the equation.
x
–2
–1
0
1
2
y = –2x + 3
(x, y)
y = –2(–2) + 3 = 7 (–2, 7)
y = –2(–1) + 3 = 5 (–1, 5)
y = –2(0) + 3 = 3
(0, 3)
y = –2(1) + 3 = 1
(1, 1)
y = –2(2) + 3 = –1 (2, –1)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
Example: Sketch the graph of y = –2x + 3.
2. Plot the points in the coordinate plane.
x
y
(x, y)
–2
–1
7
5
(–2, 7)
(–1, 5)
0
3
(0, 3)
1
2
1
–1
(1, 1)
(2, –1)
y
8
4
x
4
4
8
–4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
5
Example: Sketch the graph of y = –2x + 3.
3. Connect the points with a straight line.
y
8
4
x
4
4
8
–4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
6
Example: Sketch the graph of y = (x – 1)2.
x
–2
–1
0
1
2
y
9
4
1
0
1
(x, y)
(–2, 9)
(–1, 4)
(0, 1)
(1, 0)
(2, 1)
3
4
4
9
(3, 4)
(4, 9)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y
8
6
2
x
–2
2
4
7
Example: Sketch the graph of y = | x | + 1.
y
x
y
(x, y)
–2
–1
0
1
2
3
2
1
2
3
(–2, 3)
(–1, 2)
(0, 1)
(1, 2)
(2, 3)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
2
x
–2
2
8
The points at which the graph intersects the the xor y-axis are called intercepts.
If (x, 0) satisfies an equation, then the point (x, 0) is
called an x-intercept of the graph of the equation.
If (0, y) satisfies an equation, then the point (0, y) is
called a y-intercept of the graph of the equation.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
9
Procedure for finding the x- and y- intercepts of
the graph of an equation algebraically:
To find the x-intercepts of the graph of an
equation, substitute 0 for y in the equation and
solve for x.
To find the y-intercepts of the graph of an
equation algebraically, substitute 0 for x in the
equation and solve for y.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
10
Example: Find the x- and y-intercepts of the graph
of y = x2 + 4x – 5.
To find the x-intercepts, let y = 0 and solve for x.
0 = x2 + 4x – 5
0 = (x – 1)(x + 5)
x–1=0
x=1
Substitute 0 for y.
Factor.
x + 5 = 0 Set each factor equal to 0.
x = –5 Solve for x.
So, the x-intercepts are (1, 0) and (–5, 0).
To find the y-intercept, let x = 0 and solve for y.
y = 02 + 4(0) – 5 = –5
So, the y-intercept is (0, –5).
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
11
Procedure for finding the x- and y-intercepts of
the graph of an equation graphically:
To find the x-intercepts of the graph of an
equation, locate the points at which the graph
intersects the x-axis.
To find the y-intercepts of the graph of an
equation, locate the points at which the graph
intersects the y-axis.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
12
Example: Find the x- and y-intercepts of the graph
of x = | y | – 2 shown below.
y
The graph intersects
the x-axis at (–2, 0).
The graph intersects the
y-axis at (0, 2) and at
(0, –2).
2
x
–3
1
2
3
The x-intercept is (–2, 0).
The y-intercepts are (0, 2) and (0, –2).
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
13