Download 4.1 The Graph of a Linear Equation

4.1
The Graph of a Linear Equation
4.1
OBJECTIVES
1. Find three ordered pairs for an equation in two
variables
2. Graph a line from three points
3. Graph a line by the intercept method
4. Graph a line that passes through the origin
5. Determine domain and range
6. Graph horizontal and vertical lines
In previous algebra classes you have solved equations in one variable such as
3x 2 5x 4.
Solving such an equation required finding the value of the variable, in this case x, that
made the equation a true statement. In this case, that value is x 3, because
3(3) 2 5(3) 4
This is a true statement because each side of the equation is equal to 11; no other value
for x makes this statement true. The solution can be written in three different ways. We can
write x 3, xx 3 which is read “the set of all x such that x equals 3,” or simply
3, which is the set containing the number 3.
What if we have an equation in two variables, such as 3x y 6? The solution set is
defined in a similar manner.
Definitions: Solution Set for an Equation in Two Variables
The solution set for an equation in two variables is the set containing all ordered
pairs of real numbers (x, y) that will make the equation a true statement.
The solution set for an equation in two variables is a set of ordered pairs. Typically, there
will be an infinite number of ordered pairs that make an equation a true statement. We can
find some of these ordered pairs by substituting a value for x, then solving the remaining
equation for y. We will use that technique in Example 1.
Example 1
Finding Ordered Pair Solutions
(a) 3x y 6
We will pick three values for x, set up a table for ordered pairs, and then determine the
related value for y.
x
1
0
1
190
y
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Find three ordered pairs that are solutions for each equation.
THE GRAPH OF A LINEAR EQUATION
SECTION 4.1
Substituting 1 for x, we get
3(1) y 6
3 y 6
y9
The ordered pair (1, 9) is a solution to the equation 3x y 6.
Substituting 0 for x, we get
3(0) y 6
0y6
y6
The ordered pair (0, 6) is a solution to the equation 3x y 6.
Substituting 1 for x, we get
3(1) y 6
3y6
y3
NOTE To indicate the set of all
solutions to the equation, we
write
{(x, y) 3x y 6}
The ordered pair (1, 3) is a solution to the equation 3x y 6.
Completing the table gives us the following:
x
y
1
0
1
9
6
3
(b) 2x y 1
Let’s try a different set of values for x. We will use the following table.
x
y
5
0
5
Substituting 5 for x, we get
2(5) y 1
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10 y 1
y 11
y 11
The ordered pair (5, 11) is a solution to the equation 2x y 1.
Substituting 0 for x, we get
2(0) y 1
0y1
y 1
y 1
191
192
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
NOTE Again, the set of all
solutions is
{(x, y) 2x y 1}
The ordered pair (0, 1) is a solution to the equation 2x y 1.
Substituting 5 for x, we get
2(5) y 1
10 y 1
y 9
y9
The ordered pair (5, 9) is a solution to the equation 2x y 1.
Completing the table gives us the following:
x
y
5
0
5
11
1
9
CHECK YOURSELF 1
Find three ordered pairs that are solutions for each equation.
(a) 2x y 6
(b) 3x y 2
The graph of the solution set of an equation in two variables, usually called the graph
of the equation, is the set of all points with coordinates (x, y) that satisfy the equation.
In this chapter, we are primarily interested in a particular kind of equation in x and y and
the graph of that equation. The equations we refer to involve x and y to the first power, and
they are called linear equations.
NOTE Why can A and B not
both be zero? First, recall that,
although x and y are variables,
A, B, and C are constants. With
that in mind, look at the
equation if A and B are both
zero.
(0)x (0)y C
00C
Definitions: Linear Equations
An equation of the form
Ax By C
in which A and B cannot both be zero, is called the standard form for a line. Its
graph is always a line.
0C
00
This would be a true statement
regardless of the values of x
and y. Its graph would be every
point in the plane.
NOTE Because two points
determine a line, technically
two points are all that are
needed to graph the equation.
You may want to locate at least
one other point as a check of
your work.
Example 2
Graphing by Plotting Points
Graph the equation
xy5
This is a linear equation in two variables. To draw its graph, we can begin by assigning
values to x and finding the corresponding values for y. For instance, if x 1, we have
1y5
y4
Therefore, (1, 4) satisfies the equation and is on the graph of x y 5.
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Because zero must be a
constant, we are left with the
statement
THE GRAPH OF A LINEAR EQUATION
SECTION 4.1
193
Similarly, (2, 3), (3, 2), and (4, 1) are in the graph. Often these results are recorded in a
table of values, as shown below. We then plot the points determined and draw a line through
those points.
NOTE If you first rewrite an
equation so that y is isolated on
the left side, it can be easily
entered and graphed with a
graphing calculator. In this case,
graph the equation
y x 5
xy5
y
x
y
1
2
3
4
4
3
2
1
(1, 4)
(2, 3)
(3, 2)
(4, 1)
x
Every point on the graph of the equation x y 5 has coordinates that satisfy the
equation, and every point with coordinates that satisfy the equation lies on the line.
CHECK YOURSELF 2
Graph the equation 2x y 6.
NOTE An algorithm is a
sequence of steps that solve a
problem.
The following algorithm summarizes our first approach to graphing a linear equation
in two variables.
Step by Step: To Graph a Linear Equation
Step 1 Find at least three solutions for the equation, and write your results in
a table of values.
Step 2 Graph the points associated with the ordered pairs found in step 1.
Step 3 Draw a line through the points plotted above to form the graph of the
equation.
Two particular points are often used in graphing an equation because they are very easy
to find. The x intercept of a line is the point at which the line crosses the x axis. If the x
intercept exists, it can be found by setting y 0 in the equation and solving for x. The y
intercept is the point at which the line crosses the y axis. If the y intercept exists, it is found
by letting x 0 and solving for y.
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Example 3
Graphing by the Intercept Method
Use the intercepts to graph the equation
NOTE Solving for y, we get
1
y x3
2
To graph this result on your
calculator, you can enter
Y1 (1 2)x 3
using the x, T, u, n key for x.
x 2y 6
To find the x intercept, let y 0.
x206
x6
The x intercept is (6, 0).
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
To find the y intercept, let x 0.
0 2y 6
2y 6
y 3
The y intercept is (0, 3).
Graphing the intercepts and drawing the line through those intercepts, we have the desired graph.
y
x 2y 6
(6, 0)
x
(0, 3)
CHECK YOURSELF 3
Graph, using the intercept method.
4x 3y 12
The following algorithm summarizes the steps of graphing a line by the intercept method.
Step by Step: Graphing by the Intercept Method
Step 1
Step 2
Step 3
Step 4
Find the x intercept. Let y 0, and solve for x.
Find the y intercept. Let x 0, and solve for y.
Plot the two intercepts determined in steps 1 and 2.
Draw a line through the intercepts.
y
y intercept
(x 0)
x
x intercept
(y 0)
When can the intercept method not be used? Some lines have only one intercept. For
instance, the graph of x 2y 0 passes through the origin. In this case, other points must
be used to graph the equation.
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194
THE GRAPH OF A LINEAR EQUATION
SECTION 4.1
195
Example 4
Graphing a Line That Passes Through the Origin
NOTE Graph the equation
1
y x
2
Note that the line passes
through the origin.
Graph x 2y 0.
Letting y 0 gives
x200
x0
Thus (0, 0) is a solution, and the line has only one intercept.
We continue by choosing any other convenient values for x. If x 2:
2 2y 0
2y 2
y 1
So (2, 1) is a solution. You can easily verify that (4, 2) is also a solution. Again, plotting the points and drawing the line through those points, we have the desired graph.
y
x 2y 0
x
(2, 1)
(0, 0)
(4, 2)
CHECK YOURSELF 4
Graph the equation x 3y 0.
In Section 3.1, we defined the terms domain and range. Recall that the domain of a relation is the set of all the first elements in the ordered pairs. The range is the set of all the
second elements. Recall that a line is the graph of a set of ordered pairs. In Example 5, we
will examine the domain and range for the graph of a line.
Example 5
Finding the Domain and Range
Find the domain and range for the relation described by the equation
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xy5
We can analyze the domain and range either graphically or algebraically. First, we will
look at a graphical analysis. From Example 2, let’s look at the graph of the equation.
y
(1, 4)
(2, 3)
(3, 2)
(4, 1)
x
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
The graph continues forever at both ends. For every value of x, there is an associated point
on the line. Therefore, the domain (D) is the set of all real numbers. In set notation, we write
D xx R
This is read, “The domain is the set of every x that is a real number.”
To find the range (R), we look at the graph to see what values are associated with y. Note
that every y is associated with some point. The range is written as
R yy R
This is read, “The range is the set of every y that is a real number.”
Let’s find the domain and range for the same relation by using an algebraic analysis.
Look at the following equation.
xy5
To determine the domain, we need to find every value of x that allows us to solve for y. That
combination will result in an ordered pair (x, y). The set of all those x values is the domain
of the relation.
We can find a value for y for any real value of x. For example, if x 5,
5 y 5
y 10
The ordered pair (5, 10) is part of the relation. As in our graphical analysis, the domain
is
D xx R
By a similar argument, we can substitute any value for y and solve the equation for x. The
range is
R yy R
CHECK YOURSELF 5
Find the domain and range for the relation described by the following equation.
xy4
Two types of linear equations are worthy of special attention. Their graphs are lines that
are parallel to the x or y axis, and the equations are special cases of the general form
Ax By C
in which either A 0 or B 0.
Rules and Properties:
Vertical or Horizontal Lines
1. A line with an equation of the form
yk
is horizontal (parallel to the x axis).
2. A line with an equation of the form
xh
is vertical (parallel to the y axis).
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196
THE GRAPH OF A LINEAR EQUATION
SECTION 4.1
197
Example 6 illustrates both cases.
Example 6
Graphing Horizontal and Vertical Lines
NOTE Because part (a) is a
function, it can be graphed on
your calculator. Part (b) is not a
function and cannot be
graphed on your calculator.
(a) Graph the line with equation
y3
You can think of the equation in the equivalent form
0xy3
Note that any ordered pair of the form (__, 3) will satisfy the equation. Because x is multiplied by 0, y will always be equal to 3.
For instance, (2, 3) and (3, 3) are on the graph. The graph, a horizontal line, is shown
below.
y
y3
x
The domain for a horizontal line is every real number. The range is a single y value. We
write
D xx R
and
R 3
(b) Graph the line with equation
x 2
In this case, you can think of the equation in the equivalent form
x 0 y 2
NOTE Notice that
D 2
Now any ordered pair of the form (2, __) will satisfy the equation. Examples are (2, 1)
and (2, 3). The graph, a vertical line, is shown below.
and
y
R y y R
x 2
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x
CHECK YOURSELF 6
Graph each equation and state the domain and range.
(a) y 3
(b) x 5
CHAPTER 4
GRAPHS OF LINEAR EQUATIONS AND FUNCTIONS
CHECK YOURSELF ANSWERS
1. (a) Answers will vary, but could include (0, 6); (b) Answers will vary, but could
y
include (0, 2).
2.
x
y
0
1
2
6
4
2
x
(2, 2)
(1, 4)
(0, 6)
3.
4.
y
5. D xx R and
R y y R
y
x 3y 0
(0, 4)
4x 3y 12
(3, 1)
(0, 0)
(6, 2)
x
(3, 0)
x
6. (a)
(b)
y
y
x5
x
x
(5, 0)
(0, 3)
D xx R
R 3
y 3
D 5
R y y R
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198
Name
4.1
Exercises
Section
Date
In exercises 1 to 8, find three ordered pairs that are solutions to the given equations.
1. 2x y 5
2. 3x y 7
3. 7x y 8
4. 5x y 3
5. 4x 5y 20
6. 2x 3y 6
7. 3x y 0
8. 2x y 0
ANSWERS
1.
In exercises 9 to 26, graph each of the equations.
2.
9. x y 6
10. x y 6
11. y x 2
y
y
y
3.
4.
5.
x
x
x
6.
7.
8.
9.
12. y x 5
13. y x 1
y
y
14. y 2x 2
10.
y
11.
12.
x
x
x
13.
14.
15.
16.
17.
15. y 2x 1
16. y 3x 1
y
y
y
© 2001 McGraw-Hill Companies
1
17. y x 3
2
x
x
x
199
ANSWERS
18.
18. y 2x 4
19.
19. y x 3
20. y 2x 4
y
y
y
20.
21.
x
x
x
22.
23.
24.
25.
26.
21. x 2y 0
22. x 2y 0
y
23. x 4
y
y
x
24. x 4
25. y 4
26. y 6
y
y
y
x
x
© 2001 McGraw-Hill Companies
x
x
x
200
ANSWERS
In exercises 27 to 38, find the x and y intercepts and then graph each equation.
27. x 2y 4
28. x 3y 6
y
27.
29. 2x y 6
28.
y
y
29.
x
x
x
30.
31.
32.
33.
30. 3x 2y 12
31. 2x 5y 10
y
y
33. 5x 6y 0
34. 2x 7y 0
y
x
35. x 4y 8 0
y
y
x
x
© 2001 McGraw-Hill Companies
x
35.
y
x
x
34.
32. 2x 3y 6
201
ANSWERS
36. 2x y 6 0
36.
37. 8x 4y
y
37.
38. 6x 7y
y
y
38.
x
39.
x
x
40.
41.
42.
43.
In exercises 39 to 46, find the domain and range of each of the relations.
44.
39. 3x 2y 4
40. 5x 4y 20
41. 6x 2y 18
42. x 5y 8
43. x 4
44. 2x 10 0
45. y 3
46. 3y 12 0
45.
46.
47.
48.
49.
50.
51.
For exercises 47 to 54, select a window that allows you to see both the x and y intercepts
on your calculator. If that is not possible, explain why not.
52.
53.
47. x y 40
48. x y 80
49. 2x 3y 900
50. 5x 8y 800
51. y 5x 90
52. y 3x 450
53. y 30x
54. y 200
202
© 2001 McGraw-Hill Companies
54.
ANSWERS
Two distinct lines in the plane either are parallel or they intersect. In exercises 55 to 58,
graph each pair of equations on the same set of axes, and find the point of intersection,
where possible.
55. x y 6, x y 4
56. y x 3, y x 1
y
56.
57.
y
x
55.
58.
x
59.
60.
57. y 2x, y x 1
58. 2x y 3, 2x y 5
61.
y
y
62.
63.
x
x
64.
59. Graph y x and y 2x on the same set of axes. What do you observe?
60. Graph y 2x 1 and y 2x 1 on the same set of axes. What do
you observe?
61. Graph y 2x and y 2x 1 on the same set of axes. What do you
observe?
62. Graph y 3x 1 and y 3x 1 on the same set of axes. What do you
© 2001 McGraw-Hill Companies
observe?
1
2
63. Graph y 2x and y x on the same set of axes. What do you observe?
1
3
you observe?
64. Graph y x 7
and y 3x 2 on the same set of axes. What do
3
203
ANSWERS
65.
Use your graphing utility to graph each of the following equations.
66.
65. y 3
66. y 2
67.
67. y 3x 1
68. y 2x 2
69. Write an equation whose graph will have no x intercept but will have a y intercept at (0,
68.
6).
70. Write an equation whose graph will have no y intercept but will have an x intercept
69.
at (5, 0).
70.
Answers
3. (0, 8), (1, 1), (1, 15)
1. (0, 5), (1, 3), (1, 7)
7. (0, 0), (1, 3), (1, 3)
9. x y 6
11. y x 2
13. y x 1
y
y
y
x
x
1
2
15. y 2x 1
17. y x 3
19. y x 3
y
y
21. x 2y 0
y
x
x
23. x 4
x
x
25. y 4
y
y
204
x
y
x
x
© 2001 McGraw-Hill Companies
24
5. (0, 4), (5, 0), 1,
5
27. x 2y 4; y intercept (0, 2);
29. 2x y 6; y intercept (0, 6);
x intercept (4, 0)
x intercept (3, 0)
y
y
x
31. 2x 5y 10; y intercept (0, 2);
x
33. 5x 6y 0;
x intercept (5, 0)
intercepts: (0, 0)
y
y
x
x
35. x 4y 8 0; y intercept (0, 2);
37. 8x 4y;
x intercept (8, 0)
intercepts: (0, 0)
y
y
© 2001 McGraw-Hill Companies
x
39.
41.
43.
45.
47.
49.
51.
53.
x
D: xx R; R: yy R
D: xx R; R: yy R
D: 4; R: yy R
D: xx R; R: 3
X max 40, Y max 40
X max 450, Y max 300
X min 18, Y max 90
Any viewing window that shows the origin
205
55. Intersection: (5, 1)
57. Intersection: (1, 2)
y
y
x
x
59. The line corresponding to
61. The two lines appear to be parallel.
y 2x is steeper than that
corresponding to y x.
y
y
x
x
63. The lines appear to be
65.
y 3
4
perpendicular.
2
y
4
2
2
4
2
4
x
y 3x1
67.
69. y 6
206
2
2
4
© 2001 McGraw-Hill Companies
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