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Learning Objectives
By the end of this section, you will be able to:
- Recognize the relation between the solutions of an equation and its graph
- Graph a linear equation by plotting points
- Graph vertical and horizontal lines
Be Prepared 11.4
Before you get started, take this readiness quiz.
Evaluate: when
If you missed this problem, review Example 3.56.
Be Prepared 11.5
Solve the formula: for
If you missed this problem, review Example 9.62.
Be Prepared 11.6
Simplify:
If you missed this problem, review Example 4.28.
Recognize the Relation Between the Solutions of an Equation and its Graph
In Use the Rectangular Coordinate System, we found a few solutions to the equation . They are listed in the table below. So, the ordered pairs , , , , are some solutions to the equation. We can plot these solutions in the rectangular coordinate system as shown on the graph at right.
Notice how the points line up perfectly? We connect the points with a straight line to get the graph of the equation . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.
Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions!
Notice that the point whose coordinates are
So
This is an example of the saying,” A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation
Graph of a Linear Equation
The graph of a linear equation
- Every point on the line is a solution of the equation.
- Every solution of this equation is a point on this line.
Example 11.15
The graph of
For each ordered pair decide
- ⓐ Is the ordered pair a solution to the equation?
- ⓑ Is the point on the line?
- (a)
( 0 , 3 ) ( 0 , 3 ) - (b)
( 3 , 3 ) ( 3 , 3 ) - (c)
( 2 , − 3 ) ( 2 , − 3 ) - (d)
( − 1 , − 5 ) ( − 1 , − 5 )
- Answer
Substitute the
- andx x -values into the equation to check if the ordered pair is a solution to the equation.y y ⓐ
ⓑ Plot the points A:
B:( 0 , − 3 ) ( 0 , − 3 ) C:( 3 , 3 ) ( 3 , 3 ) and D:( 2 , − 3 ) ( 2 , − 3 ) .( − 1 , − 5 ) ( − 1 , − 5 )
The points ,( 0 , − 3 ) ( 0 , − 3 ) , and( 3 , 3 ) ( 3 , 3 ) are on the line( − 1 , − 5 ) ( − 1 , − 5 ) , and the pointy = 2 x − 3 y = 2 x − 3 is not on the line.( 2 , − 3 ) ( 2 , − 3 ) The points which are solutions to
are on the line, but the point which is not a solution is not on the line.y = 2 x − 3 y = 2 x − 3
Try It 11.29
The graph of
For each ordered pair, decide
- ⓐ is the ordered pair a solution to the equation?
- ⓑ is the point on the line?
( 0 , − 1 ) ( 0 , − 1 ) ( 2 , 2 ) ( 2 , 2 ) ( 3 , − 1 ) ( 3 , − 1 ) ( − 1 , − 4 ) ( − 1 , − 4 )
Graph a Linear Equation by Plotting Points
There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method.
Let’s graph the equation
We start by finding three points that are solutions to the equation. We can choose any value for
Since
We can organize the solutions in a table. See Table 11.2.
Table 11.2
Now we plot the points on a rectangular coordinate system. Check that the points line up. If they did not line up, it would mean we made a mistake and should double-check all our work. See Figure 11.9.
Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line. The line is the graph of
How To
Graph a linear equation by plotting points.
- Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Step 2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- Step 3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you plot only two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line. If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. See Figure 11.11.
Example 11.16
Graph the equation
- Answer
Find three points that are solutions to the equation. It’s easier to choose values for
and solve forx , x , Do you see why?y . y . List the points in a table.
y = −3 x y = −3 x x x y y ( x , y ) ( x , y ) 0 0 0 0 ( 0 , 0 ) ( 0 , 0 ) 1 1 3 3 ( 1 , −3 ) ( 1 , −3 ) −2 −2 6 6 ( −2 , 6 ) ( −2 , 6 ) Plot the points, check that they line up, and draw the line as shown.
Try It 11.30
Graph the equation by plotting points:
Try It 11.31
Graph the equation by plotting points:
When an equation includes a fraction as the coefficient of
Example 11.17
Graph the equation
- Answer
Find three points that are solutions to the equation. Since this equation has the fraction
as a coefficient of1 2 1 2 we will choose values ofx , x , carefully. We will use zero as one choice and multiples ofx x for the other choices.2 2 The points are shown in the table.
y = 1 2 x + 3 y = 1 2 x + 3 x x y y ( x , y ) ( x , y ) 0 0 3 3 ( 0 , 3 ) ( 0 , 3 ) 2 2 4 4 ( 2 , 4 ) ( 2 , 4 ) 4 4 5 5 ( 4 , 5 ) ( 4 , 5 ) Plot the points, check that they line up, and draw the line as shown.
Try It 11.32
Graph the equation:
Try It 11.33
Graph the equation:
So far, all the equations we graphed had
Example 11.18
Graph the equation
- Answer
Find three points that are solutions to the equation. Remember, you can start with any value of
orx x y . y . We list the points in a table.
x + y = 5 x + y = 5 x x y y ( x , y ) ( x , y ) 0 0 5 5 ( 0 , 5 ) ( 0 , 5 ) 1 1 4 4 ( 1 , 4 ) ( 1 , 4 ) 4 4 1 1 ( 4 , 1 ) ( 4 , 1 ) Then plot the points, check that they line up, and draw the line.
Try It 11.34
Graph the equation:
Try It 11.35
Graph the equation:
In the previous example, the three points we found were easy to graph. But this is not always the case. Let’s see what happens in the equation
The solution is the point
Now we can choose values for
Example 11.19
Graph the equation
- Answer
Find three points that are solutions to the equation.
First, solve the equation for
y . y . 3 x + y = −1 y = −3 x − 1 3 x + y = −1 y = −3 x − 1 We’ll let
bex x and0 , 1 , 0 , 1 , to find three points. The ordered pairs are shown in the table. Plot the points, check that they line up, and draw the line.−1 −1 y = −3 x − 1 y = −3 x − 1 x x y y ( x , y ) ( x , y ) 0 0 −1 −1 ( 0 , −1 ) ( 0 , −1 ) 1 1 −4 −4 ( 1 , −4 ) ( 1 , −4 ) −1 −1 2 2 ( −1 , 2 ) ( −1 , 2 ) If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the
andx - x - -axes are the same, the graphs match.y y
Try It 11.36
Graph each equation:
Try It 11.37
Graph each equation:
Graph Vertical and Horizontal Lines
Can we graph an equation with only one variable? Just
Let’s consider the equation
To make a table of solutions, we write
Then plot the points and connect them with a straight line. Notice in Figure 11.12 that the graph is a vertical line.
Vertical Line
A vertical line is the graph of an equation that can be written in the form
The line passes through the
Example 11.20
Graph the equation
- Answer
The equation has only variable,
andx , x , is always equal tox x We make a table where2 . 2 . is alwaysx x and we put in any values for2 2 y . y . x = 2 x = 2 x x y y ( x , y ) ( x , y ) 2 2 1 1 ( 2 , 1 ) ( 2 , 1 ) 2 2 2 2 ( 2 , 2 ) ( 2 , 2 ) 2 2 3 3 ( 2 , 3 ) ( 2 , 3 ) Plot the points and connect them as shown.
The graph is a vertical line passing through the
-axis atx x 2 . 2 .
Try It 11.38
Graph the equation:
Try It 11.39
Graph the equation:
What if the equation has
To make a table of solutions, write
We’ll use
Plot the points and connect them, as shown in Figure 11.13. This graph is a horizontal line passing through the
Horizontal Line
A horizontal line is the graph of an equation that can be written in the form
The line passes through the
Example 11.21
Graph the equation
- Answer
The equation
has only variable,y = −1 y = −1 The value ofy . y . is constant. All the ordered pairs in the table have the samey y -coordinate,y y . We choose−1 −1 and0 , 3 , 0 , 3 , as values for−3 −3 x . x . y = −1 y = −1 x x y y ( x , y ) ( x , y ) −3 −3 −1 −1 ( −3 , −1 ) ( −3 , −1 ) 0 0 −1 −1 ( 0 , −1 ) ( 0 , −1 ) 3 3 −1 −1 ( 3 , −1 ) ( 3 , −1 ) The graph is a horizontal line passing through the
-axis aty y as shown.–1 –1
Try It 11.40
Graph the equation:
Try It 11.41
Graph the equation:
The equations for vertical and horizontal lines look very similar to equations like
The equation
The equation
The graph shows both equations.
Notice that the equation
Example 11.22
Graph
- Answer
Find three solutions for each equation. Notice that the first equation has the variable
while the second does not. Solutions for both equations are listed.x , x , The graph shows both equations.
Try It 11.42
Graph the equations in the same rectangular coordinate system:
Try It 11.43
Graph the equations in the same rectangular coordinate system:
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Section 11.2 Exercises
Practice Makes Perfect
Recognize the Relation Between the Solutions of an Equation and its Graph
In each of the following exercises, an equation and its graph is shown. For each ordered pair, decide
- ⓐ is the ordered pair a solution to the equation?
- ⓑ is the point on the line?
39.
( 0 , 2 ) ( 0 , 2 ) ( 1 , 2 ) ( 1 , 2 ) ( − 1 , 1 ) ( − 1 , 1 ) ( − 3 , 1 ) ( − 3 , 1 )
40.
( 0 , − 4 ) ( 0 , − 4 ) ( 3 , − 1 ) ( 3 , − 1 ) ( 2 , 2 ) ( 2 , 2 ) ( 1 , − 5 ) ( 1 , − 5 )
41.
( 0 , − 3 ) ( 0 , − 3 ) ( 2 , − 2 ) ( 2 , − 2 ) ( − 2 , − 4 ) ( − 2 , − 4 ) ( 4 , 1 ) ( 4 , 1 )
42.
( 0 , 2 ) ( 0 , 2 ) ( 3 , 3 ) ( 3 , 3 ) ( − 3 , 2 ) ( − 3 , 2 ) ( − 6 , 0 ) ( − 6 , 0 )
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
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Graph Vertical and Horizontal lines
In the following exercises, graph the vertical and horizontal lines.
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In the following exercises, graph each pair of equations in the same rectangular coordinate system.
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Mixed Practice
In the following exercises, graph each equation.
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Everyday Math
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Motor home cost The Robinsons rented a motor home for one week to go on vacation. It cost them
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Weekly earning At the art gallery where he works, Salvador gets paid
Writing Exercises
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Explain how you would choose three
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What is the difference between the equations of a vertical and a horizontal line?
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?