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Prealgebra 2e

11.3 Graphing with Intercepts

Prealgebra 2e11.3 Graphing with Intercepts
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Identify the intercepts on a graph
  • Find the intercepts from an equation of a line
  • Graph a line using the intercepts
  • Choose the most convenient method to graph a line
Be Prepared 11.7

Before you get started, take this readiness quiz.

Solve: 3x+4y=−123x+4y=−12 for xx when y=0.y=0.
If you missed this problem, review Example 9.62.

Be Prepared 11.8

Is the point (0,−5)(0,−5) on the x-axisx-axis or y-axis?y-axis?
If you missed this problem, review Example 11.5.

Be Prepared 11.9

Which ordered pairs are solutions to the equation 2xy=6?2xy=6?
(6,0)(6,0)(0,−6)(0,−6)(4,−2).(4,−2).
If you missed this problem, review Example 11.8.

Identify the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line.

Intercepts of a Line

Each of the points at which a line crosses the x-axisx-axis and the y-axisy-axis is called an intercept of the line.

Let’s look at the graph of the lines shown in Figure 11.14.

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through two labeled points, “ordered pair 0, 6” and ordered pair 3, 0”.
Figure 11.14

First, notice where each of these lines crosses the x- axis:

Figure: The line crosses the x-axis at: Ordered pair of this point
42 3 (3,0)
43 4 (4,0)
44 5 (5,0)
45 0 (0,0)
Table 11.3

Do you see a pattern?

For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form (a,0)(a,0); and is called the x-intercept of the line. The x- intercept occurs when y is zero.

Now, let's look at the points where these lines cross the y-axis.

Figure: The line crosses the y-axis at: Ordered pair for this point
42 6 (0,6)
43 -3 (0,-3)
44 -5 (0,-5)
45 0 (0,0)
Table 11.4

x- intercept and y- intercept of a line

The x-interceptx-intercept is the point, (a, 0),(a,0), where the graph crosses the x-axis.x-axis. The x-interceptx-intercept occurs when yy is zero.

The y-intercepty-intercept is the point, (0, b),(0,b), where the graph crosses the y-axis.y-axis.

The y-intercepty-intercept occurs when xx is zero.

Example 11.23

Find the x- andy-interceptsx- andy-intercepts of each line:

x+2y=4x+2y=4 The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 4, 0”.
3xy=63xy=6 The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -6” and “ordered pair 2, 0”.
x+y=−5x+y=−5 The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, -5” and “ordered pair -5, 0”.
Try It 11.44

Find the x-x- and y-interceptsy-intercepts of the graph: xy=2.xy=2.

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.
Try It 11.45

Find the x-x- and y-interceptsy-intercepts of the graph: 2x+3y=6.2x+3y=6.

The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7. A line passes through the points “ordered pair 0, 2” and “ordered pair 3, 0”.

Find the Intercepts from an Equation of a Line

Recognizing that the x-interceptx-intercept occurs when yy is zero and that the y-intercepty-intercept occurs when xx is zero gives us a method to find the intercepts of a line from its equation. To find the x-intercept,x-intercept, let y=0y=0 and solve for x.x. To find the y-intercept,y-intercept, let x=0x=0 and solve for y.y.

Find the x and y from the Equation of a Line

Use the equation to find:

  • the x-intercept of the line, let y=0y=0 and solve for x.
  • the y-intercept of the line, let x=0x=0 and solve for y.
x y
0
0
Table 11.5

Example 11.24

Find the intercepts of 2x+y=62x+y=6

Try It 11.46

Find the intercepts: 3x+y=123x+y=12

Try It 11.47

Find the intercepts: x+4y=8x+4y=8

Example 11.25

Find the intercepts of 4x−3y=12.4x−3y=12.

Try It 11.48

Find the intercepts of the line: 3x−4y=12.3x−4y=12.

Try It 11.49

Find the intercepts of the line: 2x−4y=8.2x−4y=8.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Example 11.26

Graph x+2y=6x+2y=6 using intercepts.

Try It 11.50

Graph the line using the intercepts: x−2y=4.x−2y=4.

Try It 11.51

Graph the line using the intercepts: x+3y=6.x+3y=6.

How To

Graph a line using the intercepts.

  1. Step 1. Find the x-x- and y-interceptsy-intercepts of the line.
    • Let y=0y=0 and solve for xx
    • Let x=0x=0 and solve for y.y.
  2. Step 2. Find a third solution to the equation.
  3. Step 3. Plot the three points and then check that they line up.
  4. Step 4. Draw the line.

Example 11.27

Graph 4x−3y=124x−3y=12 using intercepts.

Try It 11.52

Graph the line using the intercepts: 5x−2y=10.5x−2y=10.

Try It 11.53

Graph the line using the intercepts: 3x−4y=12.3x−4y=12.

Example 11.28

Graph y=5xy=5x using the intercepts.

Try It 11.54

Graph using the intercepts: y=4x.y=4x.

Try It 11.55

Graph using the intercepts: y=x.y=x.

Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

Equation Method
#1 y=2x+1y=2x+1 Plotting points
#2 y=12x+3y=12x+3 Plotting points
#3 x=−7x=−7 Vertical line
#4 y=4y=4 Horizontal line
#5 2x+y=62x+y=6 Intercepts
#6 4x3y=124x3y=12 Intercepts

What is it about the form of equation that can help us choose the most convenient method to graph its line?

Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values.

Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.

In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form Ax+By=CAx+By=C. We substituted y=0y=0 and x=0x=0 to find the x- and y- intercepts, and then found a third point by choosing a value for x or y.

This leads to the following strategy for choosing the most convenient method to graph a line.

How To

Choose the most convenient method to graph a line.

  1. Step 1. If the equation has only one variable. It is a vertical or horizontal line.
    • x=ax=a is a vertical line passing through the x-axisx-axis at aa
    • y=by=b is a horizontal line passing through the y-axisy-axis at b.b.
  2. Step 2. If yy is isolated on one side of the equation. Graph by plotting points.
    • Choose any three values for xx and then solve for the corresponding y-y- values.
  3. Step 3. If the equation is of the form Ax+By=C,Ax+By=C, find the intercepts.
    • Find the x-x- and y-y- intercepts and then a third point.

Example 11.29

Identify the most convenient method to graph each line:

  1. y=−3y=−3
  2. 4x−6y=12 4x−6y=12
  3. x=2x=2
  4. y=25x−1y=25x−1
Try It 11.56

Identify the most convenient method to graph each line:

  1. 3x+2y=123x+2y=12
  2. y=4y=4
  3. y=15x−4y=15x−4
  4. x=−7x=−7
Try It 11.57

Identify the most convenient method to graph each line:

  1. x=6x=6
  2. y=34x+1y=34x+1
  3. y=−8y=−8
  4. 4x−3y=−14x−3y=−1

Section 11.3 Exercises

Practice Makes Perfect

Identify the Intercepts on a Graph

In the following exercises, find the x-x- and y-y- intercepts.

117.
The graph shows the x y-coordinate plane. The axes run from -10 to 10.  A line passes through the points “ordered pair 0, 3” and “ordered pair 3, 0”.
118.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 2” and “ordered pair 2, 0”.
119.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -5” and “ordered pair 5, 0”.
120.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -1” and “ordered pair 1, 0”.
121.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -2” and “ordered pair -2, 0”.
122.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, -3” and “ordered pair -3, 0”.
123.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 1” and “ordered pair -1, 0”.
124.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 5” and “ordered pair -5, 0”.
125.
The graph shows the x y-coordinate plane. The x and y-axis each run from -7 to 7.  A line passes through the points “ordered pair 0, 0” and “ordered pair 4, 2”.
126.
The graph shows the x y-coordinate plane. The x and y-axis each run from -10 to 10.  A line passes through the points “ordered pair 0, 0” and “ordered pair 1, 1”.

Find the xx and yy Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

127.

x+y=4x+y=4

128.

x+y=3x+y=3

129.

x+y=−2x+y=−2

130.

x+y=−5x+y=−5

131.

xy=5xy=5

132.

xy=1xy=1

133.

xy=−3xy=−3

134.

xy=−4xy=−4

135.

x+2y=8x+2y=8

136.

x+2y=10x+2y=10

137.

3x+y=63x+y=6

138.

3x+y=93x+y=9

139.

x−3y=12x−3y=12

140.

x−2y=8x−2y=8

141.

4xy=84xy=8

142.

5xy=55xy=5

143.

2x+5y=102x+5y=10

144.

2x+3y=62x+3y=6

145.

3x−2y=123x−2y=12

146.

3x−5y=303x−5y=30

147.

y=13x−1y=13x−1

148.

y=14x−1y=14x−1

149.

y=15x+2y=15x+2

150.

y=13x+4y=13x+4

151.

y=3xy=3x

152.

y=−2xy=−2x

153.

y=−4xy=−4x

154.

y=5xy=5x

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

155.

x+5y=10x+5y=10

156.

x+4y=8x+4y=8

157.

x+2y=4x+2y=4

158.

x+2y=6x+2y=6

159.

x+y=2x+y=2

160.

x+y=5x+y=5

161.

x+y=3x+y=3

162.

x+y=−1x+y=−1

163.

xy=1xy=1

164.

xy=2xy=2

165.

xy=−4xy=−4

166.

xy=−3xy=−3

167.

4x+y=44x+y=4

168.

3x+y=33x+y=3

169.

3xy=−63xy=−6

170.

2xy=−82xy=−8

171.

2x+4y=122x+4y=12

172.

3x+2y=123x+2y=12

173.

3x−2y=63x−2y=6

174.

5x−2y=105x−2y=10

175.

2x−5y=−202x−5y=−20

176.

3x−4y=−123x−4y=−12

177.

y=−2xy=−2x

178.

y=−4xy=−4x

179.

y=xy=x

180.

y=3xy=3x

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

181.

x=2x=2

182.

y=4y=4

183.

y=5y=5

184.

x=−3x=−3

185.

y=−3x+4y=−3x+4

186.

y=−5x+2y=−5x+2

187.

xy=5xy=5

188.

xy=1xy=1

189.

y=23x−1y=23x−1

190.

y=45x−3y=45x−3

191.

y=−3y=−3

192.

y=−1y=−1

193.

3x−2y=−123x−2y=−12

194.

2x−5y=−102x−5y=−10

195.

y=14x+3y=14x+3

196.

y=13x+5y=13x+5

Everyday Math

197.

Road trip Damien is driving from Chicago to Denver, a distance of 1,0001,000 miles. The x-axisx-axis on the graph below shows the time in hours since Damien left Chicago. The y-axisy-axis represents the distance he has left to drive.

The graph shows the x y-coordinate plane. The x and y-axis each run from - to .  A line passes through the labeled points “ordered pair 0, 1000” and “ordered pair 15, 0”.

Find the x-x- and y-y- intercepts

Explain what the x-x- and y-y- intercepts mean for Damien.

198.

Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The x-axisx-axis on the graph shows the number of miles Ozzie drove since filling up. The y-axisy-axis represents the number of gallons of gas in the truck’s gas tank.

The graph shows the x y-coordinate plane. The x and y-axis each run from - to .  A line passes through labeled points “ordered pair 0, 16” and “ordered pair 300, 0”.

Find the x-x- and y-y- intercepts.

Explain what the x-x- and y-y- intercepts mean for Ozzie.

Writing Exercises

199.

How do you find the x-interceptx-intercept of the graph of 3x−2y=6?3x−2y=6?

200.

How do you find the y-intercepty-intercept of the graph of 5xy=10?5xy=10?

201.

Do you prefer to graph the equation 4x+y=−44x+y=−4 by plotting points or intercepts? Why?

202.

Do you prefer to graph the equation y=23x−2y=23x−2 by plotting points or intercepts? Why?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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