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Elementary Algebra 2e

4.3 Graph with Intercepts

Elementary Algebra 2e4.3 Graph with Intercepts
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index
Be Prepared 4.6

Before you get started, take this readiness quiz.

Solve: 3·0+4y=−2.3·0+4y=−2.
If you missed this problem, review Example 2.17.

Identify the x- and y- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.

Intercepts of a Line

The points where a line crosses the x- axis and the y- axis are called the intercepts of a line.

Let’s look at the graphs of the lines in Figure 4.18.

Four figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line crossing the x- axis at the point (3, 0) and crossing the y- axis at the point (0, 6). The graph is labeled with the equation 2x plus y equals 6. Figure b shows a straight line crossing the x- axis at the point (4, 0) and crossing the y- axis at the point (0, negative 3). The graph is labeled with the equation 3x minus 4y equals 12. Figure c shows a straight line crossing the x- axis at the point (5, 0) and crossing the y- axis at the point (0, negative 5). The graph is labeled with the equation x minus y equals 5. Figure d shows a straight line crossing the x- axis and y- axis at the point (0, 0). The graph is labeled with the equation y equals negative 2x.
Figure 4.18 Examples of graphs crossing the x-negative axis.

First, notice where each of these lines crosses the xx negative axis. See Figure 4.18.

Figure The line crosses the x- axis at: Ordered pair of this point
Figure (a) 3 (3,0)(3,0)
Figure (b) 4 (4,0)(4,0)
Figure (c) 5 (5,0)(5,0)
Figure (d) 0 (0,0)(0,0)
Table 4.24

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 a line. The x- intercept occurs when yy is zero.

Now, let’s look at the points where these lines cross the y- axis. See Table 4.25.

Figure The line crosses the y-axis at: Ordered pair for this point
Figure (a) 6 (0,6)(0,6)
Figure (b) −3−3 (0,−3)(0,−3)
Figure (c) −5−5 (0,−5)(0,−5)
Figure (d) 0 (0,0)(0,0)
Table 4.25

What is the pattern here?

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

x- intercept and y- intercept of a line

The x- intercept is the point (a,0)(a,0) where the line crosses the x- axis.

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

No Alt Text

Example 4.19

Find the x- and y- intercepts on each graph.

Three figures, each showing a different straight line on the x y- coordinate plane. The x- axis of the planes runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. Figure a shows a straight line going through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1). Figure b shows a straight line going through the points (0, negative 6), (1, negative 3), (2, 0), (3, 3), and (4, 6). Figure c shows a straight line going through the points (negative 6, 1), (negative 5, 0), (negative 4, negative 1), (negative 3, negative 2), (negative 2, negative 3), (negative 1, negative 4), (0, negative 5), and (1, negative 6).
Try It 4.37

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

A figure showing a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 10), (negative 6, negative 8), (negative 4, negative 6), (negative 2, negative 4), (0, negative 2), (2, 0), (4, 2), (6, 4), (8, 6), and (10, 8).
Try It 4.38

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

The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 9, 8), (negative 6, 6), (negative 3, 4), (0, 2), (3, 0), (6, negative 2), and (9, negative 4).

Find the x- and y- Intercepts from an Equation of a Line

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

Find the x- and y- Intercepts from the Equation of a Line

Use the equation of the line. To find:

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

 

Example 4.20

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

Try It 4.39

Find the intercepts of 3x+y=12.3x+y=12.

Try It 4.40

Find the intercepts of x+4y=8.x+4y=8.

Example 4.21

Find the intercepts of 4x3y=124x3y=12.

Try It 4.41

Find the intercepts of 3x4y=12.3x4y=12.

Try It 4.42

Find the intercepts of 2x4y=8.2x4y=8.

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Example 4.22

How to Graph a Line Using Intercepts

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

Try It 4.43

Graph x2y=4x2y=4 using the intercepts.

Try It 4.44

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

The steps to graph a linear equation using the intercepts are summarized below.

How To

Graph a linear equation using the intercepts.

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

Example 4.23

Graph 4x3y=124x3y=12 using the intercepts.

Try It 4.45

Graph 5x2y=105x2y=10 using the intercepts.

Try It 4.46

Graph 3x4y=123x4y=12 using the intercepts.

Example 4.24

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

Try It 4.47

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

Try It 4.48

Graph y=xy=x the intercepts.

Section 4.3 Exercises

Practice Makes Perfect

Identify the x- and y- Intercepts on a Graph

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

139.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, 8), (negative 4, 7), (negative 3, 6), (negative 2, 5), (negative 1, 4), (0, 3), (1, 2), (2, 1), (3, 0), (4, negative 1), (5, negative 2) and (6, negative 3).
140.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 8), (negative 5, 7), (negative 4, 6), (negative 3, 5), (negative 2, 4), (negative 1, 3), (0, 2), (1, 1), (2, 0), (3, negative 1), (4, negative 2), (5, negative 3) and (6, negative 4).
141.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 5, negative 10), (negative 4, negative 9), (negative 3, negative 8), (negative 2, negative 7), (negative 1, negative 6), (0, negative 5), (1, negative 4), (2, negative 3), (3, negative 2), (4, negative 1), (5, 0), (6, 1), (7, 2), and (8, 3).
142.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).
143.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 7), (negative 5, negative 6), (negative 4, negative 5), (negative 3, negative 4), (negative 2, negative 3), (negative 1, negative 2), (0, negative 1), (1, 0), (2, 1), (3, 2), (4, 3), (5, 4), (6, 5), (7, 6), and (8, 7).
144.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, 3), (negative 5, 2), (negative 4, 1), (negative 3, 0), (negative 2, negative 1), (negative 1, negative 2), (0, negative 3), (1, negative 4), (2, negative 5), (3, negative 6), (4, negative 7), (5, negative 8), and (6, negative 9).
145.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 6, negative 5), (negative 5, negative 4), (negative 4, negative 3), (negative 3, negative 2), (negative 2, negative 1), (negative 1, 0), (0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), and (8, 9).
146.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 8, negative 3), (negative 7, negative 2), (negative 6, negative 1), (negative 5, 0), (negative 4, 1), (negative 3, 2), (negative 2, 3), (negative 1, 4), (0, 5), (1, 6), (2, 7), (3, 8), (4, 9), and (5, 10).
147.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the points (negative 10, 8), (negative 8, 7), (negative 6, 6), (negative 4, 5), (negative 2, 4), (0, 3), (2, 2), (4, 1), (6, 0), (8, negative 1), and (10, negative 2).
148.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the points (negative 6, 5), (negative 4, 4), (negative 2, 3), (0, 2), (2, 1), (4, 0), and (6, negative 1).
149.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 10 to 10. The y- axis of the planes runs from negative 10 to 10. The straight line goes through the plotted point (0, 0).
150.
The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from negative 7 to 7. The y- axis of the planes runs from negative 7 to 7. The straight line goes through the plotted point (0, 0).

Find the x- and y- Intercepts from an Equation of a Line

In the following exercises, find the intercepts for each equation.

151.

x+y=4x+y=4

152.

x+y=3x+y=3

153.

x+y=−2x+y=−2

154.

x+y=−5x+y=−5

155.

xy=5xy=5

156.

xy=1xy=1

157.

xy=−3xy=−3

158.

xy=−4xy=−4

159.

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

160.

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

161.

3x+y=63x+y=6

162.

3x+y=93x+y=9

163.

x3y=12x3y=12

164.

x2y=8x2y=8

165.

4xy=84xy=8

166.

5xy=55xy=5

167.

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

168.

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

169.

3x2y=123x2y=12

170.

3x5y=303x5y=30

171.

y=13x+1y=13x+1

172.

y=14x1y=14x1

173.

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

174.

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

175.

y=3xy=3x

176.

y=−2xy=−2x

177.

y=−4xy=−4x

178.

y=5xy=5x

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

179.

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

180.

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

181.

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

182.

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

183.

x+y=2x+y=2

184.

x+y=5x+y=5

185.

x+y=−3x+y=−3

186.

x+y=−1x+y=−1

187.

xy=1xy=1

188.

xy=2xy=2

189.

xy=−4xy=−4

190.

xy=−3xy=−3

191.

4x+y=44x+y=4

192.

3x+y=33x+y=3

193.

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

194.

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

195.

3x2y=63x2y=6

196.

5x2y=105x2y=10

197.

2x5y=−202x5y=−20

198.

3x4y=−123x4y=−12

199.

3xy=−63xy=−6

200.

2xy=−82xy=−8

201.

y=−2xy=−2x

202.

y=−4xy=−4x

203.

y=xy=x

204.

y=3xy=3x

Everyday Math

205.

Road trip. Damien is driving from Chicago to Denver, a distance of 1000 miles. The x- axis on the graph below shows the time in hours since Damien left Chicago. The y- axis represents the distance he has left to drive.

The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 16. The y- axis of the planes runs from 0 to 1200 in increments of 200. The straight line goes through the points (0, 1000), (3, 800), (6, 600), (9, 400), (12, 200), and (15, 0). The points (0, 1000) and (15, 0) are marked and labeled with their coordinates.
  1. Find the x- and y- intercepts.
  2. Explain what the x- and y- intercepts mean for Damien.
206.

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

The figure shows a straight line on the x y- coordinate plane. The x- axis of the plane runs from 0 to 350 in increments of 50. The y- axis of the planes runs from 0 to 18 in increments of 2. The straight line goes through the points (0, 16), (150, 8), and (300, 0). The points (0, 16) and (300, 0) are marked and labeled with their coordinates
  1. Find the x- and y- intercepts.
  2. Explain what the x- and y- intercepts mean for Ozzie.

Writing Exercises

207.

How do you find the x- intercept of the graph of 3x2y=63x2y=6?

208.

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

209.

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation y=23x2y=23x2? Why?

210.

Do you prefer to use the method of plotting points or the method using the intercepts to graph the equation y=6y=6? Why?

Self Check

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

The figure shows a table with four rows and four columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is "confidently", the third is “with some help”, “no minus I don’t get it!”. Under the first column are the phrases “identify the x and y intercepts of a graph”, “find the x and y intercepts from an equation of a line”, and “graph a line using intercepts”. Under the second, third, fourth columns are blank spaces where the learner can check what level of mastery they have achieved.

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

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