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Elementary Algebra 2e

4.4 Understand Slope of a Line

Elementary Algebra 2e4.4 Understand Slope of a Line
  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

Learning Objectives

By the end of this section, you will be able to:

  • Use geoboards to model slope
  • Use m=riserunm=riserun to find the slope of a line from its graph
  • Find the slope of horizontal and vertical lines
  • Use the slope formula to find the slope of a line between two points
  • Graph a line given a point and the slope
  • Solve slope applications
Be Prepared 4.7

Before you get started, take this readiness quiz.

Simplify: 14821482.
If you missed this problem, review Example 1.74.

Be Prepared 4.8

Divide: 04,4004,40.
If you missed this problem, review Example 1.127.

Be Prepared 4.9

Simplify: 15−3,−153,−15−315−3,−153,−15−3.
If you missed this problem, review Example 1.65.

When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?

In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

In this section, we will explore the concept of slope.

Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Exploring Slope” will help you develop a better understanding of the slope of a line. (Graph paper can be used instead of a geoboard, if needed.)

We’ll start by stretching a rubber band between two pegs as shown in Figure 4.19.

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.
Figure 4.19

Doesn’t it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in Figure 4.20

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle. The 1, 2 peg is the vertex of the 90 degree angle, while the line between the 1, 4 and 4, 2 pegs forms the hypotenuse of the triangle.
Figure 4.20

We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in Figure 4.21.

In this illustration, there are two perpendicular lines with arrows. The first line extends straight upward and is labeled “rise”. The second arrow extends straight rightward and is labeled “run”.
Figure 4.21

If our geoboard and rubber band look just like the one shown in Figure 4.22, the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2, the peg in column 1, row 4, and the peg in column 4, row 2, forming a right triangle where the 1, 2 peg is the vertex of the 90 degree angle and the line between the 1, 4 peg and the 4, 2 peg forms the hypotenuse. The line between the 1, 2 peg and the 1, 4 peg is labeled “2”. The line between the 1, 2 peg and the 4, 2 peg is labeled “3”.
Figure 4.22 The rise on this geoboard is 2, as the rubber band goes up two units.

What is the run?

The rubber band goes across 3 units. The run is 3 (see Figure 4.22).

The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter mm.

Slope of a Line

The slope of a line of a line is m=riserunm=riserun.

The rise measures the vertical change and the run measures the horizontal change between two points on the line.

What is the slope of the line on the geoboard in Figure 4.22?

m=riserunm=23m=riserunm=23

The line has slope 2323. This means that the line rises 2 units for every 3 units of run.

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.

Example 4.25

What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line.
Try It 4.49

What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 5 and the peg in column 4, row 1, forming a line.
Try It 4.50

What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 5, row 3, forming a line.

Example 4.26

What is the slope of the line on the geoboard shown?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line.
Try It 4.51

What is the slope of the line on the geoboard?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 2 and the peg in column 4, row 4, forming a line.
Try It 4.52

What is the slope of the line on the geoboard?

The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 4, row 5, forming a line.

Notice that in Example 4.25 the slope is positive and in Example 4.26 the slope is negative. Do you notice any difference in the two lines shown in Figure 4.23(a) and Figure 4.23(b)?

The figure shows two grids of evenly spaced pegs, one labeled (a) and one labeled (b). There are 5 columns and 5 rows of pegs in each grid. In the (a) grid, a rubber band is stretched between the peg in column 1, row 5 and the peg in column 5, row 2, forming a line. Below this grid is the slope of a line defined as m equals 3 fourths. In the (b) grid, a rubber band is stretched between the peg in column 1, row 3 and the peg in column 4, row 4, forming a line. Below this grid is the slope of a line defined as m equals negative 1 third.
Figure 4.23

We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in Figure 4.23(a) is going up; it has positive slope. The line in Figure 4.23(b) is going down; it has negative slope.

Positive and Negative Slopes

The figure shows two lines side-by-side. The line on the left is a diagonal line that rises from left to right. It is labeled “Positive slope”. The line on the right is a diagonal line that drops from left to right. It is labeled “Negative slope”.

Example 4.27

Use a geoboard to model a line with slope 1212.

Try It 4.53

Model the slope m=13m=13. Draw a picture to show your results.

Try It 4.54

Model the slope m=32m=32. Draw a picture to show your results.

Example 4.28

Use a geoboard to model a line with slope −14.−14.

Try It 4.55

Model the slope m=−23m=−23. Draw a picture to show your results.

Try It 4.56

Model the slope m=−13m=−13. Draw a picture to show your results.

Use m=riserunm=riserun to Find the Slope of a Line from its Graph

Now, we’ll look at some graphs on the xyxy-coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

Example 4.29

How to Use m=riserunm=riserun to Find the Slope of a Line from its Graph

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 6 and the y-axis runs from negative 4 to 2. A line passes through the points (0, negative 3) and (5, 1).
Try It 4.57

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 8 to 1 and the y-axis runs from negative 1 to 4. A line passes through the points (negative 5, 1) and (0, 3).
Try It 4.58

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 4. A line passes through the points (0, negative 1) and (4, 2).

How To

Find the slope of a line from its graph using m=riserun.m=riserun.

  1. Step 1. Locate two points on the line whose coordinates are integers.
  2. Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
  3. Step 3. Count the rise and the run on the legs of the triangle.
  4. Step 4. Take the ratio of rise to run to find the slope, m=riserunm=riserun.

Example 4.30

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 9 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 5), (3, 3), and (6, 1).
Try It 4.59

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 6 to 1. A line passes through the points (0, negative 2) and (3, negative 6).
Try It 4.60

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 3 to 6 and the y-axis runs from negative 3 to 2. A line passes through the points (0, 1) and (5, negative 2).

In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.

Example 4.31

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from 0 to 8 and the y-axis runs from 0 to 7. A line passes through the points (2, 3) and (7, 6).
Try It 4.61

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 4 to 2 and the y-axis runs from negative 6 to 2. A line passes through the points (negative 3, 4) and (1, 1).
Try It 4.62

Find the slope of the line shown.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 4 and the y-axis runs from negative 2 to 3. A line passes through the points (1, negative 1) and (3, 2).

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

Horizontal liney=bVertical linex=a y-coordinates are the same.x-coordinates are the same.Horizontal liney=bVertical linex=a y-coordinates are the same.x-coordinates are the same.

So how do we find the slope of the horizontal line y=4y=4? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 1 to 7. A line passes through the points (0, 4) and (3, 4).
What is the rise? The rise is 00.
Count the run. The run is 33.
What is the slope? m=riserun m=03 m=0 m=riserun m=03 m=0
The slope of the horizontal line y=4y=4 is 00.
Table 4.34

All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.

Slope of a Horizontal Line

The slope of a horizontal line, y=by=b, is 0.

The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.

Now, we’ll consider a vertical line, the line.

The graph shows the x y coordinate plane. The x-axis runs from negative 1 to 5 and the y-axis runs from negative 2 to 2. A line passes through the points (3, 0) and (3, 2).
What is the rise? The rise is 22.
Count the run. The run is 00.
What is the slope? m=riserun m=20 m=riserun m=20
Table 4.35

But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line x=3x=3 is undefined.

The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.

Slope of a Vertical Line

The slope of a vertical line, x=ax=a, is undefined.

Example 4.32

Find the slope of each line:

x=8x=8 y=−5y=−5.

Try It 4.63

Find the slope of the line: x=−4.x=−4.

Try It 4.64

Find the slope of the line: y=7.y=7.

Quick Guide to the Slopes of Lines

This figure shows four lines with arrows. The first line rises up and runs to the right. It has a positive slope. The second line falls down and runs to the right. It has a negative slope. The third line is neither rises nor falls, extending horizontally in either direction. It has a slope of zero. The fourth line is completely vertical, one end rising up and the other rising down, running neither to the left nor right. It has an undefined slope.

Remember, we ‘read’ a line from left to right, just like we read written words in English.


Use the Slope Formula to find the Slope of a Line Between Two Points

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Slope of Lines Between Two Points” will help you develop a better understanding of how to find the slope of a line between two points.

Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

We have seen that an ordered pair (x,y)(x,y) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol (x,y)(x,y) be used to represent two different points? Mathematicians use subscripts to distinguish the points.

(x1,y1)read ‘xsub 1,ysub 1’ (x2,y2)read ‘xsub 2,ysub 2’(x1,y1)read ‘xsub 1,ysub 1’ (x2,y2)read ‘xsub 2,ysub 2’

The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?

We will use (x1,y1)(x1,y1) to identify the first point and (x2,y2)(x2,y2) to identify the second point.

If we had more than two points, we could use (x3,y3)(x3,y3), (x4,y4)(x4,y4), and so on.

Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points (2,3)(2,3) and (7,6)(7,6).

The graph shows the x y coordinate plane. The x and y-axes run from 0 to 7. A line passes through the points (2, 3) and (7, 6), which are plotted and labeled. The ordered pair (2, 3) is labeled (x subscript 1, y subscript 1). The ordered pair (7, 6) is labeled (x subscript 2, y subscript 2). An additional point is plotted at (2, 6). The three points form a right triangle, with the line from (2, 3) to (7, 6) forming the hypotenuse and the lines from (2, 3) to (2, 6) and from (2, 6) to (7, 6) forming the legs. The first leg, from (2, 3) to (2, 6) is labeled y subscript 2 minus y subscript 1, 6 minus 3, and 3. The second leg, from (2, 3) to (7, 6), is labeled x subscript 2 minus x subscript 1, y minus 2, and 5.

Since we have two points, we will use subscript notation, (2,x1,3y1)(2,x1,3y1)(7,6x2,y2)(7,6x2,y2).

On the graph, we counted the rise of 3 and the run of 5.

Notice that the rise of 3 can be found by subtracting the y-coordinates 6 and 3.

3=633=63

And the run of 5 can be found by subtracting the x-coordinates 7 and 2.

5=725=72

We know m=riserunm=riserun. So m=35m=35.

We rewrite the rise and run by putting in the coordinates m=6372m=6372.

But 6 is y2y2, the y-coordinate of the second point and 3 is y1y1, the y-coordinate of the first point.

So we can rewrite the slope using subscript notation. m=y2y172m=y2y172

Also, 7 is x2x2, the x-coordinate of the second point and 2 is x1x1, the x-coordinate of the first point.

So, again, we rewrite the slope using subscript notation. m=y2y1x2x1m=y2y1x2x1

We’ve shown that m=y2y1x2x1m=y2y1x2x1 is really another version of m=riserunm=riserun. We can use this formula to find the slope of a line when we have two points on the line.

Slope Formula

The slope of the line between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is

m=y2y1x2x1m=y2y1x2x1

This is the slope formula.

The slope is:

yof the second point minusyof the first point over xof the second point minusxof the first point.yof the second point minusyof the first point over xof the second point minusxof the first point.

Example 4.33

Use the slope formula to find the slope of the line between the points (1,2)(1,2) and (4,5)(4,5).

Try It 4.65

Use the slope formula to find the slope of the line through the points: (8,5)(8,5) and (6,3)(6,3).

Try It 4.66

Use the slope formula to find the slope of the line through the points: (1,5)(1,5) and (5,9)(5,9).

Example 4.34

Use the slope formula to find the slope of the line through the points (−2,−3)(−2,−3) and (−7,4)(−7,4).

Try It 4.67

Use the slope formula to find the slope of the line through the points: (−3,4)(−3,4) and (2,−1).(2,−1).

Try It 4.68

Use the slope formula to find the slope of the line through the pair of points: (−2,6)(−2,6) and (−3,−4)(−3,−4).

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.

Example 4.35

How To Graph a Line Given a Point and The Slope

Graph the line passing through the point (1,−1)(1,−1) whose slope is m=34m=34.

Try It 4.69

Graph the line passing through the point (2,−2)(2,−2) with the slope m=43m=43.

Try It 4.70

Graph the line passing through the point (−2,3)(−2,3) with the slope m=14m=14.

How To

Graph a line given a point and the slope.

  1. Step 1. Plot the given point.
  2. Step 2. Use the slope formula m=riserunm=riserun to identify the rise and the run.
  3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
  4. Step 4. Connect the points with a line.

Example 4.36

Graph the line with y-intercept 2 whose slope is m=23m=23.

Try It 4.71

Graph the line with the y-intercept 4 and slope m=52.m=52.

Try It 4.72

Graph the line with the x-intercept −3−3 and slope m=34m=34.

Example 4.37

Graph the line passing through the point (−1,−3)(−1,−3) whose slope is m=4.m=4.

Try It 4.73

Graph the line with the point (−2,1)(−2,1) and slope m=3m=3.

Try It 4.74

Graph the line with the point (4,−2)(4,−2) and slope m=−2m=−2.

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

Example 4.38

The ‘pitch’ of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

This figure shows a house with a sloped roof. The roof on one half of the building is labeled "pitch of the roof". There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled "rise equals 9 feet". There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled "run equals 18 feet".
Try It 4.75

Use Example 4.38, substituting the rise = 14 and run = 24.

Try It 4.76

Use Example 4.38, substituting rise = 15 and run = 36.

Example 4.39

Have you ever thought about the sewage pipes going from your house to the street? They must slope down 1414 inch per foot in order to drain properly. What is the required slope?

This figure is a right triangle. One leg is negative one quarter inch and the other leg is one foot.
Try It 4.77

Find the slope of a pipe that slopes down 1313 inch per foot.

Try It 4.78

Find the slope of a pipe that slopes down 3434 inch per yard.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with understanding slope of a line.

Section 4.4 Exercises

Practice Makes Perfect

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

211.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 2, forming a line.
212.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 4 and the peg in column 5, row 2, forming a line.
213.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 4 and the peg in column 4, row 2, forming a line.
214.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 3, row 4 and the peg in column 5, row 1, forming a line.
215.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 1 and the peg in column 4, row 4, forming a line.
216.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 3 and the peg in column 5, row 4, forming a line.
217.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 1, row 1 and the peg in column 5, row 4, forming a line.
218.
The figure shows a grid of evenly spaced pegs. There are 5 columns and 5 rows of pegs. A rubber band is stretched between the peg in column 2, row 2 and the peg in column 4, row 5, forming a line.

In the following exercises, model each slope. Draw a picture to show your results.

219.

2323

220.

3434

221.

1414

222.

4343

223.

1212

224.

3434

225.

2323

226.

3232

Use m=riserunm=riserun to find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

227.
The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 10, negative 8), (0, negative 4), and (10, 0).
228.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, negative 8) and (2, negative 2).
229.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 4, negative 6) and (4, 4).
230.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, negative 2) and passes through the point (3, 3).
231.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 3, 3) and (3, 1).
232.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 4) and (2, 2).
233.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line intercepts the y-axis at (0, 6) and passes through the point (4, 3).
234.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 3, 1) and intercepts the y-axis at (0, negative 1).
235.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 1) and (2, 4).
236.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 1) and (2, 3).
237.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 1, 6) and (1, 1).
238.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the point (negative 1, 3) and intercepts the x-axis at (3, 0).
239.
The graph shows the x y coordinate plane. The x and y-axes run from negative 7 to 7. A line passes through the points (negative 2, 6) and (1, 4).
240.
The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (negative 1, 3) and (1, 2).
241.
The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line intercepts the x-axis at (negative 2, 0) and passes through the point (2, 1).
242.
The graph shows the x y coordinate plane. The x and y-axes run from negative 10 to 10. A line passes through the points (4, 2) and (7, 3).

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

243.

y=3y=3

244.

y=1y=1

245.

x=4x=4

246.

x=2x=2

247.

y=−2y=−2

248.

y=−3y=−3

249.

x=−5x=−5

250.

x=−4x=−4

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

251.

(1,4),(3,9)(1,4),(3,9)

252.

(2,3),(5,7)(2,3),(5,7)

253.

(0,3),(4,6)(0,3),(4,6)

254.

(0,1),(5,4)(0,1),(5,4)

255.

(2,5),(4,0)(2,5),(4,0)

256.

(3,6),(8,0)(3,6),(8,0)

257.

(−3,3),(4,−5)(−3,3),(4,−5)

258.

(−2,4),(3,−1)(−2,4),(3,−1)

259.

(−1,−2),(2,5)(−1,−2),(2,5)

260.

(−2,−1),(6,5)(−2,−1),(6,5)

261.

(4,−5),(1,−2)(4,−5),(1,−2)

262.

(3,−6),(2,−2)(3,−6),(2,−2)

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

263.

(1,−2)(1,−2); m=34m=34

264.

(1,−1)(1,−1); m=23m=23

265.

(2,5)(2,5); m=13m=13

266.

(1,4)(1,4); m=12m=12

267.

(−3,4)(−3,4); m=32m=32

268.

(−2,5)(−2,5); m=54m=54

269.

(−1,−4)(−1,−4); m=43m=43

270.

(−3,−5)(−3,−5); m=32m=32

271.

y-intercept 3; m=25m=25

272.

y-intercept 5; m=43m=43

273.

x-intercept −2−2; m=34m=34

274.

x-intercept −1−1; m=15m=15

275.

(−3,3)(−3,3); m=2m=2

276.

(−4,2)(−4,2); m=4m=4

277.

(1,5)(1,5); m=−3m=−3

278.

(2,3)(2,3); m=−1m=−1

Everyday Math

279.

Slope of a roof. An easy way to determine the slope of a roof is to set one end of a 12 inch level on the roof surface and hold it level. Then take a tape measure or ruler and measure from the other end of the level down to the roof surface. This will give you the slope of the roof. Builders, sometimes, refer to this as pitch and state it as an “xx 12 pitch” meaning x12x12, where xx is the measurement from the roof to the level—the rise. It is also sometimes stated as an “xx-in-12 pitch”.

  1. What is the slope of the roof in this picture?
  2. What is the pitch in construction terms?
    This figure shows one side of a sloped roof of a house. The rise of the roof is labeled “4 inches” and the run of the roof is labeled “12 inches”.
280.

The slope of the roof shown here is measured with a 12” level and a ruler. What is the slope of this roof?

This figure shows one side of a sloped roof of a house. The rise of the roof is measured with a ruler and shown to be 7 inches. The run of the roof is measured with a twelve inch level and shown to be 12 inches.
281.

Road grade. A local road has a grade of 6%. The grade of a road is its slope expressed as a percent. Find the slope of the road as a fraction and then simplify. What rise and run would reflect this slope or grade?

282.

Highway grade. A local road rises 2 feet for every 50 feet of highway.

  1. What is the slope of the highway?
  2. The grade of a highway is its slope expressed as a percent. What is the grade of this highway?
283.

Wheelchair ramp. The rules for wheelchair ramps require a maximum 1-inch rise for a 12-inch run.

  1. How long must the ramp be to accommodate a 24-inch rise to the door?
  2. Create a model of this ramp.
284.

Wheelchair ramp. A 1-inch rise for a 16-inch run makes it easier for the wheelchair rider to ascend a ramp.

  1. How long must a ramp be to easily accommodate a 24-inch rise to the door?
  2. Create a model of this ramp.

Writing Exercises

285.

What does the sign of the slope tell you about a line?

286.

How does the graph of a line with slope m=12m=12 differ from the graph of a line with slope m=2m=2?

287.

Why is the slope of a vertical line “undefined”?

Self Check

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

This table has seven rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “use geoboards to model slope.” The third row reads “use m equals rise divided by run to find the slope of a line from its graph.” The fourth row reads “find the slope of horizontal and vertical lines.” The fifth row reads “use the slope formula to find the slope of a line between two points.” The sixth row reads “graph a line given a point and the slope.” The last row reads “solve slope applications.” The remaining columns are blank.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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