Topics
3. Linear Functions
topic badge
United States of AmericaVirginia
Algebra 1

3.01 Slope

Lesson
Practice

Identify slope from a graph

Recall the slope of a line is a value that describes the line's steepness.

Slope

A rate of change in a proportional relationship between two quantities

We can find the slope of a line by identifying the vertical and horizontal change or: \text{slope}=\dfrac{\text{change in }y}{\text{change in }x} = \dfrac{\text{rise}}{\text{run}}

There are four types of slope:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Positive
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Negative
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Zero
-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
Undefined

To find the slope of a line from its graph we can use a slope triangle. Consider the triangle and the points A and B on the graph:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y

We can see that from point A to point B, we move 9 units down and 3 units to the right to create lines that form the sides of a triangle.

The 'rise' or vertical change is -9 and the 'run' or horizontal change is +3.

So we can write the slope as the fraction \dfrac{-9}{3} which can be simplified to \dfrac{-3}{1} or -3.

Points A and B aren't the only points we could have used to find the slope. We can use any two points on a line, however, choosing points with integer coordinates can make counting the slope much easier. Consider these points C and D instead.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
y

With this new triangle, we move 3 units down and 1 unit to the right.

The 'rise' or vertical change is -3 and the 'run' or horizontal change is +1.

So we can write the slope as the fraction \dfrac{-3}{1} or -3. This is the same slope we calculated with the larger triangle, there were just fewer steps to simplify.

If you pick two points that are so close together that no other points with integer coordinates lie between them, you will get a slope that is already simplified.

Examples

Example 1

Consider the points A, B, and C.

-1
1
2
3
4
5
6
7
x
1
2
3
4
5
6
7
8
9
10
11
12
13
y
a

Complete the directions that explain how to move from point A to point B:

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

We need to find the difference in the y-values (i.e. rise) between points A and B, and the difference between their x-values (i.e. run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 8-0Subtract the y-coordinates of points A and B
\displaystyle =\displaystyle 8\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 4-0Subtract the x-coordinates of points A and B
\displaystyle =\displaystyle 4\text{ units}Evaluate

From A, move 8 units up, and 4 units to the right.

b

Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula: m=\dfrac{\text{rise}}{\text{run}}, then convert to ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac84Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac21Evaluate
\displaystyle =\displaystyle 2\text{:}1Express as a ratio
c

Complete the directions that explain how to move from point A to point C.

From A, move ⬚ units up, and ⬚ units to the right.

Worked Solution
Create a strategy

We need to find the difference in the y-values (i.e. rise) between points A and C, and the difference between their x-values (i.e. run).

Apply the idea
\displaystyle \text{rise}\displaystyle =\displaystyle 12-0Subtract the y-coordinates of points A and C
\displaystyle =\displaystyle 12\text{ units}Evaluate
\displaystyle \text{run}\displaystyle =\displaystyle 6-0Subtract the x-coordinates of points A and C
\displaystyle =\displaystyle 6\text{ units}Evaluate

From A, move 12 units up, and 6 units to the right.

d

Express the direction of the movement in the previous question as a simplified ratio, comparing vertical movement to horizontal movement. Express the ratio in the form a\text{:}b.

Worked Solution
Create a strategy

Use the slope formula: m=\dfrac{\text{rise}}{\text{run}}, then convert to ratio.

Apply the idea
\displaystyle m\displaystyle =\displaystyle \dfrac{12}{6}Substitute the value of the rise and run
\displaystyle =\displaystyle \dfrac21Evaluate
\displaystyle =\displaystyle 2\text{:}1Express as a ratio

Example 2

Consider the graph shown:

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
10
y

Find the slope of the line represented by the graph.

Worked Solution
Create a strategy

To find the slope of a line represented by using the formula: \text{Slope}=\dfrac{\text{rise}}{\text{run}}.

Apply the idea

From point A(-2,\,10) to point B(2,\,2), move 8 units down and 4 units to the right.

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
10
y

The ratio of the rise to the run is \dfrac{-8}{4} simplified into -2, the slope of the line.

The slope of the line is m=-2.

Idea summary

The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).

\text{slope}=\dfrac{\text{change in }y}{\text{change in }x} = \dfrac{\text{rise}}{\text{run}}

The slope formula

Finding the ratio of the rise and run of the line works when it's easy to see the graph, with clearly marked points on the line. We can extend this thinking to use the coordinates of two points and construct a general formula.

A line from point (x subscript 1, y subscript 1) to point (x subscript 2, y subscript 2) is drawn on the coordinate plane. The vertical distance from the x-axis to point (x subscript 1, y subscript 1) is labeled y subscript 1. The vertical distance from the x-axis to point (x subscript 2, y subscript 2) is labeled y subscript 2.

The rise of the line is found with points that lie on a vertical line. These points share the same x-value. The distance between them is the difference in the y-values.

\text{rise} = y_2 - y_1

A line from point (x subscript 1, y subscript 1) to point (x subscript 2,y subscript 2) is drawn on the coordinate plane. The horizontal distance from the y-axis to point (x subscript 1, y subscript 1) is labeled x subscript 1. The horizontal distance from the y-axis to point (x subscript 2,y subscript 2) is labeled x subscript 2.

The run of the line is found with two points that lie on a horizontal line. These points share the same y-value. The distance between them is the difference in the x-values.

\text{run} = x_2 - x_1

We find the slope by using the formula:

\displaystyle m= \dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}
\bm{m}
slope
\bm{\left(x_1,y_1\right)}
a point on the line
\bm{\left(x_2,y_2\right)}
a second point on the line

Examples

Example 3

What is the slope of a line that passes through the points A(3,\,5) and B(-2, 10)?

Worked Solution
Create a strategy

Use the slope formula: m= \dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}. Let point A=(x_1,y_1) and point B=(x_2, y_2).

Apply the idea
\displaystyle \text{Slope}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Use the slope formula
\displaystyle =\displaystyle \dfrac{10-5}{-2-3}Substitute x_1=3,\,y_1=5,\,x_2=-2,\, and y_2=10
\displaystyle =\displaystyle \dfrac{5}{-5}Evaluate the subtraction
\displaystyle =\displaystyle -1Evaluate the division
Reflect and check

By graphing the two points and sketching the line through them, we can see that it does in fact have a negative slope.

-2
2
4
6
8
10
x
-2
2
4
6
8
10
y

Example 4

Gasoline costs a certain amount per gallon. The table shows the cost of various amounts of gasoline in dollars:

\text{Number of gallons }(x)010203040
\text{Cost of gasoline }(y)026.4052.8079.20105.60

How much does gasoline cost per gallon?

Worked Solution
Create a strategy

The cost per gallon of gasoline is the unit rate. This is the same as the ratio of the change in the cost of gasoline (y) to the change in the number of gallons (x).

We can find the unit rate by using the slope formula, m=\dfrac{y_2-y_1}{x_2-x_1}, with any two input-output pairs in the table.

Apply the idea
\displaystyle \text{Cost per gallon}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the slope formula
\displaystyle =\displaystyle \dfrac{26.40-0}{10-0}Substitue two input-output pairs
\displaystyle =\displaystyle \dfrac{26.40}{10}Evaluate the subtraction
\displaystyle =\displaystyle \$2.64Evaluate the division
Reflect and check

The relationship between the cost of gasoline and the number of gallons purchased can be modeled by the equation y=2.64x. If we use technology to graph this line, we can see that all the points in the table lie on the line.

5
10
15
20
25
30
35
40
45
50
x
10
20
30
40
50
60
70
80
90
100
110
y

Example 5

Mario wants to determine which of two slow-release pain medications is more rapidly absorbed by the body.

For the liquid form, the amount of the medication in the bloodstream is presented in the graph shown.

1
2
3
4
5
6
7
8
9
t
5
10
15
20
25
30
35
A

The results for the capsule form are presented in the table below.

\text{Time (mins)}, t\text{Amount in} \\ \text{blood (mgs)}, A
424.6
742.3
1060
1377.7
a

At what rate, in milligrams per minute, is the liquid form absorbed?

Worked Solution
Create a strategy

Chooose any two points that lie on the line, and use the formula for slope.

Apply the idea

We use the points (0,\,0)and (2,\,8) that lie on the line.

\displaystyle \text{Rate of liquid absorption}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the formula for slope
\displaystyle =\displaystyle \dfrac{8-0}{2-0}Substitute the values
\displaystyle =\displaystyle 4\text{ mg/min}Evaluate
b

At what rate, in milligrams per minute, is the capsule form absorbed?

Worked Solution
Create a strategy

Choose any two points from the table and use the slope formula.

Apply the idea

We can use the points (4,\,24.6) and (7,\,42.3) from the table.

\displaystyle \text{Rate of capsule absorption}\displaystyle =\displaystyle \dfrac{y_2-y_1}{x_2-x_1}Write the formula for slope
\displaystyle =\displaystyle \dfrac{42.3-24.6}{7-4}Substitute the values
\displaystyle =\displaystyle 5.9\text{ mg/min}Evaluate
c

In which form is the medication absorbed more rapidly?

Worked Solution
Create a strategy

Use the fact that the medication that is more quickly absorbed will have a higher rate.

Apply the idea

5.9\text{ mg/min} \gt 4\text{ mg/min}

Comparing the two rates from part (a) and part (b), the capsule form is more quickly absorbed than the liquid form as it has higher rate of absorption of 5.9\text{ mg/min}.

Idea summary

We find the slope of a line by using the formula:

\displaystyle m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}
\bm{(x_{1},\,y_{1})}
are the coordinates of the lower points
\bm{(x_{2},\,y_{2})}
are the coordinates of the upper points

We can find the slope in different ways depending on how it is represented:

  • From a graph: Count the slope as the rise over the run, or find two points and use the slope formula.
  • Table: Find the difference in two consecutive outputs and divide by the difference in the corresponding inputs, or find two input-output pairs and use the slope formula.
  • Description: The slope is the rate given in the problem.

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1a

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

What is Mathspace

About Mathspace