# 9.4: The Slope of a Line

**At Grade**Created by: CK-12

## Introduction

*Rainforest Transportation*

After a lot of debating and discussing, the class finally decided to attend the Omni Theater. The presentation on the rainforest was fascinating and all of the students were glad that this was the decision that they had made. While the film did show the animals and insects of the rainforest, it also focused on ecology and on scientists and other people working to save the rainforest.

When they returned to school, Mr. Thomas had them talk about different parts that had interested them.

“I was fascinated on the bus ride that the scientists took to get to the rainforest. It was amazing to me that the bus could travel over all of those bumpy roads and not have an accident,” Mark commented.

“Yes, if you think about it, the bus ride was pretty cool to see from the Omni point of view. I mean I felt like I was going to fall over the edge some of the time,” Karen added.

Others smiled as well. Mr. Thomas seizing the opportunity wrote the following problem on the board.

A bus leaves Boston traveling at a constant speed of 60mph. You can make a table showing the distance, \begin{align*}d\end{align*}, in miles that the train has traveled after \begin{align*}h\end{align*} hours.

**This is your task. After learning about slope, you will learn about direct and indirect variation. These concepts will all help you with Mr. Thomas’ problem.**

*What You Will Learn*

By the end of this lesson, you will be able to complete the following skills.

- Find the slope of a line passing through given points.
- Recognize and interpret positive, negative and undefined slopes.
- Compare and interpret relative slopes of given lines.
- Identify and distinguish between direct and inverse variation.

*Teaching Time*

I. **Find the Slope of a Line Passing Through Given Points**

We have described linear graphs by where the cross the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-axes. But, what about the steepness of the line? If you’ve ever been skiing, you know that the most important thing about a ski slope is how steep it is. In mathematics, we use numbers to quantify steepness so that you can describe graphs more precisely, predict its movement, and compare it with others.

**When we have a line that has been graphed on the coordinate plane, we can calculate the steepness of the line. In mathematics, we call this steepness the** *slope of the line.***The slope of the line is how steep the line is.**

Let’s look at an example.

Example

**Now we want to calculate the steepness of this line. We want to calculate the slope. The slope of the line can be calculated by using a** *ratio.***Remember that a ratio compares two quantities. In this case, we are going to compare the rise of the line with the run of the line.**

\begin{align*}Slope=\frac{rise}{run}\end{align*}

Here is a graph where the slope is highlighted.

**You can see that the rise is 2 and the run is 1. It is moving up so it is a positive slope. We can write this as the following ratio.**

\begin{align*}Slope=\frac{2}{1}=2 \end{align*}

**The slope of this line is 2.**

*Write this ratio down in your notebook.*

**Sometimes, you won’t have a graph to look at. We can also calculate the slope of a line when we have been given two sets of ordered pairs. Then we can use a formula to calculate the slope of the line.**

\begin{align*}Slope=\frac{y_2-y_1}{x_2-x_1}\end{align*}

*Write this formula down in your notebook.*

Example

Calculate the slope of a line that passes through the points (0, -2) and (1, 2).

**To start, we substitute the values of these coordinates into our formula. It doesn’t matter which value you use as \begin{align*}y_2\end{align*} or \begin{align*}y_1\end{align*} the key is that you are consistent in your choices. Here is how these values can be substituted into the formula.**

\begin{align*}Slope &=\frac{2--2}{1-0}\\ Slope &=4\end{align*}

**The slope of this line is 4.**

*Note: Sometimes, you will also see the letter “\begin{align*}m\end{align*}” used in place of the word “slope”.*

II. **Recognize and Interpret Positive, Negative, Zero and Undefined Slopes**

In the last section, you began to learn how to figure out the slope of a line. The two slopes in the last section were both positive slopes. This means that when we did our calculations, that the slope of the line was positive. In looking at the graph, you can also see that the line went up from the left corner of the graph to the right. Look at this graph of a positive slope.

We can also see what a graph with a negative slope would look like. Take a look at this example.

Notice that this line goes down from left to right. By looking at it, we can see that it is negative. We can also look at the slope of the line. Look at these arrows to see how this is a negative slope.

**We can also have lines with a slope of zero.** Take a look at this graph.

This line doesn’t go up and doesn’t go down. It has a slope of zero.

**What about a vertical line?** Let’s look at an example.

Example

Now use the slope formula on the line that goes through the points (2, 3) and (2, -3).

\begin{align*}{x_1}=2,{y_1}=3,{x_2}=2,{y_2}=-3\end{align*}

\begin{align*}m &=\frac{{y_2}-{y_1}}{{x_2}-{x_1}}\\ m &=\frac{-3-3}{2-2}\\ m &=\frac{-6}{0}\end{align*}

Slope here is undefined since it has a denominator of zero, \begin{align*}m\end{align*} is undefined. Graph this line.

This line is vertical!

**All vertical lines have an undefined slope.**

III. **Compare and Interpret Relative Slopes of Given Lines**

**We’ve seen four types of slopes.**

Positive slopes Rise up and run right |
Negative slopes Rise up and run left |
||
---|---|---|---|

Zero slopes Horizontal lines |
Undefined slopes Vertical lines |

But we have yet to compare their steepness. If you consider the ski slopes, you know that some slopes are steeper than others. In the graph below, you can also see that some lines are steeper, or have a greater slope, than others.

**In looking at these lines, we can determine that some lines are steeper than others. In the example above, lines with a greater slope are steeper than lines with a smaller slope.**

**Slope of 5 is > slope of \begin{align*}\frac{1}{2}\end{align*}**

As you continue to work with slope, you will see how the steepness of a line can be measured and how to determine the slope by looking at the equation of a line.

IV. **Identify and Distinguish Between Direct and Inverse Variation**

Example

A train leaves Boston traveling at a constant speed of 60mph. You can make a table showing the distance, \begin{align*}d\end{align*}, in miles that the train has traveled after \begin{align*}h\end{align*} hours.

\begin{align*}h\end{align*} | \begin{align*}d\end{align*} |
---|---|

0 | 0 |

3 | 180 |

6 | 360 |

9 | 540 |

12 | 720 |

**This relationship can be shown in the function \begin{align*}d = 60 h\end{align*}.**

**This type of function is called a** *direct variation***. It is a linear equation that can be written in the form \begin{align*}y=kx\end{align*}, where \begin{align*}k \neq 0\end{align*}.** This is a linear function whose \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-intercepts are always zero—it always passes through the origin. The variable \begin{align*}k\end{align*} is called the *constant of variation* which is also the slope of the line. In the case above, the distance varies directly with the time because it increases in proportion to the time. That is, if the time doubles, the distance doubles, and so on. In a direct variation, as one variable increases, the other increases, too. In a direct variation, for any ordered pair \begin{align*}(x, y), k=\frac{y}{x}\end{align*},.

**Good question. We call this an inverse variation.**

**An** *inverse variation***can be written in the form \begin{align*}y=\frac{k}{x}\end{align*} where \begin{align*}k\end{align*} is still the constant of variation as in direct variations but the product of the ordered pairs is \begin{align*}k\end{align*}.**

**A plane flying is an excellent example of an inverse variation. When the speed of the plane increases, the time that the plane is in the air actually decreases.**

## Real-Life Example Completed

*Rainforest Transportation*

**Here is the original problem once again. Reread it and then solve the problem at the end.**

After a lot of debating and discussing, the class finally decided to attend the Omni Theater. The presentation on the rainforest was fascinating and all of the students were glad that this was the decision that they had made. While the film did show the animals and insects of the rainforest, it also focused on ecology and on scientists and other people working to save the rainforest.

When they returned to school, Mr. Thomas had them talk about different parts that had interested them.

“I was fascinated on the bus ride that the scientists took to get to the rainforest. It was amazing to me that the bus could travel over all of those bumpy roads and not have an accident,” Mark commented.

“Yes, if you think about it, the bus ride was pretty cool to see from the Omni point of view. I mean I felt like I was going to fall over the edge some of the time,” Karen added.

Others smiled as well. Mr. Thomas seizing the opportunity wrote the following problem on the board.

A bus leaves Boston traveling at a constant speed of 60mph. You can make a table showing the distance, \begin{align*}d\end{align*}, in miles that the train has traveled after \begin{align*}h\end{align*} hours.

*Solution to Real – Life Example*

**Now here is the table and the graph to represent the data.**

\begin{align*}h\end{align*} | \begin{align*}d\end{align*} |
---|---|

0 | 0 |

3 | 180 |

6 | 360 |

9 | 540 |

12 | 720 |

**This relationship can also be shown by the function \begin{align*}d = 60 h\end{align*}.**

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Slope
- the steepness of the line, calculated with the ratio \begin{align*}\frac{rise}{run}\end{align*}.

- Ratio
- a comparison between two quantities.

- Positive Slope
- a slope that goes up from the left to the right.

- Negative Slope
- a slope that goes down from the left to the right.

- Zero Slope
- the slope of a horizontal line

- Undefined Slope
- the slope of a vertical line

## Time to Practice

Directions: Answer true or false for each of the following questions.

- True or false. The equation of a line is always linear.
- True or false. A linear equation will be shown as a straight line on a graph.
- True or false. The \begin{align*}x\end{align*} – intercept is where the line crosses the \begin{align*}y\end{align*} axis.
- True or false. The \begin{align*}y\end{align*} – intercept is where the line crosses the \begin{align*}y\end{align*} axis.
- True or false. A vertical line has an undefined slope.
- True or false. A horizontal line has a slope of 0.
- True or false. Slope is the distance that the line travels on the coordinate graph.
- True or false. Slope is found using a ratio.
- True or false. You can figure out the slope of a line if you have been given one set of points.
- True or false. You will need two sets of points that a line passes through to figure out the slope.

Directions: Figure out the slope of a line that passes through each of the following pairs of points.

- (2, 3) (3, 4)
- (4, 5) (2, 3)
- (2, 1) (-1, 3)
- (3, 1) (4, 3)
- (5, 7) (3, 6)
- (3, 0) (4, 1)
- (6, 4) (2, 7)
- (2, 0) (0, 1)
- (6, 1) (1, 6)
- (4, 4) (5, 0)

Directions: Solve each problem.

- The caloric intake of a hummingbird varies directly with the amount of nectar that it consumes. For each gram the hummingbird consumes, it takes in 5 calories. Write an equation that shows the direct variation. Identify \begin{align*}k\end{align*}. Create a t-table and a graph.
- Use the data in this table to model the inverse variation with an equation and graph.

\begin{align*}& x \qquad \ 5 \qquad 10 \qquad 20 \qquad 25 \qquad 50\\ & y \qquad 20 \qquad 10 \qquad \ 5 \qquad \ 4 \qquad \ 2\end{align*}