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# Rates of Change

## Understand slope as rate of change

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Slopes of Lines from Graphs

Joseph drove from his summer home to his place of work. To avoid the road construction, Joseph decided to travel the gravel road. After driving for 20 minutes, he was 62 miles away from work, and after driving for 40 minutes, he was 52 miles away from work. This situation is shown in the graph below. Determine the slope of the line, and describe what it means in this situation.

License: CC BY-NC 3.0

### Slope

The slope of a line is the steepness, slant or gradient of that line. Slope is often defined as riserun\begin{align*}\frac{\text{rise}}{\text{run}}\end{align*} (rise over run). The slope of a line is represented by the letter m.\begin{align*}m.\end{align*}

You can determine the slope of a line from a graph by counting. Choose two points on the line that are exact points on the Cartesian grid. Exact points mean points that are located on the corner of a box or points that have coordinates that do not have to be estimated. On the graph below, two exact points are indicated by the blue dots.

License: CC BY-NC 3.0

Begin with the point that is farthest to the left and 'run' to the right until you are directly below (in this case) the second indicated point. Count the number of spaces that you had to run to be below the second point and place this value in the 'run' position (the denominator) of the slope. Next, count the number of spaces you have to move to reach the second point. In this case you have to rise upward which indicates a positive move. This value must be placed in the 'rise' position (the numerator) of the slope.

Here, you had to run 5 spaces to the right, which indicates moving 5 spaces in a positive direction. You now have m=rise5\begin{align*}m=\frac{\text{rise}}{{\color{blue}5}}\end{align*}. To reach the point directly above involved moving upward 6 spaces in a positive direction. You now have m=65\begin{align*}m=\frac{{\color{magenta}6}}{{\color{blue}5}}\end{align*}. The slope of the above line is 65\begin{align*}\frac{6}{5}\end{align*}.

License: CC BY-NC 3.0

In the above graph, there are not two points on the line that are exact points on the Cartesian grid. Therefore, the slope of the line cannot be determined by counting. The coordinates of points on this line would only be estimated values. When this occurs, the task of calculating the slope of the line must be presented in a different way. The slope would have to be determined from two points that are on the line and these points would have to be given.

#### Let's practice finding the slope in the following graphs:

1.

License: CC BY-NC 3.0

Two points have been indicated. These points are exact values on the graph. From the point to the left, run one space in a positive direction and rise upward 2 spaces in a positive direction.

mm=riserun=21\begin{align*}m &=\frac{\text{rise}}{\text{run}}\\ m &= \frac{2}{1}\end{align*}

1.

License: CC BY-NC 3.0

Two points have been indicated. These points are exact values on the graph. From the top-left point, 'run' two spaces in a positive direction, then 'rise' downward 5 spaces in a negative direction.

mm=riserun=52\begin{align*}m &=\frac{\text{rise}}{\text{run}}\\ m &= \frac{-5}{2}\end{align*}

1.

License: CC BY-NC 3.0

Two points on this line are (–5, 5) and (4, 5). The rise is 0 and the run is 9. The slope is m=09=0\begin{align*}m=\frac{0}{9}=0\end{align*}.

All lines perpendicular to the y-axis (horizontal lines) will have a slope of 0.

1.

License: CC BY-NC 3.0

Two points on this line are (–3, 5) and (–3, –10). The rise is 15 and the run is 0. The slope is m=150=undefined.\begin{align*}m=\frac{15}{0}= \text{undefined}.\end{align*}

All lines perpendicular to the x-axis (vertical lines) will have a slope that is undefined, since dividing by zero is undefined.

Note that having a slope of 0 is different from having a slope that is undefined.

### Examples

#### Example 1

License: CC BY-NC 3.0

Earlier, you were asked to find the slope of the line in the graph above.

If the slope is calculated by counting, caution must be used to determine the correct values for both rise and run. The scale on both the x\begin{align*}x\end{align*}-axis and y\begin{align*}y\end{align*}-axis of this graph is in increments of ten. The x\begin{align*}x\end{align*}-axis represents the driving time in minutes. The y\begin{align*}y\end{align*}-axis represents the driving distance, in miles.

Two points have been specified. These points are exact values on the graph. From the point to the left, run 20 spaces spaces in a positive direction and move downward 10 spaces.

mm=riserun=-1020=-12\begin{align*}m &=\frac{\text{rise}}{\text{run}}\\ m &= \frac{\text{-}10}{20}=\text{-}\frac{1}{2}\end{align*}

The slope means that for every two minutes that Joseph is driving, he has one fewer mile remaining to get to work.

#### Example 2

Identify the slope for the following graph.

License: CC BY-NC 3.0

License: CC BY-NC 3.0

Two exact points on the above graph are (0, 4) and (16, –2). From the point to the left, run sixteen spaces in a positive direction and move downward six spaces in a negative direction.
mmm=riserun=-616=-38\begin{align*}m&=\frac{\text{rise}}{\text{run}}\\ m&=\frac{\text{-}6}{16}\\ m&=\frac{\text{-}3}{8}\end{align*}

#### Example 3

Identify the slope for the following graph.

License: CC BY-NC 3.0

License: CC BY-NC 3.0

Two exact points on the above graph are (–2, –2) and (8, 4). From the point to the left, run ten spaces in a positive direction and move upward six spaces in a positive direction.
mmm=riserun=610=35\begin{align*}m&=\frac{\text{rise}}{\text{run}}\\ m&=\frac{6}{10}\\ m&=\frac{3}{5}\end{align*}

#### Example 4

What is the slope of the line that passes through the point (2, 4) and is perpendicular to the x\begin{align*}x\end{align*}-axis?

You are not given the coordinates of two points. Sketch the graph according the information given.

License: CC BY-NC 3.0
A line that is perpendicular to the x\begin{align*}x\end{align*}-axis is parallel to the y\begin{align*}y\end{align*}-axis. The slope of a line that is parallel to the y\begin{align*}y\end{align*}-axis has a slope that is undefined.

#### Example 5

What is the slope of the line that passes through the point (–6, 8) and is perpendicular to the y\begin{align*}y\end{align*}-axis?

You are not given the coordinates of two points. Sketch the graph according the information given.

License: CC BY-NC 3.0
A line that is perpendicular to the y\begin{align*}y\end{align*}-axis is parallel to the x\begin{align*}x\end{align*}-axis. The slope of a line that is parallel to the x\begin{align*}x\end{align*}-axis has a slope of zero.

### Review

1. Explain how to find the slope of a line from its graph.

2. What does the slope of a line represent?

3. From left to right, a certain line points upwards. Is the slope of the line positive or negative?

4. How can you tell by looking at a graph if its slope is positive or negative?

5. What is the slope of a horizontal line?

6. What is the slope of a vertical line?

Find the slope of each of the following lines.

1. .

License: CC BY-NC 3.0

1. .

License: CC BY-NC 3.0

1. .

License: CC BY-NC 3.0

1. .

License: CC BY-NC 3.0

1. .

License: CC BY-NC 3.0

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License: CC BY-NC 3.0

For questions 15 and 16, plot the pairs of points and then find the slope of the line connecting the points. Can you come up with a way to find the slope without graphing?

1. (2,4)\begin{align*}(2, 4)\end{align*} and (-1,3)\begin{align*}(\text{-}1, 3)\end{align*}
2. (-4,-2)\begin{align*}(\text{-}4, \, \text{-}2)\end{align*} and (2,7)\begin{align*}(2, 7)\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.1.

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### Vocabulary Language: English

Horizontally

Horizontally means written across in rows.

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$

Vertically

Vertically means written up and down in columns.

### Image Attributions

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