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# 4.4: Slope and Rate of Change

Difficulty Level: At Grade Created by: CK-12

The pitch of a roof, the slant of a ladder against a wall, the incline of a road, and even your treadmill incline are all examples of slope.

The slope of a line measures its steepness (either negative or positive).

For example, if you have ever driven through a mountain range, you may have seen a sign stating, “10% incline.” The percent tells you how steep the incline is. You have probably seen this on a treadmill too. The incline on a treadmill measures how steep you are walking uphill. Below is a more formal definition of slope.

The slope of a line is the vertical change divided by the horizontal change.

In the figure below, a car is beginning to climb up a hill. The height of the hill is 3 meters and the length of the hill is 4 meters. Using the definition above, the slope of this hill can be written as 3 meters4 meters=34\begin{align*}\frac{3 \ meters}{4 \ meters}=\frac{3}{4}\end{align*}. Because 34=75%\begin{align*}\frac{3}{4}=75\%\end{align*}, we can say this hill has a 75% positive slope.

Similarly, if the car begins to descend down a hill, you can still determine the slope.

Slope=vertical changehorizontal change=34\begin{align*}Slope=\frac{vertical \ change}{horizontal \ change}=\frac{-3}{4}\end{align*}

The slope in this instance is negative because the car is traveling downhill.

Another way to think of slope is: slope=riserun\begin{align*}slope=\frac{rise}{run}\end{align*}.

When graphing an equation, slope is a very powerful tool. It provides the directions on how to get from one ordered pair to another. To determine slope, it is helpful to draw a slope-triangle.

Using the following graph, choose two ordered pairs that have integer values such as (–3, 0) and (0, –2). Now draw in the slope triangle by connecting these two points as shown.

The vertical leg of the triangle represents the rise of the line and the horizontal leg of the triangle represents the run of the line. A third way to represent slope is:

slope=riserun\begin{align*}slope=\frac{rise}{run}\end{align*}

Starting at the left-most coordinate, count the number of vertical units and horizontal units it took to get to the right-most coordinate.

slope=riserun=2+3=23\begin{align*}slope=\frac{rise}{run}=\frac{-2}{+3}=-\frac{2}{3}\end{align*}

Example 1: Find the slope of the line graphed below.

Solution: Begin by finding two pairs of ordered pairs with integer values: (1, 1) and (0, –2).

Draw in the slope triangle.

Count the number of vertical units to get from the left ordered pair to the right.

Count the number of horizontal units to get from the left ordered pair to the right.

Slope=riserun=+3+1=31\begin{align*}Slope=\frac{rise}{run}=\frac{+3}{+1}=\frac{3}{1}\end{align*}

A more algebraic way to determine a slope is by using a formula. The formula for slope is:

The slope between any two points (x1,y1)\begin{align*}(x_1,y_1 )\end{align*} and (x2,y2)\begin{align*}(x_2,y_2)\end{align*} is: slope=y2y1x2x1\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}\end{align*}.

(x1,y1)\begin{align*}(x_1,y_1)\end{align*} represents one of the two ordered pairs and (x2,y2)\begin{align*}(x_2,y_2)\end{align*} represents the other. The following example helps show this formula.

Example 2: Using the slope formula, determine the slope of the equation graphed in Example 1.

Solution: Use the integer ordered pairs used to form the slope triangle: (1, 1) and (0, –2). Since (1, 1) is written first, it can be called (x1,y1)\begin{align*}(x_1,y_1)\end{align*}. That means (0,2)=(x2,y2)\begin{align*}(0,-2)=(x_2,y_2)\end{align*}

Use the formula: slope=y2y1x2x1=2101=31=31\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-1}{0-1}=\frac{-3}{-1}=\frac{3}{1}\end{align*}

As you can see, the slope is the same regardless of the method you use. If the ordered pairs are fractional or spaced very far apart, it is easier to use the formula than to draw a slope triangle.

## Types of Slopes

Slopes come in four different types: negative, zero, positive, and undefined. The first graph of this lesson had a negative slope. The second graph had a positive slope. Slopes with zero slopes are lines without any steepness, and undefined slopes cannot be computed.

Any line with a slope of zero will be a horizontal line with equation y=some number\begin{align*}y = some \ number\end{align*}.

Any line with an undefined slope will be a vertical line with equation x=some number\begin{align*}x = some \ number\end{align*}.

We will use the next two graphs to illustrate the previous definitions.

To determine the slope of line A\begin{align*}A\end{align*}, you need to find two ordered pairs with integer values.

Sample: (–4, 3) and (1, 3). Choose one ordered pair to represent (x1,y1)\begin{align*}(x_1,y_1)\end{align*} and the other to represent (x2,y2)\begin{align*}(x_2,y_2)\end{align*}.

Now apply the formula: slope=y2y1x2x1=331(4)=01+4=0\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{3-3}{1-(-4)}=\frac{0}{1+4}=0\end{align*}.

To determine the slope of line B\begin{align*}B\end{align*}, you need to find two ordered pairs on this line with integer values and apply the formula.

Sample: (5, 1) and (5, –6)

slope=y2y1x2x1=6155=70=Undefined\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{-6-1}{5-5}=\frac{-7}{0}=Undefined\end{align*}

It is impossible to divide by zero, so the slope of line B\begin{align*}B\end{align*} cannot be determined and is called undefined.

## Finding Rate of Change

When finding the slope of real-world situations, it is often referred to as rate of change. “Rate of change” means the same as “slope.” If you are asked to find the rate of change, use the slope formula or make a slope triangle.

Example 3: Andrea has a part-time job at the local grocery store. She saves for her vacation at a rate of 15 every week. Find her rate of change. Solution: Begin by finding two ordered pairs. You can make a chart or use the Substitution Property to find two coordinates. Sample: (2, 30) and (10, 150). Since (2, 30) is written first, it can be called (x1,y1)\begin{align*}(x_1,y_1)\end{align*}. That means (10,150)=(x2,y2)\begin{align*}(10,150)=(x_2,y_2)\end{align*}. Use the formula: slope=y2y1x2x1=15030102=1208=151\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{150-30}{10-2}=\frac{120}{8}=\frac{15}{1}\end{align*}. Andrea’s rate of change is15/1 week.

Multimedia Link: For more information regarding rates of change, visit NCTM’s website for an interactive – http://standards.nctm.org/document/eexamples/chap6/6.2/part2.htm – rate of change activity.

Example 4: A candle has a starting length of 10 inches. Thirty minutes after lighting it, the length is 7 inches. Determine the rate of change in the length of the candle as it burns. Determine how long the candle takes to completely burn to nothing.

Solution: Begin by finding two ordered pairs. The candle begins at 10 inches in length. So at time “zero”, the length is 10 inches. The ordered pair representing this is (0, 10). 30 minutes later, the candle is 7 inches, so (30, 7). Since (0, 10) is written first, it can be called (x1,y1)\begin{align*}(x_1,y_1)\end{align*}. That means (30,7)=(x2,y2)\begin{align*}(30,7)=(x_2,y_2)\end{align*}.

Use the formula: slope=y2y1x2x1=710300=330=110\begin{align*}slope=\frac{y_2-y_1}{x_2-x_1}=\frac{7-10}{30-0}=\frac{-3}{30}=-\frac{1}{10}\end{align*}.

The candle has a rate of change is –1 inch/10 minutes. To find the length of time it will take for the candle to burn out, you can create a graph, use guess and check, or solve an equation.

You can create a graph to help visualize the situation. By plotting the ordered pairs you were given and by drawing a straight line connecting them, you can estimate it will take 100 minutes for the candle to burn out.

Example: Examine the following graph. It represents a journey made by a large delivery truck on a particular day. During the day, the truck made two deliveries, each one taking one hour. The driver also took a one-hour break for lunch. Identify what is happening at each stage of the journey (stages A\begin{align*}A\end{align*} through E\begin{align*}E\end{align*}).

Truck’s Distance from Home by Time

Here is the driver's journey.

A. The truck sets off and travels 80 miles in 2 hours.

B. The truck covers no distance for 1 hour.

C. The truck covers (12080)=40\begin{align*}(120 - 80) = 40\end{align*} miles in 1 hour.

D. The truck covers no distance for 2 hours.

E. The truck covers 120 miles in 2 hours.

Solution: To identify what is happening at each leg of the driver’s journey, you are being asked to find each rate of change.

The rate of change for line segment A\begin{align*}A\end{align*} can be found using either the formula or the slope triangle. Using the slope triangle, vertical change=80\begin{align*}vertical \ change=80\end{align*} and the horizontal change=2\begin{align*}horizontal \ change=2\end{align*}.

slope=riserun=80 miles2 hours=40 miles/1 hour\begin{align*}slope=\frac{rise}{run}=\frac{80 \ miles}{2 \ hours}= 40 \ miles/1 \ hour\end{align*}.

Segments B\begin{align*}B\end{align*} and D\begin{align*}D\end{align*} are horizontal lines and each has a slope of zero.

The rate of change for line segment C\begin{align*}C\end{align*} using the slope formula: \begin{align*}\text{Rate of change} = \frac{\triangle y}{\triangle x} = \frac{(120-80) \ miles}{(4-3) \ hours}= 40 \ \text{miles per hour}\end{align*}.

The rate of change for line segment \begin{align*}E\end{align*} using the slope formula: \begin{align*}\text{Rate of change} = \frac{\triangle y}{\triangle x} = \frac{(0-120) \ miles}{(8-6) \ hours}= \frac{-120 \ miles}{2 \ hours}=-60 \ \text{miles per hour}\end{align*}. The truck is traveling at negative 60 mph. A better way to say this is that the truck is returning home at a rate of 60 mph.

## Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both. CK-12 Basic Algebra: Slope and Rate of Change (13:42)

1. Define slope.
2. How is slope related to rate of change? In what ways is it different?
3. Describe the two methods used to find slope. Which one do you prefer and why?
4. What is the slope of all vertical lines? Why is this true?
5. What is the slope of all horizontal lines? Why is this true?

Using the graphed coordinates, find the slope of each line.

In 9 – 21, find the slope between the two given points.

1. (–5, 7) and (0, 0)
2. (–3, –5) and (3, 11)
3. (3, –5) and (–2, 9)
4. (–5, 7) and (–5, 11)
5. (9, 9) and (–9, –9)
6. (3, 5) and (–2, 7)
7. \begin{align*}\left (\frac{1}{2},\frac{3}{4}\right )\end{align*} and (–2, 6)
8. (–2, 3) and (4, 8)
9. (–17, 11) and (4, 11)
10. (31, 2) and (31, –19)
11. (0, –3) and (3, –1)
12. (2, 7) and (7, 2)
13. (0, 0) and \begin{align*}\left (\frac{2}{3},\frac{1}{4}\right )\end{align*}
14. Determine the slope of \begin{align*}y=16\end{align*}.
15. Determine the slope of \begin{align*}x=-99\end{align*}.
16. The graph below is a distance-time graph for Mark’s 3.5-mile cycle ride to school. During this particular ride, he rode on cycle paths but the terrain was hilly. His speed varied depending upon the steepness of the hills. He stopped once at a traffic light and at one point he stopped to mend a tire puncture. Identify each section of the graph accordingly.

1. Four hours after she left home, Sheila had traveled 145 miles. Three hours later she had traveled 300 miles. What was her rate of change?
2. Jenna earns 60 every \begin{align*}2\frac{1}{2} \ weeks\end{align*}. What is her rate of change? 3. Geoffrey has a rate of change of 10 feet/1 second. Write a situation that could fit this slope. Mixed Review 1. Find the intercepts of \begin{align*}3x-5y=10\end{align*}. 2. Graph the line \begin{align*}y=-6\end{align*}. 3. Draw a line with a negative slope passing through the point (3, 1). 4. Draw a graph to represent the number of quarter and dime combinations that equal4.00.
5. What is the domain and range of the following: \begin{align*}\left \{(-2,2),(-1,1),(0,0),(1,1),(2,2)\right \}\end{align*}?
6. Solve for \begin{align*}y: 16y-72=36\end{align*}.
7. Describe the process used to solve an equation such as: \begin{align*}3x+1=2x-35\end{align*}.
8. Solve the proportion: \begin{align*}\frac{6}{a}=\frac{14}{2a+1}\end{align*}.

## Quick Quiz

1. Find the intercepts of \begin{align*}3x+6y=25\end{align*} and graph the equation.
2. Find the slope between (8, 5) and (–5, 6).
3. Graph \begin{align*}f(x)=2x+1\end{align*}.
4. Graph the ordered pair with the following directions: 4 units west and 6 units north of the origin.

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