4.4: Slope and Rate of Change
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
Similarly, if the car begins to descend down a hill, you can still determine the slope.
The slope in this instance is negative because the car is traveling downhill.
Another way to think of slope is:
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 slopetriangle.
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:
Starting at the leftmost coordinate, count the number of vertical units and horizontal units it took to get to the rightmost coordinate.
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.
A more algebraic way to determine a slope is by using a formula. The formula for slope is:
The slope between any two points
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
Use the formula:
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
Any line with an undefined slope will be a vertical line with equation
We will use the next two graphs to illustrate the previous definitions.
To determine the slope of line
Sample: (–4, 3) and (1, 3). Choose one ordered pair to represent
Now apply the formula:
To determine the slope of line
Sample: (5, 1) and (5, –6)
It is impossible to divide by zero, so the slope of line
Finding Rate of Change
When finding the slope of realworld 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 parttime 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
Use the formula:
Andrea’s rate of change is $15/1 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
Use the formula:
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 onehour break for lunch. Identify what is happening at each stage of the journey (stages
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
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
Segments
The rate of change for line segment
The rate of change for line segment
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. CK12 Basic Algebra: Slope and Rate of Change (13:42)
 Define slope.
 How is slope related to rate of change? In what ways is it different?
 Describe the two methods used to find slope. Which one do you prefer and why?
 What is the slope of all vertical lines? Why is this true?
 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.
 (–5, 7) and (0, 0)
 (–3, –5) and (3, 11)
 (3, –5) and (–2, 9)
 (–5, 7) and (–5, 11)
 (9, 9) and (–9, –9)
 (3, 5) and (–2, 7)

(12,34) and (–2, 6)  (–2, 3) and (4, 8)
 (–17, 11) and (4, 11)
 (31, 2) and (31, –19)
 (0, –3) and (3, –1)
 (2, 7) and (7, 2)
 (0, 0) and
(23,14)  Determine the slope of
y=16 .  Determine the slope of
x=−99 .  The graph below is a distancetime graph for Mark’s 3.5mile 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.
 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?
 Jenna earns $60 every
212 weeks . What is her rate of change?  Geoffrey has a rate of change of 10 feet/1 second. Write a situation that could fit this slope.
Mixed Review
 Find the intercepts of
3x−5y=10 .  Graph the line
y=−6 .  Draw a line with a negative slope passing through the point (3, 1).
 Draw a graph to represent the number of quarter and dime combinations that equal $4.00.
 What is the domain and range of the following:
{(−2,2),(−1,1),(0,0),(1,1),(2,2)} ?  Solve for
y:16y−72=36 .  Describe the process used to solve an equation such as:
3x+1=2x−35 .  Solve the proportion:
6a=142a+1 .
Quick Quiz
 Find the intercepts of
3x+6y=25 and graph the equation.  Find the slope between (8, 5) and (–5, 6).
 Graph
f(x)=2x+1 .  Graph the ordered pair with the following directions: 4 units west and 6 units north of the origin.
 Using the graph below, list two “trends” about this data. A trend is something you can conclude about the given data.