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# 4.8: Rates of Change

Difficulty Level: Basic Created by: CK-12
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What if 1 week after a gym opened, it had 20 members, 2 weeks after it opened, it had 40 members, and 3 weeks after it opened, it had 60 members? How would you calculate the rate of change in the number of gym members? How is this different than the slope? If this rate continues, how long will it take for the gym to have 300 members? In this Concept, you'll learn the meaning of rate of change and how to calculate it. You'll also learn how to make predictions about the future based on a rate of change so that you can answer questions about real-world scenarios such as the gym.

### Try This

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.

### Guidance

Finding the 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 A

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.

#### Example B

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 the ordered pair is (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 of –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 C

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: Rate of change=yx=(12080) miles(43) hours=40 miles per hour\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 E\begin{align*}E\end{align*} using the slope formula: Rate of change=yx=(0120) miles(86) hours=120 miles2 hours=60 miles per hour\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.

### Guided Practice

Adel spent $125 on groceries in one week. Use this to predict how much Adel will spend on groceries, if she keeps buying them at the same rate, after about a month. Solution: Adel's weekly rate is$125 per week. Since a month is about 4 weeks, multiply $125/week times 4 weeks:$1251 week4 weeks=$1251 week4 weeks=$1254=500\begin{align*} \frac{125}{1 \text{ week}} \cdot 4 \text{ weeks}= \frac{125}{1 \cancel{\text{ week}}} \cdot 4 \cancel{\text{ weeks}}=125\cdot 4=500\end{align*} Adel will spend about500 on groceries in about four weeks or a month.

### Practice

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. How is slope related to rate of change? In what ways is it different?
1. 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 212 weeks\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 3x5y=10\begin{align*}3x-5y=10\end{align*}. 2. Graph the line y=6\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: {(2,2),(1,1),(0,0),(1,1),(2,2)}\begin{align*}\left \{(-2,2),(-1,1),(0,0),(1,1),(2,2)\right \}\end{align*}?
6. Solve for y:16y72=36\begin{align*}y: 16y-72=36\end{align*}.
7. Describe the process used to solve an equation such as: 3x+1=2x35\begin{align*}3x+1=2x-35\end{align*}.
8. Solve the proportion: 6a=142a+1\begin{align*}\frac{6}{a}=\frac{14}{2a+1}\end{align*}.

#### Quick Quiz

1. Find the intercepts of 3x+6y=25\begin{align*}3x+6y=25\end{align*} and graph the equation.
2. Find the slope between (8, 5) and (–5, 6).
3. Graph f(x)=2x+1\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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Color Highlighted Text Notes

### Vocabulary Language: English Spanish

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.

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.

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