Rate of Change & Slope

Rate of Change

Rate of change allows us to look at the relationship between different quantities. Rate of change is expressed as a ratio of the dependent variable (y) and the independent variable (x).

Rate of change =

You can find the rate of change over an interval by finding the difference. You can compare the rate over change for different intervals to determine where the least and greatest rate of change occurs.

change in dependent variablechange in independent variable

Review: The slope of a line is its “Constant Rate of Change”

Recall that: Rate of change =

Therefore,

Slope = m =

(where )

(x1, y1) and (x2, y2) are simply an easy

way to keep track of the first point, (x1, y1), you

use and the second point, (x2, y2), you use.

012 xx

change horizontalchange vertical

12

12change horizontal

change verticalxxyy

runrise

Let’s look at a tableNumber of Pages Read

5 15 25 35 45

Amount of Time in Minutes

8 12 16 20 24Joanna recorded the number of minutes she read and how many pages she read during that time. What does the slope tell us about the number of pages she reads?

We will use the first two values in the table to find the slope.

Slope = 12 – 8 = 4/5 = 0.8/115 – 5

So that means she reads 4 pages in five minutes or 0.8 pages in one minute.

Let’s try another one!Number of texts 100 200 300 400 500

Jarrod’s Bill $20 $30 $40 $50 $60Mike’s Bill $17 $29 $41 $53 $65

What can be determined about which person pays the most money per text? How much more?

We are going to find the slope for Jarrod and Mike’s bill, and then compare those slopes to answer the question.

Let’s Compare Slopes Jarrod’s Bill

30.00 – 20.00200 – 100

10 dollars/100 texts$0.10/text

Mike’s Bill

29.00 – 17.00200 – 100

12 dollars/100 texts$0.12/text

Even though Mike paid less money for the smaller number of texts, he is still paying $0.02 more per text!

Understanding Slope In the graph to the left, we

have a comparison of how long it takes to bike vs. how long it takes to walk around a local city.

We find two points on the graph and find the slope. We can use (5, 15) and (10, 30)

The slope is (30-15)/(10-5) and then we reduce to get 3/1.

This means that in this city for every one minute biked, it would take 3 minutes to walk.

What do we do with slope? We can use the slope to understand the relationship between the

variables.

Example: When Marvin works 8 hours, he earns $78 He earns $68.25 when he works for 7 hours. What is the rate of change?

So, we use our slope formula:78 – 68.25 = 9.75 (dollars) 8 – 7 1 (hour)

If we think about the units of the slope, we realize that this is saying he makes $9.75 for working one hour. So Marvin makes $9.75 an hour.

In terms of an equation A local cab company charges by the mile to drive

passengers from one location to another. The equation can be written as C(m) = 0.85m + $3.00 where m stands for the number of miles and C(m) stands for the total cost. What can we conclude about the equation according to its slope? If we think about this equation, we have two numbers

in play. The $3.00 is just a flat rate. You would need to pay that $3.00 just for using the cab.

We notice that the $0.85/1 is multiplied by the number of miles, so that is our slope and tells us that each mile travelled will cost $0.85.

Balloon ExampleJoshua let go of his balloon and its height is shown as a function of time. What does the slope tell you?

Well, let’s look at two points and find the slope:10 sec, height is 25 feet20 sec, height is 50 feetSlope: 50 – 25 = 25 = 2.5/1

20 – 10 10

So what does this mean?It means that it takes the

balloon 10 seconds to travel 25 feet, or one

second to take 2.5 feet. So, we can say that the

balloon travels 2.5 feet/second.

Bird ExampleA bird descends from the sky according to the graph at the left. What does the slope tell us?

Well, let’s look at two points and find the slope:0 seconds, 200 feet15 seconds, 125 feetSlope: 125 - 200 = -75 = -5/1

15 – 0 15

So what does this mean?It means that over a time of

15 seconds, the bird descends 75 feet. Or, after

one second, the bird descends 5 feet. So the

bird travels down 5 feet/second.