copyright 2014 scott storla average rate of change some vocabulary

Copyright 2014 Scott Storla

Average Rate of ChangeSome Vocabulary


The words increasing, decreasing or constant discuss the behavior of y values as x values increase (move to the right) on the number line.

We discuss where a function is increasing, decreasing or constant using intervals on x.


The function is increasing on the interval x = 0 to x = 2.

Incr

easi

ng

Constant

Decreasing

The function is constant on the interval x = 2 to x = 4.

The function is decreasing on the interval x = 4 to x = 6.

Use intervals on x to describe where the function is increasing, decreasing or constant.


The average rate of change begins with two ordered pairs and uses a quotient (a fraction) to compare the change (as a difference) in range values (the “y’s”) to the change in domain values (the “x’s”).

On a graph the average rate of change begins with two points and compares the change in the values of y to the change in the values of x as we go from one point to the other.

The average rate of change

The difference in x valuesThe difference in y values


The average rate of change is positive from x = 0 to x = 2.

Pos

itive

Zero

Negative

The average rate of change is 0 from x = 2 to x = 4.

The average rate of change is negative from x = 4 to x = 6.

We describe an average rate of change as positive, negative or 0.


Average Rate of ChangeSome Vocabulary


Average Rate of ChangeFrom a Graph

120 140



When x =10, y =140

When x =20, y =120

2

1

The difference in x values

20 1020

10

The difference in y values

2

140 120 22

10 20 1

Find and describe the average rate of change as the function goes from x = 10 to x =20.

120 140



Find and describe the average rate of change as the function goes from x = 10 to x =20.

When x =10, y =140

When x =20, y =120

2

1

The difference in x values

20 1020

10


2

The decreasing function has a negative average rate of change.


Find, and describe the average rate of change as the function goes from x = 2 to x = 4.

When x =4, y =8

When x =2, y =2

8 2 6 33

4 2 2 1

Average rate of change

The increasing function has a positive average rate of change.

2 8 6 33

2 4 2 1


Find, and describe, the average rate of change as the function goes from x = 0 to x =7.

When x =7, y =1

When x =0, y =9

1 9 8

7 0 7


The decreasing function has a negative average rate of change.


Find, and describe, the average rate of change as the function goes from x = 10 to x =20.

When x =20, y =6

When x =10, y =6

6 6 00

20 10 10


The constant function has an average rate of change of 0.


Average Rate of ChangeFrom a Graph


Applying Average Rate of ChangeTo a Graph – Getting Ready


Applying the Average Rate of Change

When applying the average rate of change make sure to include the units (units are what the numbers are counting or measuring) along with the final values.

We usually divide to get a denominator of 1.


The difference in x valuesm

2 1

2 1The "slope" formula

y ym

x x


What’s the general meaning for the average rate of change?

The specific average rate of change

For the first two hours the distance from home was increasing on average by fifty miles per hour.

What’s the specific meaning for the average rate of change for the interval x = 0 to x =2.

100

2

50

1

100 0

2 0m

miles

hour

(0,0)

(2,100)

How the distance from home changes over time.


Applying Average Rate of ChangeTo a Graph – Getting Ready


Applying Average Rate of ChangeTo a Graph


What’s the specific meaning of the average rate of change between 1980 and 1985?

What’s the specific meaning of the average rate of change between 1990 and 1995?

What’s the general meaning of the average rate of change?

Describe the trend in the average rate of change for the two time periods.

36 17 19 3.8 subscribers

5 0 5 1 yearm

Between 1980 and 1985 the number of subscribers on average was increasing by 3.8 million per year.

59 50 9 1.8 subscribers

15 10 5 1 yearm

Between 1990 and 1995 the number of subscribers on average was increasing by 1.8 million per year.

Although the number of subscribers is continuing to increase the rate of increase is slowing down.

The change in the number of subscribers over time.



What’s the specific meaning for the average rate of change over the first ten years (from year 0 to year 10)?

What’s the specific meaning for the average rate of change between years 40 and 50.


33 20 13 $1.3

10 0 10 1 yearm

On average for the first 10 years the value of the account is increasing by $1,300 per year.

244 148 96 $9.6

50 40 10 1 yearm

On average between years 40 and 50 the value of the account is increasing by $9,600 per year.

The longer you leave your money in the account the faster the rate of increase in the value of the account.

The change in the value of the account over time.


What’s the specific meaning for the average rate of change between midnight and 4 a.m.?

What’s the specific meaning for the average rate of change between 4 p.m. and 8 p.m.?


The change in the outside temperature over time.

10 2 8 2 degrees

0 4 4 1 hourm

On average the temperature is increasing 2 degrees per hour between midnight and 4 a.m..

8 6 2 0.5 degrees

16 20 4 1 hourm

On average the temperature is decreasing half a degree per hour between 4 p.m. and 8 p.m..


Estimate the specific meaning of the average rate of change between purchase and the beginning of year 2.

Estimate the specific meaning of the average rate of change between year 8 and year 10.



32,000 20,000 12,000 $6,000

0 2 2 1 yearm

Between purchase and the beginning of year 2 the value is decreasing on average by $6,000 per year.

6,000 4,000 2,000 $1,000

8 10 2 1 yearm

Between years 8 and 10 the value of the vehicle is decreasing on average by $1,000 per year.

Although the value of the vehicle is steadily decreasing over time the rate at which it’s decreasing is slowing down.

The change in the value of a hybrid vehicle over time.


Applying Average Rate of ChangeTo a Graph


Applying Average Rate of ChangeTo a Data Table


Find, and discuss the meaning of, the average rate of change for the first 10 months.

Find, and discuss the meaning of, the average rate of change for month 10 to month 15.

Find, and discuss the meaning of, the average rate of change for month 0 to month 15.


How the outstanding debt is changing over time.

2000 5200 3200 $320

10 0 10 1 monthm

On average the loan amount is decreasing by $320 per month.

2000 400 1600 $320

10 15 5 1 monthm

5200 400 4800 $320

0 15 15 1 monthm



Describe the trend in the average rate of change.

Over time the average rate of change remains constantly decreasing at $320 per month.

MonthsOutstanding

paying on adebt ($)

loan

0 5,200

10 2,000

15 400


Minutes since the Number of carsparking ramp opened in parking lot

5 22715 46730 82760 154790 2267

2 1

2 1

827 227 60024

30 5 25

y ym

x x

Find the slope

copyright 2014 scott storla average rate of change some vocabulary

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