Warm-upFind the slope

y

xRun=6-2=4

Rise=3-1=2

= Rise

Run

2 1

4 2

(2,1)

(6,3)

Find the slope

Math I

UNIT QUESTION: How do we graph functions, and what can be done to change the way they look?Standard: MM1A1

Today’s Question:How do we find the rate of change of a function?Standard: MM1A1g

Constant Rate of ChangeConstant Rate of Change

The The slopeslope of a nonvertical line is the of a nonvertical line is the ratio of the vertical change (rise) to ratio of the vertical change (rise) to the horizontal change (run) between the horizontal change (run) between any two points on the line.any two points on the line.

Find the slope between (2, 4) and (4, Find the slope between (2, 4) and (4, 8).8).

2 1

2 1

y ychange in ym

changein x x x

Rate of ChangeRate of Change

The The rate of changerate of change is the ratio of the is the ratio of the change of one quantity to a change in change of one quantity to a change in another quantity.another quantity.

Example:Example:

- The table shows the amount of water - The table shows the amount of water evaporating from a swimming pool on a evaporating from a swimming pool on a hot day. Find the rate of change in hot day. Find the rate of change in gallons with respect to time.gallons with respect to time.Time (hours) 2 6 12

Gallons evaporated 4.5 13.5 27

Rate of ChangeRate of Change

Where is the greatest rate of Where is the greatest rate of change on the graph? What is the change on the graph? What is the value?value?

Partner PracticePartner Practice

Calculating rate of change from a Calculating rate of change from a table.table.

Example 4Example 4

Example 5Example 5

2.09 1.990.05

3 1dollars per day

43 382.5 deg

2 0rees per hour

Average Rate of ChangeAverage Rate of Change

The The average range of changeaverage range of change between any two points (xbetween any two points (x11,f(x,f(x11)) and )) and (x(x22,f(x,f(x22)) is the slope of the line )) is the slope of the line through the 2 points.through the 2 points.

2 1

2

2

1

1

2 1

( ) ( )y ym becomes

x x

f x f x

x x

Example 1Example 1

Find the average rate of change of Find the average rate of change of f(x) = 2xf(x) = 2x22 – 3 when x – 3 when x11= 2 and x= 2 and x22 = 4. = 4.

2 1

2 1

( ) ( ) (4) (2)

4 2

29 5 2412

4 2 2

f x f x f f

x x

Example 2Example 2

Find the average rate of change of Find the average rate of change of f(x) = xf(x) = x33 – 3x for -2 – 3x for -2 << x x << 0. 0.

2 1

2 1

( ) ( ) (0) ( 2) 0 ( 2) 21

0 ( 2) 2 2

f x f x f f

x x

Class workClass work

p. 72 #9-13p. 72 #9-13