Types of Functions, Rates of Change

Lesson 1.4

Constant Functions

Consider the table of ordered pairs

The dependent variable is the same It is constant

The graph is a horizontal line

Month 1 2 3 4 5 6

Rent Paid $735 $735 $735 $735 $735 $735

Rent Paid

$0$100$200$300$400$500$600$700$800

0 1 2 3 4 5 6 7( ) 735R m

Linear Function

Can be represented by Where a and b are constants

See Geogebra example

( )f x a x b

Slope and Y-Intercept

Considering y = m * x + b The b is the y-intercept

Where on the y-axis, the line intersects On your calculator

Go to Y= screen Enter at Y1 (2/3) * x + 5 Predict what the graph will look like before you

specify F2, 6 for standard zoom

Family of Linear Functions

Slope = Rate of Change

change in y

change in x

constant rate of change

y m x b

ym slope

x

y=3x + 5• Slope = m = 3

• y-intercept = b = 5

Slope and Y-Intercept

The function y = (2/3) * x + 5

Slope = 2/3 (up to the right) Y-intercept = 5

•

Linear Functions

Consider this set ofordered pairs

If we plot the pointsand join them wesee they lie in aline

x y

0 5

1 8

2 11

3 14

4 17

••

••

Rate of Change

Given function y = 3x + 5

••

••

x y

0 5

1 8

2 11

3 14

4 17

Change in y = y = 66

Change in x = = 2x

3

change in x 6Average rate of change = 3

change in y 2

y

y

Rate of Change

Try calculating for differentpairs of (x, y) points

You should discover that the rate of change is constant … in this case, 3

x y

0 5

1 8

2 11

3 14

4 17

1 2

1 2

change in xAverage rate of change =

change in y

y

y

y y

x x

GeogebraDemo

Slope

When slope =

Try y = -7x – 3(predict the results before you graph)

2 change in y

3 change in x

y

x

3x

2y

Family of Linear Functions

Calculating slope with two ordered pairs

•

•(X1, Y1)

(X2, Y2)

xy

1 2

1 2

y yslope

x x

Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line

through these two points?

Given two ordered pairs, (7,5) and (-3,12). What is the slope of the line

through these two points?

5 12 70.7

7 ( 3) 10

Rate of Change

Consider the function Enter into Y= screen of calculator

View tables on calculator (♦ Y)

( )f x x

You may need to specify the

beginning x value and the increment

Rate of Change

As before, determine therate of change fordifferent sets of orderedpairs

x sqrt(x)

0 0.00

1 1.00

2 1.41

3 1.73

4 2.00

5 2.24

6 2.45

7 2.651 2

1 2

change in xAverage rate of change =

change in y

y

y

y y

x x

Rate of Change (NOT a constant)

You should find that the rate of change is changing – NOT a constant.

Contrast to thefirst functiony = 3x + 5 ( )f x x

GeogebraDemo

Function Defined by a Table

Consider the two functions defined by the table The independent variable is the year.

Predict whether or not the rate of change is constant

Determine the average rate of change for various pairs of (year, sales) values

Year 1982 1984 1986 1988 1990 1992 1994

CD sales 0 5.8 53 150 287 408 662

LP sales 244 205 125 72 12 2.3 1.9

Warning

Not all functions which appear linear will actually be linear!!

Consider the set of ordered pairs Graph them Decide whether graph

is linear Check slope for different

pairs

t P

0 67.38

1 69.13

2 70.93

3 72.77

4 74.67

5 76.61

6 78.60

Results

Graph appearsstraight

But …rate of change is not a constant

66

68

70

72

74

76

78

80

0 2 4 6 8

t P slope

0 67.38  

1 69.13 1.75

2 70.93 1.8

3 72.77 1.84

4 74.67 1.9

5 76.61 1.94

6 78.6 1.99

Assignment

Lesson 1.4 Page 53 Exercises 1 – 65 EOO