Types of Functions,
Rates of Change
Lesson 1.4
Constant Functions


Consider the table of ordered pairs
1
2
3
4
5
6
Rent Paid
$735
$735
$735
$735
$735
$735
The dependent variable is the same


Month
It is constant
The graph is a
horizontal line
R (m)  735
Rent Paid
$800
$700
$600
$500
$400
$300
$200
$100
$0
0
1
2
3
4
5
6
7
Linear Function

Can be represented by


f ( x)  a  x  b
Where a and b are constants
See Geogebra example
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
y  m x b
y change in y
m  slope 

x change in x
 constant rate of change
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 of
ordered pairs
If we plot the points
and join them we
see they lie in a
line
•
x
y
0
5
1
8
2
11
3
14
4
17
•
•
•
Rate of Change

•
Given function y = 3x + 5
•
•
•
Change in y = y = 6
Change in x = x = 2
Average rate of change =
3
x
y
0
5
1
8
2
11
3
14
4
17
change in x y 6

 3
change in y y 2
6
Rate of Change

Try calculating for different
pairs of (x, y) points
change in x y
Average rate of change =

change in y y
y1  y2
Geogebra
x1  x2
Demo

x
y
0
5
1
8
2
11
3
14
4
17
You should discover that the rate of change
is constant … in this case, 3
Slope

2 change in y y


3 change in x x
When slope =
y  2
x  3

Try y = -7x – 3
(predict the results before you graph)
Family of Linear Functions

Calculating slope with two ordered pairs
(X2, Y2)
•
(X1, Y1)
•
x
y
Given two ordered pairs, (7,5) and
(-3,12). What is the slope of the line
through these two points?
y1  y2
 slope
x1  x2
5  12
7

 0.7
7  (3) 10
Rate of Change

Consider the function f ( x )  x
Enter into Y= screen of calculator

View tables on calculator (♦ Y)

You may need to
specify the
beginning x value
and the increment
Rate of Change

As before, determine the
rate of change for
different sets of ordered
pairs
change in x y
Average rate of change =

change in y y
y1  y2
x1  x2
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
2.65
Rate of Change (NOT a constant)


You should find that the rate of change
is changing – NOT a constant.
Contrast to the
first function
y = 3x + 5
f ( x)  x
Geogebra
Demo
Function Defined by a Table
Year


CD sales
0
5.8
53
150
287
408
662
LP sales
244
205
125
72
12
2.3
1.9
Consider the two functions defined by the table


1982 1984 1986 1988 1990 1992 1994
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
Warning


Not all functions which appear linear will actually
be linear!!
t
P
Consider the set of
0 67.38
ordered pairs
1 69.13



Graph them
Decide whether graph
is linear
Check slope for different
pairs
2
3
4
5
6
70.93
72.77
74.67
76.61
78.60
Results
80
78


76
Graph appears
straight
But …
rate of change is
not a constant
74
72
70
68
66
0
2
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
4
6
8
Assignment



Lesson 1.4
Page 53
Exercises 1 – 65 EOO