EXPONENTIAL RELATIONSHIPS
(AN INVESTIGATION)
INTRODUCTION
What would you choose?
One million dollars today…
or
the final amount after doubling
1 penny for 30 days.
Complete the chart below:
Day
1
2
3
4
5
6
7
8
9
10
Total ¢
Day
11
12
13
14
15
16
17
18
19
20
Total ¢
Day
21
22
23
24
25
26
27
28
29
30
Total ¢
EXPONENTIAL RELATIONS
A relation that can be represented by the form y = ax,
where a is a positive constant and a  1, is called an
exponential relation.
Examples of exponential relations include:
 the growth of bacteria
 compound interest earned on an investment
 the rate of decay of radioactive materials
In an exponential relation, the ratios of consecutive
y-values are constant.
Unit 4 Lesson 3
Page 1 of 3
The graph of an exponential relation is a curve that is approximately horizontal at one
end and increases or decreases rapidly at the other end.
Exponential Growth
y = 2x
x
–3
Exponential Decay
y
–2
1
y =  
2
x
x
–3
–2
–1
–1
0
0
1
1
2
2
3
3
Exponential Growth
y
Exponential Decay
 non–linear growth and the
graph is an upward curve
 non–linear decay and the
graph is a downward curve
 it is an increasing function
 it is a decreasing function
 the base is a whole number
greater than 1 (ex. y = 2x)
 the base is a fraction less than 1
(ex. y = (½)x)
Unit 4 Lesson 3
Page 2 of 3
Example 
Example 
Decide whether the given relation is exponential:
x
y
0
1
2
3
4
5
2
6
18
54
162
486
Ratio of Successive
y–values
Sketch each of the following exponential relations on the axes below:
y = 2x
1
y =  
2
y = 3x
x
1
y = 
3
y
0
y = 5x
x
1
y = 
5
x
y
x
1.
Explain the trend exhibited by the graphs of exponential equations
(ie) what happens as the value of the base changes.
2.
What is the point of intersections of all the exponential graphs? Why???
Unit 4 Lesson 3
Page 3 of 3