

Course 1, Unit 5  Exponential Functions
Overview
In the Exponential Functions unit, students analyze situations
that can be modeled well by rules of the form y = a(b)^{x}.
They construct and use data tables, graphs, and equations in the form y = a(b)^{x} to
describe and solve problems about exponential relationships such as population
growth, investment of money, and decay of medicines and radioactive materials.
Key Ideas from Course 1, Unit 5

Exponential growth or decay relationship: In the rule y = a(b)^{x}, b is
the constant growth or decay factor. In tables where x is
increasing in uniform steps, the ratios of succeeding y values
will always be b. If b is greater than 1, the pattern
will be exponential growth; if b is between 0 and 1, the pattern
will be exponential decay. The value of a indicates the yintercept
(0, a) of the graph of the relationship.
Example 1: y = 4(1.3)^{x} represents
an exponential growth relationship between x and y,
where the initial value of y (when x = 0)
is 4, and the y values increase by 30% for each increase of
1 in x values. The table would be as follows:
x

0

1

2

3

y

4

5.2

6.76

8.788

Notice that each y value is 130% of the preceding y value.
Example 2: y = 4(0.5)^{x} represents
an exponential decay relationship between x and y,
where the initial value of y (when x = 0)
is 4, and the y values decrease by 50% for each increase of
1 in x values. The table would begin as follows:

Asymptote: The graph of an exponential relationship will
be asymptotic to the xaxis, getting closer and closer to
the axis without ever touching or crossing it.

NOWNEXT equations: Since exponential growth involves
repeated multiplication by a constant factor, those patterns can
be represented by equations in the general form NEXT = b * NOW, starting
at a. For example, the pattern of change in a population growing
at a rate of 20% per year from a base of 5 million in the year
2000 can be expressed as NEXT = 1.20NOW, starting
at 5.
Year 
2000

2001

2002

2003

Population
(in millions) 
5

6

7.2

8.64


Rewriting exponential expressions: See the unit summary below
for the exponent rules. Practice using these rules is distributed
throughout the Review tasks in subsequent units.
Example:
