# Exponential Functions

**GOAL:** Learn exponential functions with different
bases and use them to model real-world situtations.

** Exponential functions** are of the form
:
where b > 0 is called base, like f(x) = 2^{x}.

**Q1: **Where do they appear?

**A1: Everywhere!** For example, if we put $1 in an account paying 5% interest,
compounded annually,

then t years later it will become f(t) = (1.05)^{t}, which is an **exponential
function** with base b = 1.05.

** The laws of exponents. ** For b > 0 and u and v any numbers, we have

and

and

for any real number r; and

**Example 1** If and
then

** ** Graph of

Case 1: b > 1For example, y = 2x. (i) Complete the table below:
Truncate answers to 2 decimal places (iii) • Asymptote: |
Case 2: 0 < b < 1For example, y = (1/2) ^{x}.(i) Complete the table below: Truncate
answers to 2 decimal places (iii) • Asymptote: |

**Three applications of the exponential function**

**1** Compound interest

**Example 1** If $1,000 is invested in an account paying 5% interest, how much will
it grow to in 10 year

if the interest is compounded monthly?

• Annual
(in** decimals**)

• Compounding per

• Compounding

• Time(in years)

At the end of 1st period have:

At the end of 2nd period have:

At the end of 3th period have:

...

At the end of nth period have:

Interest compounded 12 times a year over t** years
**

At the end of 1 year (12 periods) have:

At the end of 2 years (24 periods) have:

...

At the end of t years have:

General formula:

**Example 2** If $8,000 is invested in an account paying 3%
interest, how much will it grow to in 15

years if the interest is compounded quarterly?

**2** Population Growth (with unlimited resources)

**Example 3 **A certain bacteria culture grows exponentially. In 1 hour the
population grows from

300,000 to 500,000. Write a formula expressing the population P as a function of
the time t in hours.

**3** Decay of radioactive substances:

**Example 4** Radon gas decays according to the formula
, where t is
measured in days.

If there are 500 cubic centimeters left after 7 days, how much was there to
begin with?