Download Unit 1: Relationships Between Quantities

Warm-Up
A
population of mice quadruples every 6
months. If a mouse nest started out with 2
mice, how many mice would there be
after 2 years? Write an equation and then
use it to solve the problem.
Introduction to
Exponential
Functions
Exponential Functions
 An
exponential function is a function in
which the independent variable appears
in an exponent.
 An exponential function has the form
𝑓 𝑥 = 𝑎𝑏 𝑥 , where 𝑎 ≠ 0, 𝑏 ≠ 1, and 𝑏 > 0.
a
represents the initial/original/principal
amount
 b represents the rate of increase or decrease
 x represents the time period
Exponential Growth and
Decay
 Exponential
growth occurs when a
quantity increases by the same rate in
each time period.
 Exponential decay occurs when a
quantity decreases by the same amount
in each time period.
Example
A
population of insects doubles every
month. This particular population started
out with 20 insects. Find the population
after 6 months.
𝑦 = 20 ∙ 2𝑥
𝑦 = 20 ∙ 26
𝑦 = 20 ∙ 64
𝑦 = 1280
Example
 Flourine-20
has a half-life of 11 seconds.
Find the amount of Flourine-20 left from a
40-gram sample after 44 seconds.
𝐴 = 40(0.5)4
𝐴 = 2.5 grams
Growth/Decay Rates Given as
Percentages
 Note:
When the rate is given as a
percentage, you must convert the
percent to a decimal. Add or subtract the
rate from 1, depending on if it is a growth
or decay.
 For exponential growth, use the formula
𝑦 = 𝑎(1 + 𝑟)𝑡
 For exponential decay, use the formula
𝑦 = 𝑎(1 − 𝑟)𝑡
What changes about the
formula when the rate is given
as a percentage?
Example:
The original value of a painting is $1400, and
the value increases by 9% each year. Write
an exponential growth function to model
this situation. Then find the value of the
painting after 25 years.
𝑦 = 1400(1 + 0.09)𝑥
𝑦 = 1400(1.09)25
𝑦 = 1400 8.62
𝑦 =$12,072.31
Exponential Growth/Decay
and Money
 Money
is not free to borrow 
 Interest is how much is paid for the use of
money.
So how much does it cost to
borrow money??
 Different
places charge different amounts
at different times.
 In general, interest is charged as a
percent of the amount borrowed.
Example



Let’s say that Alex wants to borrow $1000 and
the local bank offers the loan with 10%
interest.
To borrow the $1000 for 1 year will cost
1000 + 1000 0.10
So Alex borrows $1000, but must pay back
$1100.
 There
are generally special words used
when borrowing money, as shown
below…
 Alex
is the Borrower. The bank is the
Lender. The Principal of the loan is $1000.
The Interest is $100.
More Than One Year…
 What
if Alex wanted to borrow the money
for 2 years?
 If the bank charges “Simple Interest” then
Alex just pays another 10% for the extra
year. So after 2 years, Alex will pay $1200
 But
what if the bank says “If you paid me
everything back after one year, and then
I loaned it to you again… I would be
loaning your $1100 for the second year!”
 Then Alex would pay $110 interest in the
second year, not jus $100.
 Such a way of calculating interest is
called compounding.
Compound Interest
 Compound
interest is the interest earned
or paid on both the principal and
previously earned interest.
How much would Alex owe on
a 5 year loan with 10% interest
compounded annually??
Year
Loan at Start
Interest
Loan at End
0 (Birth of the
loan)
$1000
$100
$110
1
$1100
Formula for Compound
Interest

The formula for compound interest is as
follows:

Growth:

A is the balance after t years
P is the principal/original amount
r is the rate (given as a decimal)
n is the number of times interest is
compounded per year
t is the time in years




𝑟
𝑛
A = 𝑃(1 + )𝑛𝑡
Periodic Compounding



It is also possible to have yearly interest with
several compounding’s within the year.
For example, 6% interest with monthly
compounding does not mean 6% per month,
it means 0.5% per month (6% divided by 12
months)
For a $1000 loan, the final amount owed
could be worked out as follows:
0.06 12
1000(1 +
) = $1,061.68
12
Example
 $1000
is invested at a rate of 3%
compounded quarterly for 5 years
.03 4(5)
𝐴 = 1000(1 +
)
4
𝐴 = 1000(1.0075)20
𝐴 ≈ 1161.18
Example
 $18,000
is invested at a rate of 4.5%
compounded annually for 6 years.
0.045 1∙6
𝐴 = 18,000(1 +
)
1
𝐴 = 18,000(1.045)6
𝐴 ≈ $23,440.68