Aim #80: How do we write exponential equations?
Homework: Handout
Do Now: Write the equation that best represents the given tables.
x
y
x
y
0
5
0
100
1
10
1
50
2
20
2
25
3
40
3
12.5
4
80
4
6.25
Exponential Growth and Decay
y = a(b)x
In general exponential functions represent many real world applications of growth
and decay. To determine whether an explicit formula models exponential decay or
growth, we look at the value of the growth factor b. If b > 1, we have exponential
__________. If b < 1, we have exponential __________.
(This is true for formulas where our exponent is positive).
What happens to our output if the growth factor of the formula is 1?
A more specific form is the form where the rate of growth "r" is input as a
decimal into the equation. "a" represents the amount you started with.
GROWTH: y = a(1 + r)t
DECAY: y = a(1 - r)t
Identify the initial value, "a", in each formula below, and state whether the formula
models exponential growth or exponential decay. Justify your responses.
Applications of exponential functions:
1) Bacteria is growing in a petri dish at a rate of 2% per hour. If therewere 100
bacteria present at the start, write an equation that can be used to find the
number of bacteria after x hours.
2) $200 is put in a banking account that gives you 1.5% interest compounded annually.
a) Write the formula that will help you determine how much money you will have
after t years.
b) How much money will you have after 26 years, to the nearest cent?
3) Jen was given $1000 when she turned 4 years old. Her parents invested it at a
1.5% interest rate compounded annually. No deposits or withdrawals were made.
Which expression can be used to determine how much money Jen had in the
account when she turned 21?
4) Caffeine is absorbed into the blood stream so that the amount in you body
is decreasing at a rate of 20% per hour. If you drank a cup of coffee with 50mg of
caffeine...
a) Write the formula that will help you determine how much caffeineyou will have in
your body after t hours.
b) To the nearest tenth, how much caffeine will be in your body after 3 hours?
5) Mr. Smith buys his "dream car" the Porsche 911 for $80,000. The value of the
car decreases by 10.5% each year.
a) Write an exponential decay model that gives the value of the carafter t years.
b) To the nearest dollar, how much more is the car worth after 3 years than after
4 years?
6) Mr. and Mrs. Newton buy a house for $400,000. If the value of the house
appreciates at a rate of 2% per year.
a) Write an exponential model that gives the value of the houseafter t years.
b) To the nearest dollar, use the model to estimate the value after 5 years.
7) Kelli‛s mom takes a 400 mg dose of aspirin. Each hour, the amount ofaspirin in a
person‛s system decreases by about 29%.
a) Write an explicit formula to model this situation.
b) To the nearest tenth, how much aspirin is left in her system after 6 hours?
8) A huge ping-pong tournament is held in Beijing with 65,536 participants at
the start of the tournament. Each round eliminates half the players.
a) If p(r) represents the number of players remaining after r rounds of play,
write a formula to model the number of participants remaining.
b) Use your model to determine how many players remain after 10 rounds.
c) How many rounds of play will it take to determine the champion ping-pong
player?
9) The function w(t) = 1200(1.015)t represents the value w(t), in dollars, of a piece
of artwork t years after its purchase. What is the yearly rate of appreciation of
the piece of artwork written as a percentage?
Sum It Up!
A more specific form is the form where the rate of growth "r" is input as a
decimal into the equation. "a" represents the amount you started with.
GROWTH: y = a(1 + r)t
DECAY: y = a(1 - r)t