Aim #81: How can we determine if a set of data can be
modeled by a linear equation or an exponential equation?
Do Now: The growth of a certain population from 2003 through 2015 can be
modeled by the equation f(t) = 1543(1.018)t where 1543 represents the population
(in millions) in the year 2003 and t represents the number of years after 2003.
What was the population in 2011?
1) Complete the chart comparing linear to exponential functions.
2) A river has an initial minnow population of 40,000 that is growing at 5% per
year. Due to environmental conditions, the amount of algae that minnows use for
food is decreasing, supporting 1000 fewer minnows each year. Currently, there is
enough algae to support 50,000 minnows.
a) Is the minnow population increasing linearly or exponentially?
b) Is the amount of algae decreasing at a linear or exponential rate?
c) In what year will the minnow population exceed the amount of algae available?
3) Using a calculator, Joanna made the following table and then made the following
conjecture: 3x is always greater than (1.02)x. Is Joanna correct? Explain.
4) Mr. Smith has an apple orchard. He hires his daughter, Lucy, to
pick apples and offers her two payment options.
Option A: $1.50 per bushel of apples picked
Option B: $0.01 for picking one bushel, $0.03 for picking two bushels,
$0.09 for picking three bushels, and so on, with the amount
of money tripling for each additional bushel picked.
a) Write a function to model each option.
b) If Lucy picks six bushels, which option should she choose?
c) If Lucy picks 12 bushels, which option should she choose?
d) How many bushels does Lucy need to pick to make option B better
for her than option A?
5) Tim deposits money in a Certificate of Deposit account. The
balance (in dollars) in his account t years after making the deposit is
given by T(t) = 1000(1.06)t for t ≥ 0.
a) Explain, in terms of the structure of the expression used to define
T(t), why Tim's balance can never be $999.
b) By what percent does the value of T(t) grow each year?
c) By what percentage does the value of T(t) grow every two years?
6) For each of the following tables, determine if it should be
modeled with a linear equation or an exponential equation, then
write the equation.
a)
c)
x f(x)
1
b)
x f(x)
8
1
256
3
2
0
64
9
32
1
16
12
47
2
4
30
137
3
1
x f(x)
5
d)
x f(x)
1
2
9
6
1.5
3
27
7
2
4
81
8
2.5
5
243
9
3
6
729
Sum It Up!!
Linear equations grow at a constant difference.
Exponential equations grow at a constant rate.
HW #81 Answers
1. exponential; f(x) = (.5)x
2. linear; f(x) = 1.1x + .3
3. neither
4. exponential; f(x) = 10(2)x
5. linear; f(x) = -7x + 2
6. neither
7. a. exponentially; f(x) = 100000(1.02)x
b. linear; f(x) = -1500x + 200000
c. year 25
8. Plan 1 is a better choice
Review: x = 26