3.1 Exponential Functions and Their Graphs
Problems
1
Examples and Practice
Finding the Balance for Compound Interest
1. A total of $9000 is invested at an annual interest rate of 2.5%,
compounded annually. Find the balance in the account after 1 year,
5 years, 10 years, 25 years, 50 years, t years.
2. A total of $4000 is invested at an annual interest rate of 5.25%,
compounded monthly. Find the balance in the account after 1 year, 5
years, 10 years, 25 years, 50 years, t years.
Finding Compound Interest
3. A total of $12,000 is invested at an annual interest rate of 3%.
Find the balance after 5 years if the interest is compounded (a)
quarterly and (b) continuously.
4. On the day of a child’s birth, a deposit of $25,000 is made in a
trust fund that pays 8.25% interest. Determine the balance in this
account on the child’s 18th birthday if the interest is compounded
(a) quarterly, (b) monthly, and (c) continuously.
Practice:
3.1 Exponential Functions and Their Graphs
Problems
2
Examples and Practice
Without using a calculator, match each graph to its function. (Determine
how the function moves compared to
.)
1.
i.
ii.
iii.
iv.
2.
3.
4.
Use the graph of f to describe the transformation that yields the graph
of g.
5.
,
g
8.
6.
,
9.
7.
,
10.
,
,
,
Use a calculator to evaluate the function at the indicated value of x.
Round your result to the nearest thousandth.
11.
,
14.
12.
13.
,
,
,
15. Compound Interest
There are three options for investing $500. The
first earns 7% compounded annually, the second earns 7% compounded
quarterly, and the third earns 7% compounded continuously.
3.1 Exponential Functions and Their Graphs
Problems
3
Examples and Practice
a. Find equations that model each investment growth and use a graphing
utility to graph each model in the viewing window over a 20-year
period.
b. Use the graph from part (a) to determine which investment yields
the highest return after 20 years. What is the difference in
earning between each investment?
16. Population Growth
The projected populations of California for the
years 2015 and 2030 can be modeled by
where P is the population (in millions) and t is the time (in years),
with
corresponding to 2015. (Source: U.S. Census Bureau)
a.
Use a graphing utility to graph the function for the years 2015
through 2030. What do you notice about the graph?
b.
Determine an estimated slope of the
graph by finding the difference
between
and
. What does this
represent?
c.
Use the table feature of a
graphing utility to create a table of
values for the same time period as in
part (a).
d.
According to the model, when will
the population of California exceed 50
million?
Year
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
Population
3.1 Exponential Functions and Their Graphs
Problems
4
Examples and Practice
17. Bacteria Growth A certain type of bacteria increases according to
the model
,
where t is the time in hours.
a.
Use a graphing utility to graph the model.
b.
Use a graphing utility to approximate
c.
Verify your answers in part (b) algebraically.
,
, and
.
18. Inflation If the annual rate of inflation averages 4% over the next
10 years, the approximate cost C of goods or services during any year
in that decade will be modeled by
, where t is the time
(in years) and P is the present cost.
a.
The price of an oil change for your car is presently $23.95. Use
to find the approximate price of an oil change 10 years from
now.
b.
The price of a gallon of milk is $3.69. Use
to approximate
the price of a gallon of milk in 5 years, 10 years, and 50 years
from now.
3.1 Exponential Functions and Their Graphs
Problems
5
c.
Examples and Practice
Find the price of an item you currently purchase. Use
to
approximate the price of the item in 5 years, 10 years, and 50
years from now.
19. Depreciation
In early 2006, a new Jeep Wrangler Sport Edition had
a manufacturer’s suggested retail price of $23,970. After t years the
Jeep’s value is given by
.
(Source: DaimlerChrysler Corporation)
a.
b.
Create a table of values that shows the value V for
years.
to
According to the model, when will the Jeep have no value?
Year
1
2
3
4
5
6
7
8
9
10
Value