MATHEMATICAL PROBLEMS, EQUATIONS AND ANWERES
A certificate of deposit (CD) is a type of savings account offered by banks, typically offering
a higher interest rate in return for a fixed length of time you will leave your money invested.
If a bank offers a 24 month CD with an annual interest rate of 1.2% compounded monthly,
how much will a $1000 investment grow to over those 24 months?
Solution
First, we must notice that the interest rate is an annual rate, but is compounded monthly,
meaning interest is calculated and added to the account monthly. To find the monthly interest
rate, we divide the annual rate of 1.2% by 12 since there are 12 months in a year: 1.2%/12 =
0.1%. Each month we will earn 0.1% interest. From this, we can set up an exponential
function, with our initial amount of $1000 and a growth rate of = 0.001, and our input
measured in months.
r m
m
f(m) = 1000 (1 + )
f(m) = 1000(1 + 0.001)m
After 24 months, the account will have grown to f(24) = 1000(1 + 0.001)24 = $1024.28
Exercise
Looking at these two equations that represent the balance in two different savings accounts,
which account is growing faster, and which account will have a higher balance after 3 years?
A(t) = 1000(1.05)t
B(t) = 900(1.075)t
Answer
is growing faster ( = 0.075 > 0.05), but after 3 years A(t) still has a higher account balance
B(t) r
In all the preceding examples, we saw exponential growth. Exponential functions can also be
used to model quantities that are decreasing at a constant percent rate. An example of this is
radioactive decay, a process in which radioactive isotopes of certain atoms transform to an
atom of a different type, causing a percentage decrease of the original material over time.
2.4.3 https://math.libretexts.org/@go/page/146724
Example
Bismuth-210 is an isotope that radioactively decays by about 13% each day, meaning 13% of
the remaining Bismuth-210 transforms into another atom (polonium-210 in this case) each
day. If you begin with 100 mg of Bismuth-210, how much remains after one week?
Solution
With radioactive decay, instead of the quantity increasing at a percent rate, the quantity is
decreasing at a percent rate. Our
initial quantity is = 100 mg, and our growth rate will be negative 13%, since we are
decreasing: = -0.13. This gives the r
equation:
Q(d) = 100(1 − 0.13)d = 100(0.87)d
This can also be explained by recognizing that if 13% decays, then 87 % remains.
After one week, 7 days, the quantity remaining would be
Q(7) = 100(0.87)7 = 37.73 mg of Bismuth-210 remains.
Exercise
A population of 1000 is decreasing 3% each year. Find the population in 30 years.
Answer
P (t) = 1000(1 − 0.03)t = 1000(0.97)t
P(30) = 1000(0.97)30 = 401.0071
Example
represents the total number of Android smart phone contracts, in thousands, held by a certain
Verizon store region
T (q) .
measured quarterly since January 1, 2016, interpret all the parts of the equation 2
T(2) = 86(1.64) = 231.3056
Solution
Interpreting this from the basic exponential form, we know that 86 is our initial value. This
means that on Jan. 1, 2016 this
region had 86,000 Android smart phone contracts. Since , we know that every quarter the
number of smart
= 1 + r = 1.64
phone contracts grows by 64%. means that in the nd quarter (or at the end of the second
quarter) there were T(2) = 231.3056
approximately 231,306 Android smart phone contracts.
Finding Equations of Exponential Functions
Exponential Functions:Finding Equations
In the previous examples, we were able to write equations for exponential functions since we
knew the initial quantity and the growth rate. If we do not know the growth rate, but instead
know only some input and output pairs of values, we can still construct an exponential
function.
Example
2.4.5
In 2009, 80 deer were reintroduced into a wildlife refuge area from which the population had
previously been hunted to elimination. By 2015, the population had grown to 180 deer. If this
population grows exponentially, find a formula for the function.
Solution
By defining our input variable to be , years after 2009, the information listed can be written
as two input-output pairs: (0,80) t
and (6,180). Notice that by choosing our input variable to be measured as years after the first
year value provided, we have effectively “given” ourselves the initial value for the function:
= 80. This gives us an equation of the form
a
2.4.4 https://math.libretexts.org/@go/page/146724
f(x) = a
f(t) = 80 t
Substituting in our second input-output pair allows us to solve for :
6
Divide by 80
=80=
Take the 6th root of both sides.
9
6
b = √ = 1.1447
This gives us our equation for the population:
f(t) = 80(1.1447)t
Recall that since , we can interpret this to mean that the population growth rate is =
0.1447, and so the population is
= 1 + r
growing by about 14.47% each year.
In this example, you could also have used (9/4)^(1/6) to evaluate the 6th root if your
calculator doesn’t have an th root button. n
In the previous example, we chose to use the f(x) = a x form of the exponential function
rather than the f(x) = a(1 + r)x form.
This choice was entirely arbitrary – either form would be fine to use.
When finding equations, the value for or will usually have to be rounded to be
written easily. To preserve accuracy, it is
b r
important to not over-round these values. Typically, you want to be sure to preserve at least 3
significant digits in the growth rate.
For example, if your value for was 1.00317643, you would want to round this no
further than to 1.00318.
b
In the previous example, we were able to “give” ourselves the initial value by clever
definition of our input variable. Next, we consider a situation where we can’t do this.
Example
Find a formula for an exponential function passing through the points (-2,6) and (2,1).
Solution
Since we don’t have the initial value, we will take a general approach that will work for
parameters: we will substitute in both given input-output pairs in the function form values,
and .
Substituting in (-2, 6) gives 6 = ab−2
Substituting in (2, 1) gives
2
= ab
any function form with unknown
and solve for the unknown
We now solve these as a system of equations. To do so, we could try a substitution approach,
solving one equation for a variable, then substituting that expression into the second equation.
Solving −2 for :
6 = ab a
a = −2 = 6 2
In the second equation, 2 , we substitute the expression above for :
1 = ab a
1=(6 2) 2
1=6 4
4
4
b = √ 6 ≈ 0.6389
2.4.5 https://math.libretexts.org/@go/page/146724
Going back to the equation
2 lets us find :
a = 6b2 = 6(0.6389)2 = 2.4492
Putting this together gives the equation x
f(x) = 2.4492(0.6389)
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
2.4.5
Recalculate example 2 from above with quarterly compounding.
Answer
24 months = 2 years. .012 4(2)= $1024.25
1000(1 + )
Because of compounding throughout the year, with compound interest the actual increase in a
year is more than the annual percentage rate. If $1,000 were invested at 10%, the table below
shows the value after 1 year at different compounding frequencies:
Frequency Value after 1 year
Annually $1100
Semiannually $1102.50
Quarterly $1103.81
Monthly $1104.71
Daily $1105.16
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
Exercise
Given the two points (1, 3) and (2, 4.5) find the equation of an exponential function that
passes through these two points.
Answer
3 = ab1 , so a = 3 ,
4.5 = ab2 , so 4.5 = 3 b2 . 4.5 = 3b
b
3
b = 1.5 a = 1.5 = 2
f(x) = 2(1.5)x
Example
Find an equation for the exponential function graphed.
Solution
The initial value for the function is not clear in this graph, so we will instead work using two
clearer points. There are three clear points: (-1, 1), (1, 2), and (3, 4). As we saw in the last
example, two points are sufficient to find the equation for a standard exponential, so we will
use the latter two points.
Substituting in (1,2) gives 1
Substituting in (3,4) gives 3
Solving the first equation for gives .
=
Substituting this expression for a into the second equation:
= ab3
4 = b3 = 2b3
b b
Simplify the right-hand side
2
2.4.6 https://math.libretexts.org/@go/page/146724
2 = b2
b = ±√2
Since we restrict ourselves to positive values of , we will use . We can then go back and
find :
b b = √2 a
a = =√ =√
This gives us a final equation of f(x) = √ (√ )x .
Compound Interest
In the bank certificate of deposit (CD) example earlier in the section, we encountered
compound interest. Typically bank accounts and other savings instruments in which earnings
are reinvested, such as mutual funds and retirement accounts, utilize compound
interest. The term comes from the behavior that interest is earned not on the original
value, but on the accumulated compounding
value of the account.
In the example from earlier, the interest was compounded monthly, so we took the annual
interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by
12, the number of compounds in a year, to find the monthly interest. The exponent was then
measured in months.
Generalizing this, we can form a general formula for compound interest. If the APR is written
in decimal form as , and there are
r K
compounding periods per year, then the interest per compounding period will be . Likewise,
if we are interested in the value
after years, then there will becompounding periods in that time. r/k
t kt
Definition: Compound Interest Formula
Compound Interest can be calculated using the formula
kt
A(t) = a(1 + ) (2.4.1)
Where is the account value
A(t)
is measured in years
t
is the starting amount of the account, often called the principal
a
is the annual percentage rate (APR), also called the nominal rate
r
is the number of compounding periods in one year
k
Example
If you invest $3,000 in an investment account paying 3% interest compounded quarterly, how
much will the account be worth in 10 years?
Solution
Since we are starting with $3000,
a = 3000
Our interest rate is 3%, so
r = 0.03
Since we are compounding quarterly, we are compounding 4 times per year, so
We want to know the value of the account in 10 years, so we are looking for , the value
when .
A(10) t = 10
4(10)
A(10) = 3000(1 + ) = $4045.05
The account will be worth $4045.05 in 10 years.
2.4.7 https://math.libretexts.org/@go/page/146724
Example
A 529 plan is a college savings plan in which a relative can invest money to pay for a child’s
later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for
her new granddaughter, wants the account to grow to $40,000 over 18 years, and she believes
the account will earn 6% compounded semi-annually (twice a year), how much will Lily need
to invest in the account now?
Solution
Since the account is earning 6%,
r = 0.06
Since interest is compounded twice a year,
In this problem, we don’t know how much we are starting with, so we will be solving for a,
the initial amount needed. We do
know we want the end amount to be $40,000, so we will be looking for the value of a so that
. A(18) = 40, 000
0.06 2(18)
40, 000 = A(18) = a(1 + 2 )
40, 000 = a(2.8983)
40, 000
a = ≈ $13, 801
Lily will need to invest $13,801 to have $40,000 in 18 years.
If we were to compute the actual percentage increase for the daily compounding, there was an
increase of $105.16 from an original
amount of $1,000, for a percentage increase of = 10.516% increase. This quantity is
called the annual
= 0.10516
percentage yield (APY).
Notice that given any starting amount, the amount after 1 year would be
k . To find the total change, we would subtract the original amount, then to find the
percentage change we
A(1) = a(1 + )
would divide that by the original amount:
k
a(1 + k ) − a k
= (1 + ) − 1
