2. Lesson Objective
At the end of the lesson, the students must be
able to represent real-life situations using
exponential functions.
3. Graph of Exponential Function
The graph of an exponential function defined by
f(x) = bˣ where b > 0 and b ≠ 1 indicates that:
• It is an increasing function,
sometimes called an exponential
growth function if b > 1.
• It is a decreasing function,
sometimes called an
exponential decay function if 0 < b < 1.
4. Activity 1
1. When a quantity increases by 20%, how does its
new value compare to its original value? That is,
what is the ratio of the new value to the original
value? Complete the table below to discover your
answer.
5. Activity 1
2. What is the ratio of the new value to the original
value of any quantity that increases by 20%?
3. Use the common ratio you obtained in the table
to fine the new value of the quantities in Column
1. What did you do?
a. _________________________________
b. _________________________________
c. _________________________________
d. _________________________________
6. Activity 1
4. Using the procedure you have done in No. 3,
find the new value of a television set that
increased by 20% over the original value of
₱4, 500.00.
_________________________________
_________________________________
_________________________________
7. Growth Factor
Growth factor - formed by adding the specified
percent increase to 100% and then changing to
its decimal form
The growth factor in Activity 1 that corresponds
to a 20% increase is
100% + 20% = 120%
But, 120% = 1.20 in decimal form.
8. Example 1
Determine the growth factor of the quantity
that increases by the given percent.
a. 50% b. 75% c. 10% d. 12.5%
9. Solution to Example 1
a. For 50% : 100% + 50% = 150% = 1.50
b. For 75%: 100% + 75% = 175% = 1.75
c. For 10%: 100% + 10% = 110% = 1.10
d. For 12.5%: 100% + 12.5% = 112.5% = 1.125
10. When a quantity increases by a specified
percent, like 20% in Activity 1, it’s new value can
be obtained by multiplying the original value by
the corresponding growth factor. That is,
Original value ∙ growth factor = new value
11. Example 2
Emerson deposits ₱50,000 in a savings account.
The account pays 6% annual interest. If he
makes no more deposits and no withdrawals,
calculate his new balance after 10 years.
12. Solution to Example 2
The interest represents a 6% rate of growth
per year, so the growth factor or the constant
multiplied is 100% + 6% = 106% or 1.06. To
find an equation that can be used to find the
new balance after any number of years by
considering the yearly calculations, we have:
13. Solution to Example 2
We can now use this equation y = 50 000 (1.06)ˣ, where x
represents time in years and y represents the new balance in
pesos. To find the new balance after 10 years, we have:
y = 50 000 (1 .06)¹⁰
= 89 542.38
The new balance after 10 years is ₱89 542.38
14. Exponential Growth
The rule for exponential growth can be
modeled by
Y = abˣ
where a is the starting number, b is the growth
factor, and x is the number of intervals
(minutes, years, and so on).
15. Example 3
A bacteria grows at a rate of 25% each day.
There are 500 bacteria today. How many will
there be?
a. tomorrow?
b. one week from now?
c. one month from now?
16. Solution to Example 3
a. Using the formula y = abˣ where a = 500, b =
1.25, and x = 1 then
y = 500(1.25)¹ Substitute 500 for a,
= 625 1.25 for b, and 1 for x.
There will be 625 bacteria tomorrow.
17. Solution to Example 3
b. Using the formula y = abˣ where a = 500, b =
1.25, and x = ?, then
y = 500(1.25)⁷ Substitute 500 for a,
= 2 384.19 1.25 for b, and 7 for x.
There will be 2,384 bacteria one week from
now.
18. Solution to Example 3
c. Using the formula y = abˣ where a = 500,
b = 1.25, and x = ?, then
y = 500(1.25)ᶟ⁰ Substitute 500 for a,
= 403,896.78 1.25 for b, and 30 for x.
There will be 403,897 bacteria one month from
now.
19. Exponential Growth Models
• If a certain quantity increases by a fixed percent
each year (or any other time period), the amount
y of that quantity after t years can be modeled by
the equation:
y = a(1+r)ᵗ
where:
a = initial amount r = is the percent
l + r = growth factor increased expressed
as decimal
20. Exponential Growth Models
• A real-life application of exponential growth
occurs in the computation of compound
interest. The formula for the compound
interest is:
A = P
where:
A = compound amount t = time in years
P = principal n = period per year
r = interest rate
nt
n
r
1
21. Example 4
Ellaine invested ₱50,000.00 at an annual rate of
6% compound yearly. Find the total amount in
the account after 10 years if no withdrawals and
no additional deposits are made.
22. Solution to Example 4
Use the compound interest formula with P =
₱50,000.00, r = 0.06, n = 1, and t = 10.
A = P
= 50,000
= 50,000 (1.06)¹⁰
= 89 542.38
There would be ₱89 542.38 in the account at
the end of 10 years.
nt
n
r
1
10
.
1
1
06
.
0
1
23. Example 5
Determine the amount of money that will be
accumulated if a principal of ₱100,000 is
invested at an annual rate of 8% compounded:
a. yearly for 10 years
b. semi-annually for 10 years
c. monthly for 10 years
24. Solution to Example 5
a. Use the compound interest formula with
P = ₱50 000, r = 0.06, n = 1, and t = 10.
A = 100 000
= 100 000 (1.08) ¹⁰
= 215 892.50
There would be ₱215 892.50 after 10 years
10
.
1
1
08
.
0
1
25. Solution to Example 5
b. Use the compound interest formula with P, r,
and t the same as in (a) and n=2.
A = 100 000
= 100 000 (1.04)²⁰
= 219 112.31
There would be ₱219 112.31 after 10 years
10
.
2
2
08
.
0
1
26. Solution to Example 5
c. Given: P = 100 000, r = 0.08, n = 12, and t = 10.
Substituting these values in the formula, we
have
A = 100 000
= 100 000 (1.007)¹²⁰
= 230 959.84
There would be ₱230 959.84 after 10 years
10
.
12
12
08
.
0
1
27. When the principal P in pesos is invested
at an annual interest rate r (in percent),
compounded continuously, the amount A
accumulated after t years is given by the
formula:
A =Peʳᵗ
28. Example 6
Determine the amount of money that will be
accumulated if a principal of ₱100,000 is
invested at an annual rate of 8% compounded
continuously after 10 years if no withdrawal are
made.
29. Solution to Example 5
Apply the continuous compound formula with P = 100
000, r = 0.08, and t = 10.
A = Peʳᵗ
A = 100 000
A = 100 000
A = 100 000 (2.2255409)
A = 222 554.09
There will be ₱222 554.09 in the account after 10 years.
)
10
(
08
.
0
e
08
.
0
e
30. Decay Factor
The ratio is called the decay factor
associated with the specified percent decrease.
The decay factor is formed by subtracting the
specified decrease from 100% and then
changing this percent to its decimal form.
value
original
value
new
31. Exercise A
Match each equation with a table of values.
1. y = 3(2)ˣ 2. y = 3. y = 2(0.5)ˣ 4. y = 2(3)ˣ
x
2
1
3
32. Exercise B
Answer the following.
1. Mrs. Lacap bought an antique dresser for
₱15 000. She estimates that it will increase its
value by 5% per year.
a. Formulate an equation to calculate the value,
y, of Mrs. Lacap’s dresser after x years.
b. Find the value of the dresser after 8 years.
33. Exercise B
Answer the following.
2. When you use credit card to purchase an item,
you are actually making a loan. A constant
percent interest is added to the balance.
Maricar buys a microwave oven worth ₱7,500
with her credit card. How much will she owe if
makes no payment in 6 months?
