The document discusses exponential equations and their applications. Exponential equations take the form y=abx, where a and b are constants. When b>1, b is the growth factor, and when 0<b<1, b is the decay factor. Examples are provided to demonstrate how to write an exponential equation that passes through two given points. The concepts are then applied to modeling growth and decay scenarios using exponential equations. Compound interest is also discussed, with the formula A(t)=P(1+r/n)nt provided, where n is the number of times interest is compounded per year.

1. 8.2 Writing Exponential
Models

2. What is an exponential equation?
An exponential equation has the general form
y=abx
where a ≠ 0, b〉 0 and b ≠ 1

3. Growth Factor, Decay Factor
Given the general form
y=abx
 When
b > 1, b is the
growth factor
 When
0 < b < 1, b is
the decay factor

4. Growth or Decay???
y = 10(1.2)
y = 5(.9)
x
Growth
x
Decay
y = 50(1.54)
y = 5.2(.70)
y = 4(2)
x
y = 100(.07)
x
x
Growth
Decay
Growth
x
Decay

5. Writing Exponential Equations
 Find
the exponential equation passing through
the points (3,20) and (1,5).
y = ab
20 = ab
20
=a
3
b
x
3
Start with the general form.
Choose a point. Substitute for x and y using (3, 20)
Solve for a
20 1
5= 3 b
b
1−3
5 = 20b
Substitute x and y using (1, 5) and a using
Division property of exponents
2
=a
b3

6. Writing Exponential Equations
 Find
the exponential equation passing through
the points (3,20) and (1,5).
5 = 20b −2
Simplify
20
5= 2
b
20
2
b =
=4
Solve for b
5
b=2
20 20 20 5
a= 3 = 3 =
=
b
2
8 2
Go back to the equation for a; substitute
in b and solve for a

7. Writing Exponential Equations
 Find
the exponential equation passing through
the points (3,20) and (1,5).
y = ab
x
5 x
y = (2)
2
Going back to the general form,
substitute in a and b
The exponential equation passing through
5
the points (3,20) and (1,5) is
y = ( 2) x
2

8. Let’s Try One
 Find
the exponential equation passing through
the points (2,4) and (3,16).
y = ab
x
4 = ab
4
=a
2
b
2
4 3
16 = 2 b
b
16 = 4b
3− 2
Start with the general form.
Choose a point. Substitute for x and y using (2, 4)
Solve for a
Substitute x and y using (3, 16) and a using
Division property of exponents
4
=a
b2

9. Writing Exponential Equations
16 = 4b
1
Simplify
b=4
y = ab
Solve for b
x
4 1
a = 2 = = 0.25
4
4
y = 0.25(4)
x
Go back to the equation for a; substitute in b
and solve for a
Going back to the general form,
substitute in a and b
The exponential equation passing through
x
the points (2,4) and (3,16) is y = 0.25(4)

10. Putting it all together . . .
 Find
the equation of the exponential
function that goes through
(1,6) and (0,2). Graph the function.

13. Modeling Growth with an
Exponential Equation
 The
growth factor can be found in word
problems using
b=1+r
where r = rate or amount of increase.
You can substitute your new b into your
general equation to find the exponential
function.

14.  EX-
a guy puts $1000 into a simple 3%
interest account. What is the exponential
equation?
r = rate 3% (write as 0.03)
b = 1 + r = 1.03
x = time
a = amount put into the account
($1,000)
y = ab
x
y = 1000 (1.03) x

15.  EX
– a colony of 1000 bacteria cells
doubles every hour. What is the
exponential equation?
r = 1 (why not 2?)
b=r+1=2
x = time (in hours)
a = the original number in the
colony (1,000 bacteria )
y = ab
x
y = 1000 (2) x
b = r + 1, where r is the amount of
increase. We are increasing by 100%
each time something doubles, so r = 1

16.  EX-
a $15000 car depreciates at 10% a
year. What is the exponential equation?
r = - 10% (the car is worth 10%
less each year)
b = 1 - r = 1 – 0.1 = 0.9
x = time (in years)
a = amount put into the account
($15,000)
y = ab
x
y = 15000 (0.9) x

17. Compound Interest
 The
formula for compound interest:
nt
 r
A(t ) = P 1 + ÷
 n
Where n is the number of times per
year interest is being compounded
and r is the annual rate.

18. Compound Interest - Example
 Which
plan yields the most interest?
Plan A: A $1.00 investment with a 7.5% annual
rate compounded monthly for 4 years
 Plan B: A $1.00 investment with a 7.2% annual
rate compounded daily for 4 years

12(4)
A:11 + 0.075 
≈1.3486

÷
12 

$1.35
365(4)
 B: 
0.072 
11 +
≈1.3337
÷
365 


$1.34