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