1. 7.2 Exploring Exponential
Models

2. What is an exponential equation?
An exponential equation has the general form
y=abx
1band0b,0wherea

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???
x
)2.1(10y
x
)9(.5y
x
)54.1(50y
x
)70(.2.5y
x
y )2(4
x
)07(.100y
Growth
Decay
Growth
Decay
Growth
Decay

5. Graphs of Exponential Functions
x
)2(10y

6. What is an asymptote?
x
)2(10y“Walking halfway to the wall”
An Asymptote is a line that
a graph approaches as x
or y increases in absolute
value.
In this example, the
asymptote is the x
axis.

7. Graphing Exponential Functions
x
)5(.100y
X .5x Y=100(.5)x
-3
-2
-1
0
1
2
3
Complete the table using the
integers -3 through 3 for x.

8. Let’s graph one together
x
)5(.100y
X .5x Y=100(.5)x
-3 .5-3 800
-2 .5-2 400
-1 .5-1 200
0 .50 100
1 .51 50
2 .52 25
3 .53 12.5

9. Let’s try one
x
y )5(.2
X .5x Y=2(.5)x
-3
-2
-1
0
1
2
3
Complete the table using the
integers -3 through 3 for x. Then
graph the function.

10. Let’s try one
x
y )5(.2
X .5x Y=2(.5)x
-3
-2
-1
0
1
2
3

11. Let’s try one
x
y )10(5
X 10x Y=5(10)x
-3
-2
-1
0
1
2
3
Complete the table using the
integers -3 through 3 for x. Then
graph the function.

12. Let’s try one
x
y )10(5
X 10x Y=5(10)x
-3
-2
-1
0
1
2
3

13. Writing Exponential Equations
 Find the exponential equation passing through
the points (3,20) and (1,5).
x
aby
a
b3
20
1
3
20
5 b
b
31
205 b
3
20 ab
Start with the general form.
Choose a point. Substitute for x and y using (3, 20)
Solve for a
Substitute x and y using (1, 5) and a using a
b3
2
Division property of exponents

14. Writing Exponential Equations
 Find the exponential equation passing through
the points (3,20) and (1,5).
2
4
5
20
20
5
205
2
2
2
b
b
b
b
Simplify
Solve for b
2
5
8
20
2
2020
33
b
a
Go back to the equation for a; substitute
in b and solve for a

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

16. Let’s Try One
 Find the exponential equation passing through
the points (2,4) and (3,16).
x
aby
a
b2
4
3
2
4
16 b
b
23
416 b
2
4 ab
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 a
b2
4
Division property of exponents

17. Writing Exponential Equations
1
416 b
4b
Simplify
Solve for b
25.0
4
1
4
4
2
a
Go back to the equation for a; substitute in
b and solve for a
x
aby
x
y )4(25.0 Going back to the general form,
substitute in a and b
The exponential equation passing through
the points (2,4) and (3,16) is
x
y )4(25.0

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

21. 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.

22.  EX- a guy puts $1000 into a simple 3%
interest account. What is the exponential
equation?
x
y )03.1(1000
r = rate 3% (write as 0.03)
b = 1 + r = 1.03
x = time
a = amount put into the account
($1,000)
x
aby

23.  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 )
x
y )2(1000
x
aby
b = r + 1, where r is the amount of
increase. We are increasing by 100%
each time something doubles, so r = 1

24.  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)
x
y )9.0(15000
x
aby