Post on 24-Dec-2015
8.1 Exploring Exponential Models
What is an exponential equation?
An exponential equation has the general form
y=abx
1b and 0b ,0 where a
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
Growth or Decay???
x)2.1(10y
x)9(.5y x)54.1(50y
x)70(.2.5y xy )2(4
x)07(.100y
Growth
Decay
Growth
Decay
Growth
Decay
Graphs of Exponential Functions
x)2(10y
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.
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.
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
Let’s try one
xy )5(.2X .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.
Let’s try one
xy )5(.2X .5x Y=2(.5)x
-3
-2
-1
0
1
2
3
Let’s try one
xy )10(5X 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.
Let’s try one
xy )10(5X 10x Y=5(10)x
-3
-2
-1
0
1
2
3
Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5). xaby
ab
3
20
13
205 b
b
31205 b
320 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 ab
3
2
Division property of exponents
Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5).
2
45
20
205
205
2
2
2
b
b
b
b
Simplify
Solve for b
2
5
8
20
2
202033
b
aGo back to the equation for a; substitute in b and solve for a
Writing Exponential Equations Find the exponential equation passing through
the points (3,20) and (1,5).
xaby
xy )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 xy )2(
2
5
Let’s Try One Find the exponential equation passing through
the points (2,4) and (3,16). xaby
ab
2
4
32
416 b
b
23416 b
24 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 ab
2
4
Division property of exponents
Writing Exponential Equations
1416 b4b
Simplify
Solve for b
25.04
1
4
42
aGo back to the equation for a; substitute in b and solve for a
xaby
xy )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
xy )4(25.0
Putting it all together . . .
Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.
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.
EX- a guy puts $1000 into a simple 3% interest account. What is the exponential equation?
xy )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)
xaby
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 )
xy )2( 1000
xaby
b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1
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)
xy )9.0( 15000
xaby