Transcript b < 1 - Algebra1GeomEssentials

Review:
An exponential function is any function of
the form:
y  a bx
where a ≠ 0, b ≠ 1, and b > 0.
•
•
•
•
If b > 1, the graph is increasing.
If 0 < b < 1, the graph is decreasing.
If b = 1, the graph is a horizontal line.
The farther b gets from 1, the steeper the
graph.
Section 8.2
Exponential Growth
Exponential functions that are increasing
are called exponential growth functions.
y  a bx
• a is the initial amount.
• b (the base) is the growth factor.
• x (the exponent) is the number of
increases.
In many real-life problems, there
is a percentage
increase/decrease.
Growth Factor
y  a bx
• The growth factor (b) is equal to:
1 + the percent increase
Conversely, the percent increase can be
found by subtracting 1 from the growth
factor.
What does that mean?
Example 1: Given the equation:
y  201.45x
x
y  a b
a = 20, which is the initial amount.
b = 1.45, which is the growth factor.
To find the percent, subtract 1:
1.45 – 1
= .45
= 45%
On a separate sheet of paper, find the
following. Keep the paper.
x
y  a b
1.)
y  10 1.15
2.)
y  100 1.5
3.)
y2
x
initial amount: 10
percent increase: 15%
x
initial amount: 100
percent increase: 50%
x
initial amount: 1
percent increase: 100%
Write an exponential equation:
Example #2
Suppose the population in a village
is 50 people. If the population is
increasing at a rate of 13% every year, what is
the equation that represents the situation?
x
y  a b
Initial amount = 50 = a
Percentage of growth = 13% = .13
y  501.13x
b = 1 + percent = 1 + .13 = 1.13
On the same paper as before, write an
exponential function with the following
characteristics:
4.)
Initial amount = 5
Percent of increase = 3%
5.)
A population starts with 100 people
and grows at 5% per year. y  (100)(1.05) x
6.)
Initial amount = 112
Percent of increase = 200%
y  (5)(1.03)
x
y  (112)(3.00)
x
Example 3:
Suppose that the rate of inflation over the
past 10 years has been 3% per year. If 10
years ago an item cost $5, how much
should it cost today?
x
y  a b
Initial amount = 5 = a
Percentage of growth = 3% = .03
b = 1 + percent = 1 + .13 = 1.03
y  51.03x
Suppose that the rate of inflation over the
past 10 years has been 3% per year. If 10
years ago an item cost $5, how much
should it cost today?
y  a bx
y  51.03x
Time = 10 years = x
y  5(1.03)10 = 5(1.3439)
= 6.71958
= $6.72
Write an exponential equation to model the
growth function in the situation and then
solve the problem.
7.) Suppose that the number of bacteria in a
shoe increases by 20% every day. If there are
5000 bacteria in the shoe on Monday, how many
bacteria will be in the shoe on Friday (five days
later)?
Please continue to use the same
paper, and keep your work.
If you need a calculator, you may exchange
your ID for one during this class period.
Linear or Exponential:
Decide if each graph/table is linear, exponential or
neither. Explain your reasoning.
x y
y
8) x
9) x
10)
y
1 10
1
30
1
4
2 20
2
90
2
8
3 30
3
270
3
16
4 40
4
810
4
32
11)
12)
13)
Turn in your papers when you are done!
Homework 8.2 part 1
(Due at the beginning of next class.)
Page 370-372
(1-15 odds,16, 23-29 odds)