Transcript Alg II (6.14) Finding an exponential equ. 1

Finding an exponential
growth or decay equation
(6.14)
Using two points
to derive y = abx
The General Idea
Just like with linear and parabolic
models, we find two variables,
determine which is independent
and dependent, and derive an
equation.
The General Idea
With exponential equations, you have some
flexibility in determining the initial point.
If a population is growing and you want to
predict what it will be in a few years, you have
to pick some point to start the comparison-that is your initial amount.
For this initial amount, x = 0, and that is handy.
An example
Suppose you have a bacterial culture in a dish in
the lab, and the population increases
exponentially with time. On Tuesday, there
are 2000 bacteria per square millimeter and
on Thursday the number has increased to
4500.
Besides wondering how in the world a person
measures bacteria per square millimeter, you
want to estimate what the population will be
on the following Tuesday. When did we start?
Find the variables
To begin with, you need to know what you’re
dealing with. Find the variables and
determine which is independent and which is
dependent.
In this case, the independent variable (x) is the
day and the dependent variable (y) is the
population.
Find the points
For exponential growth and decay equations, you need
only two points.
We know the y values will be 2000 and 4500.
What are the x values?
Since the first count is 2000, let’s set that as our initial
amount, and make the point (0, 2000). In other words,
at the start, time equals 0 and population equals 2000.
The other count is 2 days later so the second point is (2,
4500).
Using the points
Use (0, 2000) and (2, 4500) to complete the equation
y = abx.
In other words, find a and b.
a is easy: 2000 = a(b0)
2000 = a Notice how a
represents our initial
amount?
Plug in the second point to find b:
4500 = 2000(b)2
2.25 = b2
b = 1.5
b can be seen as a rate of change per
unit of time (in this case, days)
In seven days
Put all the pieces together to form the equation
y = 2000(1.5)x
On the following Tuesday, x = what?
In seven days
Put all the pieces together to form the equation
y = 2000(1.5)x
On the following Tuesday, x = 7, and the
population is
2000(1.5)7 = 34172 bacteria
per mm2 (assuming our
model holds)
What are the limitations for x and y?
We started on
Put all the pieces together to form the equation
y = 2000(1.5)x
We started with, say, 1 bacteria. When was
this?
We started on
Put all the pieces together to form the equation
y = 2000(1.5)x
We started with, say, 1 bacteria. When was
this?
1 = 2000(1.5)x
1/2000 = (1.5)x
x = log1.5(1/2000)
= log(1/2000)/ log(1.5) = -19
19 days before we began the count
The final answer
What does the graph of the equation look like?
y = 2000(1.5)x
What is the y-intercept?
What does that point
correspond to in our
scenario?
Can you tell by looking at the graph what the bacteria
population would have been a week before you started
counting? Find it algebraically.
Examples of exponential
models
I’m asking you to do this with real world data.
But what sorts of data would work?
Think of classes you’ve already had in other
subjects, especially science. Or consider
money in a savings account over time.
In what sorts of places could we find examples
of exponential data?