Finding an exponential growth or decay equation (6.14)

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Presentation transcript:

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?