05.07 Reasoning about Exponential Graphs (part 2)

unit 5

Reasoning about Exponential Graphs (part 2)

 Goal

You will identify the initial value and growth factor of an exponential function given a graph showing two points with non-consecutive input values. Then, apply that knowledge by answering questions. 

  Estimated completion time: 35 minutes

 Watch

If we have enough information about a graph representing an exponential function f, we can write a corresponding equation. Here is a graph of .

An equation defining an exponential function has the form . The value of a is the starting value or , so it is the y-intercept of the graph. We can see that  is 500 and that the function is decreasing.

The graph cannot be easily described. If you need an explanation of this image, please ask your teacher for help.

The value of b is the growth factor. It is the number by which we multiply the function’s output at x to get the output at . To find this growth factor for f, we can calculate , which is or . So an equation that defines f is:

We can also use graphs to compare functions. Here are graphs representing two different exponential functions, labeled g and h. Each one represents the area of algae (in square meters) in a pond, x days after certain fish were introduced.

• Pond A had 40 square meters of algae. Its area shrinks to  of the area on the previous day.
• Pond B had 50 square meters of algae. Its area shrinks to  of the area on the previous day.

The graph cannot be easily described. If you need an explanation of this image, please ask your teacher for help.

Can you tell which graph corresponds to which algae population?

We can see that the y-intercept of g's graph is greater than the y-intercept of h's graph. We can also see that g has a smaller growth factor than h because as x increases by the same amount, g is retaining a smaller fraction of its value compared to h. This suggests that g corresponds to Pond B and h corresponds to Pond A.

Watch the following video of reasoning about Exponential Graphs (part 2).

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