Unit 9, Lesson 13
Reasoning about Exponential Graphs (Part 2)
Integrated Math 1
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Here is a graph representing an exponential function, . The function, , gives the value of a computer, in dollars, as a function of time, , measured in years since the time of purchase.
Graph of a decreasing exponential function, x y plane, origin O. Horizontal axis, scale 0 to 4, by 2’s. Vertical axis, scale 0 to 400, by 200’s. The function starts at the vertical axis at (0 comma 400) and goes through (2 comma 100) and (4 comma 25).
Based on the graph, what can you say about the following?
Here are graphs of two exponential functions. For each, write an equation that defines the function, and find the value of the function when is 5.
Here are graphs representing two functions, and descriptions of two functions.
Graph of two exponential functions, origin O. Horizontal axis, time(months). Vertical axis, area (square inches). The functions are growing exponentially. The red function has a higher start on the vertical axis than the green function. They meet at point (p comma q) and then the red function ends lower than the green function.
If we have enough information about a graph representing an exponential function , we can write a corresponding equation.
Here is a graph of .
An equation defining an exponential function has the form . The value of is the starting value or , so it is the -intercept of the graph. We can see that is 500 and that the function is decreasing.
The value of is the growth factor. It is the number by which we multiply the function’s output at to get the output at . To find this growth factor for , we can calculate , which is (or ).
Graph of a function on grid, origin O. Horizontal axis, x, from 0 to 4, by 2's. Vertical axis, y. from 0 to 600, by 200's. Plotted points as follows: 0 comma 500, 1 comma 300. Line is drawn through the points, moving downward and to the right.
So an equation that defines is:
We can also use graphs to compare functions. Here are graphs representing two different exponential functions, labeled and . Each one represents the area of algae (in square meters) in a pond, days after certain fish were introduced.
Graph of 2 functions on grid, origin O. Horizontal axis, x, Vertical axis, y. Function h, blue line, starts on the vertical axis, moves downward and to the right, almost linearly.. Function g, green line, starts above function h, moves downward and to the right, exponentially.
Can you tell which graph corresponds to which algae population?
We can see that the -intercept of 's graph is greater than the -intercept of 's graph. We can also see that has a smaller growth factor than because as increases by the same amount, is retaining a smaller fraction of its value compared to . This suggests that corresponds to Pond B, and corresponds to Pond A.