Unit 9, Lesson 13
Reasoning about Exponential Graphs (Part 2) (Optional)
Integrated Math 1
Lesson Synthesis
In this lesson, we interpreted graphs and reasoned about their equations. Show the graph of for an exponential function with some points labeled.
Tell students that the -coordinate of is 1 and the -coordinate of is 2. Ask students what information would give us insight into the situation and help us write an equation corresponding to the graph.
- “What do you need to know in order to describe how the function is decreasing?” (coordinates of two of the labeled points)
- “If you could find out the coordinates of two of these points, which two would you choose? Why?” ( and , or and , because the quotient of their -coordinates gives the growth factor.)
- “How would you find the equation using these two coordinates?” (use the -coordinate of and the growth factor)
Consider showing the graph of along with the graph of for another exponential function .
Ask students to compare the two functions:
- “Which function is decreasing faster? How do you know?” ( because it has a larger -intercept than but then takes smaller values than as grows)
- (If the graphs cross) “What does the intersection of the two graphs mean?” (The outputs of and are the same for this value of .)
Student Lesson Summary
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.
- 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.
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.