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
Tell students that the
- “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
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
Here is a graph of
An equation defining an exponential function has the form
The value of
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
We can also use graphs to compare functions. Here are graphs representing two different exponential functions, labeled
- 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