Applications of Power Functions
Lesson 2 of 13
Objective: SWBAT use power functions to model the relationships between two quantities.
Power Functions Graphically
I begin this lesson by building on the homework from the previous lesson, Intro to Power Functions. The students have already completed graphs of some power functions, and they've begun thinking about the differences between odd and even functions.
Now, we'll use GeoGebra to bring it all together and take a look at the whole family of power functions. (You can download the GeoGebra files in the resources, or you can make your own. They're pretty simple.) I want the class to see (and then explain) a few things as we play with the two parameters in the generic function .
- By changing the value of a we can change the "steepness" of the graph. Ask students to explain why this parameter has this effect on the graph; it may help them to relate it to the slope of a linear function.
- Notice that even as the value of a changes, the function still always contains the origin. Students should be able to explain why this is.
- By changing the value of b we can change the end behavior of the function. Students should recognize that end behavior depends on the parity (odd/even) of b, and should explain why this is. (For odd/even comparison, you can use the other GeoGebra setup.)
- Notice that even as b changes, the graph still always contains the origin and the point (1, 1), as well as either (-1, 1) or (-1, -1).
- Notice that as b increases, the graph becomes nearly horizontal around x = 0 and nearly vertical around x = 1 and x = -1. In fact, if we let b keep increasing, we could almost make a right angle at (1, 0) or (-1, 0)!
Do your best to let the students do the talking in this section of the lesson. If you give them the time, and simply ask, "What do you notice?", they will probably come up with everything on this list. In fact, they'll probably surprise you with some things you never noticed yourself!
Setting the Stage
Next, we'll be using power functions of the form y = ax^b to describe the relationships between different quantities in some real world scenarios. I selected these problems during the summer, so they're all about water!
Before handing out the problems, I'll ask if anyone can remind me what it means to say that two quantities are "directly proportional" or that one "varies directly" with the other. Of course, what I'm looking for is an informal explanation that "if one increases, the other one does, too." Pointing out the term "proportional", we'll recall that the two always maintain the same ratio. This kind of relationship may be expressed either as y/x = a or as y = ax.
With this brief reminder, I'll break the class into groups of about three students and hand out the problem set Modeling with Power Functions.
The various groups of students should have arrived at solutions to all (or nearly all) of the problems, and I like to take 5 minutes at the end of class for any general comments or questions.
This is a good time to discuss strategy for approaching real-world applications. We can discuss the notion of decontextualizing from physical quantities to mathematical quantities. How are they different? Do the mathematical quantities and equations "do things" that the physical one's don't or can't? For instance, the equations we arrive at could include negative values from a mathematical standpoint, but turning our attention back to the physical referents, we can see that this domain should be excluded.
Tonight's homework will be to wrap up any unfinished problems and to make the solutions beautiful so they will be ready to hand in tomorrow!