In this section, I'm going to use some very simple questions to draw out a common misconception, then I'll use this misconception as a springboard to launch the class into a quick study of power functions (y = x^n) that will the foundation for their study of polynomials (HSA.APR 1 - 7). Technically, this is review of previous material, but I've found it's well worth the time.
See the short video Intro to Power Functions for an idea of how I kick off this lesson.
Using the Power Point slides Intro to Power Functions, I'll present pairs of numbers involving exponents. The first few are just warm ups and should be very easy for the students. After a couple that should give them a moment's pause, I cut to the chase and ask whether x^2 is greater or less than x^3.
My expectation is that many students will immediately say that x^2 is less than x^3. Once they do, we're ready for the next section of the lesson! (MP 6)
Once I have several students on the hook by claiming that x^2 is less than x^3, I will innocently ask if that's always true. I expect a conversation will follow in which students begin by proposing specific values of x that make the square less than the cube. Pretty quickly, however, other students will point out that the relationship is reversed if x is negative. Other students will notice that the two are equal when x is one or zero.
The goal of all of this is to determine precisely the domain on which x^2 is less than x^3, equal to it, and greater (if that ever happens). The hardest part of this task is recognizing that when 0 < x < 1, the higher power will actually have the lesser value. (MP 6) If students are failing to recognize this fact, I'll ask them explicitly what the relationship is when x = 0.5. Also, it helps to think of a number like 0.9^2 as taking 90% of 90% or of 0.5^2 as taking half of one half.
Without dwelling too much on this first comparison, I'll move on with the slide show to compare two even functions and then two odd functions. We'll discuss them in the same way, paying special attention to the domain -1 < x < 1.
By now the class has started to get the hang of this, so it's time to formalize it a bit.
First, I'll define power function as any function of the form where a is any real number and b is any positive integer. I'll also point out that these functions are going to act as building blocks for other functions (e.g. quadratic equations and other polynomials). By understanding how they behave on their own, we can better understand how they'll act when they're combined. (MP 6, 7)
Next, I'll make a quick sketch of the function f(x) = x^2 on the domain [-2, 2]. The class should be very familiar with this one, but they are probably not familiar with the graphs of higher power functions.
Together, we'll add g(x) = x^3 to the sketch, but just on the domain [0, 2]. I'll call out x-values and the class will give me the g(x) values so I can plot them. For comparison, I'll be sure to ask about each one, "Is this point above or below f(x)?"
They've already seen numerically that x^3 is sometimes less than x^2, but I'll explicitly point out how the graph illustrates the same thing. The graph is useful because we can see at a glance how x^2 and x^3 compare not just at one value of x, but for all values of x on the given domain.
Without finishing the graph of x^3, it's time to hand things over to the students.
At this point, I'll tell the class that they're going to be investigating the graphs of the other power functions and I'll pass out Intro to Power Functions Homework. For the remainder of the class period, they should use their calculators to accurately produce graphs of the indicated functions. The students may work either independently or in small groups, but each students must produce his or her own graphs. You can view the solutions here.
The purpose of the assignment is to help the students make special note of the following features:
The ultimate goal is to use these patterns to predict the behavior of other functions, especially higher order polynomials. (MP 7)