Over the course of the last few lessons, the students have really gotten to know cubic functions pretty well and have seen some very close connections to quadratic functions. Now it's time to introduce them formally to the family to which these functions belong: polynomials.
In this section, we'll discuss a formal definition and what it means in specific cases (MP 6).
See the video for a detailed narrative.
OK, so we have a definition of "cubic polynomial" and we have some experience working with it. The goal of this section is to glean as much as we can about the behavior of the function from the values of its coefficients. I'll load up the GeoGebra file I've shared as a resource. It will allow us to observe the changes in the graph that result from changing the values of the coefficients (MP 5).
Objective: Observe & explain the effect that each term (in isolation) has on the shape of the graph. Observe the possibility of up to 3 real roots and the guarantee of at least 1. (This will be important for the Fundamental Theorem of Algebra!) [Please not, the goal is NOT to be able to predict precisely the graph of a given cubic based on a visual inspection of its equation. Do you know what y = 3x^3 - 2.5x^2 + 8.17x - 0.35 looks like? Neither do I.]
Use the sliders one at a time in the GeoGebra applet to change the function.
Note to Self: TO ENCOURAGE MP1 AND MP3, DO NOT LECTURE. ASK STUDENTS WHAT THEY SEE, THEN ASK THEM TO EXPLAIN IT.
1. Observe & explain effect of a. This builds on recent work with power functions. Draw out the analogy to the slope of a line and the influence of a on a quadratic function. Reset a to 1.
2. Observe & explain effect of d. A constant amount is being added to the function everywhere. Perfect analogy with linear & quadratic functions. Prove d must be the y-intercept. Reset d to 0.
3. Observe & explain effect of c. Point out that the y-value of the line is being added to the cubic function at every point. When c is negative, the line and the cubic have opposite signs, and this can draw the graph out of its original path for a time. Soon, however, the cubic term overpowers the linear term and the graph reverts to its cubic shape. This is how new vertices are formed! Notice & explain the "straightening" effect around x = 0 when c is positive. Reset c to 0.
4. Observe & explain effect of b. A quadratic term is being added to the cubic. New vertices are formed where the quadratic and cubic terms have opposite signs. For example, near x = 0, the (+) quadratic overpowers the (-) cubic and draws the graph into the positive range. Before long, the (-) cubic overcomes the (+) quadratic, and the graph is ultimately drawn forever downward! This is why it was so important to notice that lower powers may sometimes be greater than higher powers - but only near zero!
5. Put it all together. Adjust all four sliders to observe the many variations on the cubic parabola. Pay special attention to things that are invariant (odd symmetries) and to the limits to the variation. For instance, there are only either 2 or 0 vertices; we'll never be able to get 3 vertices. There is always at least 1 root. There are at most 3 roots. Et cetera.
To wrap things up, I like to have an open popcorn-style conversation about what the class has seen and learned today. Do they see connections with other kinds of functions? Does the equation make more sense to them now? Have fun & be surprised by how much your students can explain!