Saturday, March 28, 2009

The "Higher" Polynomials

Here is a lesson I love because of graphing calculators. My goals for my students were to be able to look at a graph of a polynomial and know its degree and find an equation given various points on the polynomial. I also wanted them to be able to look at an equation and be able to give a quick sketch of the graph with the intercepts and shape correct.

At the start of class before I mentioned anything about shapes of non parabola polynomials, I handed out graphing calculators and we all got the same window and turned the grid on. In Y=, I had them type in y=(x-2)(x+1)(x-1) (or something similar previously checked out to make sure it fits in the window). I told them not to graph it but to think: what are the x-int, y-int, and make a conjecture about the shape. THEN they could graph and see if they were right. We then had a discussion to link the intercepts with the equation.

Then they typed in y=2(x-2)(x+1)(x-1). I asked them to think about how this might affect the graph and intercepts and shape and THEN graph and confirm their reasoning. We did one more, and then I discussed the shape and degree.

Then same process with y=(x+2)(x-1)^2. They had to think and do the same analysis as above. We then did y=(x-1)(x+2)^2 and such.

Then the fun part: I typed in an equation and just showed them the graph, and they had to match it. We did that a few times. I made sure to change the constant in front sometimes and to make it "upside down" and to have various powers. Then I asked them to find an equation for a 5th degree polynomial with 2 "bounces" off the x-axis going in the same direction (both below the axis or both above). Then for the ones that finished that early: opposite bounce direction.

Then we took notes, and I believe they had a good sense of polynomial graphs.


  1. Nice example of putting the learning before the telling. Good work!

  2. I spend a lot of time on this sort of thing, problem is always that with enough roots and a high enough leading coefficient, on the graphing calculator the graphs just look like straight lines up and down. How do you overcome this?

  3. Anonymous4:31 PM

    I have had the straight line thing before, so I make sure to test things out ahead of time. I stick to 1's, 2's, 0.5's. My initial window was (-6,6) x (-6,6).

    I think it's important for them to also see the straight line situation. Towards the end I mention that one of the big reasons to get a sense of what these equations look like is that so when you're graphing something and you see (for example) straight lines, you can reason it out and know it what it should look like. Then you know it's a window problem (this situation came up on an AP Calculus exam question and it threw the kids).

    I also show them that they can go to the table and see what the y values should be, and they can use that to adjust their window.

    Ms. Cookie

  4. This is a great lesson! What grade level students are you dealing with? I feel like my 9th graders may have some serious issues following along with the concept. Also, how long were your periods that you taught this lesson?

  5. Anonymous8:58 PM

    Hi Meridyth,

    This was for a precalculus class, and our periods are 1.5 hours long.

    I think the same idea could be used with 9th grade algebra 1 students when teaching lines and changing either the slope for a bunch of graphs while you/they stop and think and then changing the y-intercept, etc.

    Ms. Cookie

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