Reader William Carey via email:

Last year I realized that Pre-Calculus is really a class about moving from the particular to the general. We take particular skills and ideas students are comfortable with — like solving a quadratic equation — and generalize them to as many mathematical objects as we can — solving all polynomial equations. As we worked our way through polynomials, we wanted to move from reasoning about particular quadratic equations like

y = xto reasoning about all quadratic equations:^{2}+ 2x + 1y = ax. For homework, the students had to graph about twenty quadratics with varying^{2}+ bx + ca,b, andc.Then we got together to discuss the results in class. They remembered that

acontrols the “fatness” or “narrowness” of the parabola and sometimes flips it upside down. They remembered thatcmoves the parabola up and down. They weren’t totally sure whatbdid. A few students adamantly maintained that it moved the parabola left and right (with supporting examples). After about fifteen minutes of back and forth, we decided to go to Desmos and just animateb.

Shock and disbelief: the vertex traces out what looks like a parabola as. Furious math and argument ensue. Ten minutes later, a student has what seems to be the parabola the vertex traces graphed in Desmos. Is it the right parabola? Why? Can we prove that? (We could and did!)bchanges

**Previously**: WTF Math Problems.

## 18 Comments

## William Carey

December 13, 2016 - 7:09 pm -And one student is writing her junior thesis generalizing these results to the maxima and minima of cubic and quartic polynomials. We worked out the vertex-tracing function for the cubic as a class; it too was surprising!

## John Golden

December 13, 2016 - 7:26 pm -When my preservice teachers discover this it is total shock and awe. Always a highlight. Because that b is already mysterious and then *this*? (In GeoGebra, though. Will Desmos ever add tracing?)

## Dan Anderson

December 14, 2016 - 6:50 am -Until tracing is added(?), here is one way you can show the effect of “b” using desmos (and lists): https://www.youtube.com/watch?v=vkxIsr-iioc

## education realist

December 13, 2016 - 9:16 pm -I teach parabolas as three different operations.

1) Take a line, square it, stretch it, and move it up and down. So sketch the line and the x-intercept is the line of symmetry. Very visual. (Vertex)

2) Take two lines and multiply them together. The product will be zero where the x-intercepts of the lines are zero, because zero is the death lord, Shiva the destroyer of multiplication. And because we know about the line of symmetry, we know its right between the zeros. Then plug in ot find y. (looking for a way around that.) Also very visual.

3) This is the new one: a parabola in standard form is the sum of a parabola centered at the origin and a line. Not visual. But the one thing that’s clear is that the origin plus the line’s y-intercept is going to be the y-intercept. And a bit of investigation shows that there’s a clear and predictable slope from the vertex of a parabola to the y-intercept (aka, rate of change). This last has really been useful in unlocking the mystery of graphing in standard form, other than negative b over 2a. But it’s also given me a much clearer sense of how the linear term affects the parabola. The kids, too.

## Katrin

December 14, 2016 - 2:18 am -I tried your steps in Desmos. It’s great! I’ve also noticed, which I’m sure you have as well, that the tangent to the quadratic in the y-intercept is the linear term bx+c. Wouldn’t it be more interesting to investigate that instead of the line through the vertex and the y-intercept (b/2 x+c)?

## Katrin

December 14, 2016 - 1:50 am -It is beautiful to see how the extremum moves when you vary coefficient b, but if you want to study the quadratic function graphically, isn’t it much more revealing to use the completed square form, m(x-p)^2 + q? It’s really nice to see it in Desmos with sliders for m, p and q.

## William Carey

December 14, 2016 - 4:51 am -Katrin,

I suppose it depends on what you want to reveal about the quadratic. For an Algebra I or II class, probably the completed square form reveals more about the parabola. But for a Pre-Calculus class, what I’m trying to do is ruthlessly move from the particular to the general. And the vertex form doesn’t really generalize.

I don’t know that the completed square form generalizes to higher degree polynomials. Is there a completed cube form? A quick google search suggests “no”. That makes the completed square form, in some ways, a dead end.

But if you look at the properties of the coefficients of standard form, you can do similar things for higher degree polynomials. There’s a cubic equation that describes the curve traced out by the extrema of y=ax^3+bx^2+cx+d is surprising and interesting (and requires a metric crap-ton of algebra to work out; if you want to get kids to practice algebraic manipulation, make them yearn to know things that require it in great volume). Why is it cubic? Why isn’t it lower degree? It’s also a nice segue to Calculus, because it makes the location of maxima and minima a headache that kids want aspirin for.

## Anna Scholl

December 14, 2016 - 7:24 am -I’m curious about the polynomial generalization… and the possible implications. Does that mean with the polynomial one, we might have a way to use the extrema to find an equation without factors? (for example… it’s hard to find an equation from a polynomial graph that has invisible complex roots… )

## Dan Anderson

December 14, 2016 - 7:50 am -Cool question Anna. It led me to look for an answer (to a different question). I was wondering if when you change the second coefficient in a cubic, if the “vertex” (really the inflection point) of the cubic also traced out a cubic graph. like how the second coefficient in a quadratic traces out another quadratic. Here’s what I came up with: https://www.desmos.com/calculator/imsllrnfvn

I think I’ve made a deeper connection between the neat fact that William and Dan shared, and calculus. Thanks for the thought Anna, even if I mostly ignored it!

## William Carey

December 14, 2016 - 8:11 am -A good question. The other question I’m wrestling with is this: there are cubic and quartic formulas, but per Abel-Ruffini, we know there’s no quintic formula. Is the result that the extrema of a curve are functions (of the same degree!) of the coefficients of the curve a sufficiently weaker result such that you could express the coordinates of the extrema of a sixth degree polynomial as another sixth degree polynomial?

## Laurie Hailer

December 14, 2016 - 8:33 am -Yes, I love that Desmos function. I was amazed when I first saw that and show as many students and colleagues as I can. I think it’s really cool that you are proving that in Precalc. Go Math!

## Randy Revels

December 14, 2016 - 1:00 pm -This same thing happened in my Algebra 1 class the last time we studied transformations of parabolas. So cool. :-)

## James Key

December 14, 2016 - 4:13 pm -Since we are on cool facts about quadratic functions, here’s an insight I learned from James Tanton: all the textbooks teach that the axis of symmetry has equation x = -b/(2a), but the reason is seldom explored in a satisfactory way. Check this out:

y = ax^2 + bx + c = x(ax+b) + c

From this we can read off the solutions (0,c) and (-b/a, c), and the symmetry line lies halfway between any two points with the same y-coordinate, so its equation must be x = -b/(2a). Neat stuff! This is a great way to graph parabolas that are presented in standard form. But the cool thing for students to learn is the *technique,* not the formula.

## CCSSIMath

December 14, 2016 - 5:13 pm -If a quadratic function is given by y=ax²+bx+c, then the x-coordinate of the vertex is −b/2a. That’s basic.

If you substitute −b/2a for x and solve for y, the y-coordinate of the vertex becomes −b²/4a+c.

Since the coordinates of the vertex are (−b/2a, −b²/4a+c), if you leave a and c constant and vary b, then the y-coordinate changes as the square of the x-coordinate, in other words, a parabola.

The result that seems to be impressing people is not so remarkable if you make an effort to do simple math.

Yes, desmos can be used to stimulate an effort to prove this result, but it might instead be used to confirm the result, which is more in line with the scientific process.

Incidentally, not that many seem to give too much credence to international assessments, but in nations such as Japan, analyzing quadratic functions is in the standard curriculum for 9th grade.

## William Carey

December 14, 2016 - 5:16 pm -Sweet. Now talk through how to get your students to that for the cubic over their lunch break. Go.

## Yongyi

January 6, 2017 - 5:22 pm -I’ve proved that the graph of the minima of a general polynomial of any degree traces out a polynomial of the same degree! There’s a lot of algebra and mathematical notation in the proof, so I’ve written it up in LaTeX and you can see the proof here: http://latex.artofproblemsolving.com/miscpdf/huzvyugx.pdf?t=1483752108866

## Yongyi

January 6, 2017 - 5:38 pm -And here’s a Desmos graph showing that my formula in the cubic case works! In fact it goes through both extrema of all the graphs!

https://www.desmos.com/calculator/3s3obopabq

## Ethan Weker

January 18, 2017 - 8:27 am -I’m late to this discussion clearly, but can’t wait to start playing with all these ideas! I’ve taught parabolas as products of lines in the past (including with this Desmos activity: https://teacher.desmos.com/activitybuilder/custom/56c39561a72db32f0a6f0e28), but there’s clearly so much more rich content that I can and should get into here. I especially love the description (from EducationRealist – I feel like I should know who you are) of a quadratic in standard form as the sum of a quadratic with a vertex at the origin and a line – that should be a nice way to add to/expand my previous activity.