Sketching a graph of f(x) = x

^{2}+ 6x + 7

is the same as sketching f(x) = x(x + 6) + 7

which is the same as sketching g(x) = x(x + 6) and then shifting g up 7 units.

Since the points (0, 0) and (-6, 0) are on the graph of g, the points (0, 7), (-6, 7) must be on the graph of f. This allows us to quickly see the symmetry line at x = -3 without memorizing x=-b/(2a).

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 1)(x + 5) + 2

So, if h(x) = (x + 1)(x + 5), then we can imagine the points (-1, 0) and (-5, 0) from h being translated up 2 units to get the points (-1, 2), (-5, 2) on f.

Another way of "partially" factoring f to get the middle term 6x is f(x) = (x + 2)(x + 4) - 1

So, if j(x) = (x + 2)(x + 4), by thinking about the relationship between j and f, we can deduce that f must have the points (-2, -1), (-4, -1).

Similarly, we can get the partial factorization f(x) = (x + 3)(x + 3) - 2.

If we assume m(x) = (x + 3)(x + 3) and consider the relationship between m and f, we can deduce that (-3, -2) must exist on the graph of f.

So, we can pull together all those points so far to get (0, 7), (-6, 7), (-1, 2), (-5, 2), (-2, -1), (-4, -1), and (-3, -2) as points that must be on f. This way of thinking about graphing quadratics ties together strongly the ideas of factorization and transformation. They're no longer two separate concepts but integrated as one. Since I've never seen this connection in a textbook before, I decided to call it flexible factorization.

One distinct advantage of flexible factorization is that as soon as you are given y = x

^{2}+ kx + m, you can quickly factor it partially into y = x(x + k) + m, which allows you to quickly determine two points on the graph, (0, m) and (-k, m) and to find the axis of symmetry at x=-k/2. You can sketch the graph roughly in about 30 seconds for any standard quadratic function (this extends to y = ax

^{2}+ bx + c, as it factors into y = x(ax + b) + c, which means that (0, c) and (-b/a, c) are two points on this graph and the parabola opens in the direction as indicated by the leading coefficient "a".)

Of course, this does not mean that the kids won't have to learn the standard analysis techniques, but I think being able to connect factorization with transformation gives them another tool when modeling and thinking about graphs.

I'm going to keep playing around with this idea, possibly turning it into an end-of-year project in Grade 8. Stay tuned!

James Tanton is a big fan of this approach. You should check out his curriculum newsletters for April and May 2012 at http://www.jamestanton.com/?p=1072

ReplyDeleteTotally awesome! I'm pretty amazed by how his approach is pretty exactly my line of thinking. Thanks for the link, Tessa!

ReplyDeleteI was just going to mention Tanton too. Have you seen his videos on this? I love that you figured it out independently. And your way of describing it really works for me. (I think I like print better than video.)

ReplyDeleteHaven't seen it, but I sure will have to check it out! My mind has already turned towards how to structure this into lessons leading up to a modeling/explanatory project though....

ReplyDeleteAs an opponent of factoring, the title of this post intrigued me. I may be less opposed to it now that there's more coolness involved.

ReplyDeleteAlso, I'm off to look at Tanton's things.

I'm not sure how you can be against factoring... To me it's just like being against "breaking down a diagram into smaller parts" or being against "analyzing one variable at a time".

ReplyDeleteFactorization fits into the big math theme of breaking a complex problem (ie. polynomial of degree n) into smaller parts (n linear factors) and analyzing each smaller part separately?!