Saturday, March 10, 2007

Parabolas, SATs, Quadratic Functions, Symmetry, Oh My!

The new SAT and other state math assessments are or will be including more Algebra 2 types of questions, particularly those involving quadratic functions. The following was inspired by a recent SAT math problem. As usual, my goal here is not to give conundrums and 'puzzlers'. I'll leave that to the expertise of Jonathan over at jd2718! My intent is to provide enrichment and extensions of questions that students are doing in class. More time is required for these than is normally given for an example presented by the teacher. Hopefully these can be used in the classroom.
The original question on the SAT gave a particular length for segment PQ (see below) and that may be a more reasonable start for most Algebra 2 students. The objective here is to have students apply and extend their knowledge of quadratic functions, graphs, coordinates, symmetry, etc. There are several approaches to this question. If instruction enables students to investigate this problem for 10-15 minutes, students may discover alternate methods that will deepen their understanding of the material. The teacher's role is to gauge the ability level and background of the group to determine how much structure/guidance is needed. This is not obvious at all and requires considerable pedagogical skill and experience.

Consider the graph of the quadratic function f(x) = x2. Assume P, Q are points on the graph so that segment PQ is parallel to the x-axis and let the length of segment PQ be denoted by 2k.
If the graph of g(x) = b - x
2, intersects the graph of f(x) at P and Q, express the value of b in terms of k.

Encourage several methods, i.e., pair students and require that they find at least two different methods. This is critical to develop that quick thinker who always has the answer before anyone else and does not want to deepen his/her insight. Many students will need to start with a numerical value for the length of segment PQ, say 4. Symmetry is a key idea in this problem, not only with respect to the y-axis, but also with respect to segment PQ! Some will see this quickly, others won't. It is our obligation to think this through in advance and be prepared to guide the investigation. Those who believe this kind of activity is a waste of precious time (so much more content could be covered) will never understand why I believe 'less is more' when it comes to learning math. Profound understanding can never be rushed. Short-cuts, IMO, are PART of a discussion, not the objective. Try it! Can you find at least THREE ways?


jonathan said...

Well, we could talk about the coordinates of P and Q.
(-sqrt(b/2), b/2) and
(sqrt(b/2), b/2)

2k = 2sqrt(b/2)
k = sqrt(b/2)
k^2 = b/2
and b = 2k^2

As we solved non-linear systems a few times last week, and then this morning, perhaps we will play with a variation. Now, that would not count as problem solving, since they are expecting this sort of question, and will not think twice about approach.

Dave Marain said...

nice straightforward solution, jonathan...
now consider how it can be done using only y = x^2, the length 2k and the reflection of the graph of f(x) over the line PQ. Ask your students why the graph of g(x) must be the reflection image of the graph of f(x) over this line. After all, there are many 'inverted' parabolas with vertices on the y-axis that pass through P and Q! Perhaps this is obvious to some...

Dave Marain said...

More on the symmetry approach...
Using only the function y = x^2 to start:
From symmetry to the y-axis, the point Q has coordinates (k,k^2).
Since the graph of y = -x^2 + b is 'congruent' to the graph of y = x^2 (abs values of coeff of square terms are equal!), it follows that the graph of y = b - x^2 is the reflection of the graph of y = x^2 over line PQ. Therefore, b = 2k^2. QED!
I believe Jonathan's solution is the one most of us would use, since it directly address the point of intersection of the 2 graphs. However, also suggesting the symmetry approach may be beneficial. Students need to develop comfort with this, since, in calculus, they will frequently use symmetry arguments when integrating. Does anyone think a couple of students would have done this?