## Tuesday, March 31, 2009

### Another Quadratic Function SAT Problem

Have you noticed the SAT Tips of the Week in the sidebar? These are intended both for math teachers and students.

The "new" SAT has a few 2nd year algebra questions and, typically, there is at least one 'parabola' problem usually expressed in function form. Here are two different versions - one multiple choice and one "grid-in" (student-generated response).

Can you predict which one might give most students more difficulty?
It might be interesting to list all of the skills, knowledge and concepts being tested here. Are all of these typically included in your Algebra 2 course? Do students get enough exposure to these kinds of problems?

Version I
For some constant r, the graph of the quadratic function f(x) = -x
2 + 2rx is a parabola with x-intercepts at P and Q and vertex V. What is the area of ΔPQV, in terms of r?

(A) r2 (B) r3 (C) 2r2

(D) 2r3
(E) 4r3

Version II

For some constant r, the graph of the quadratic function f(x) = -x2 + 2rx is a parabola with x-intercepts at P and Q and vertex V. If the area of ΔPQV equals 27, what is the value of r?

Version I

Possible Solution (no frills):
Factoring, we have f(x) = -x(x-2r); x-intercepts are 0 and 2r. Therefore base of triangle has length 2r.
The x-coordinate of the vertex is r (why?), so y-coord = f(r) = -r(r-2r) = r2.
Area of triangle = (1/2)(2r)(r2) = r3.

Version II

Possible Solution:
From Version I, we obtain r3 = 27, so r = 3.