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) = -x² + 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?

Answers, solutions, strategies and comments will appear below the Read more...



Answers/Solutions/Comments/...

Version I
Answer: (B) r³

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
Answer: r = 3

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

Comments

• These kinds of questions typically appear among the last 3-4 problems on a section, meaning they are of above-average difficulty. Students who are in Algebra 2 or beyond should definitely attempt it. After reviewing it, most students may conclude it's not very hard at all!
  
• Testmakers are more frequently using a parameter like 'r' to make it more difficult to merely punch it into the graphing calculator and read off the intercepts and vertex.
• This kind of question could also appear on standardized tests like the Algebra 2 End of Course Exam from Achieve/ADP or other state tests.
• Pick up a copy of 10 Real SATs from the College Board to find several other practice problems like this.
• One could modify and extend these problems in many ways. For example inscribe the parabolic region (bounded by the cruve and the x-axis) in a rectangle and determine its area, a simple variation. More interestingly is to note that the ratio of the area enclosed by the parabola to the area of the rectangle is 2:3, a famous result proved in Calculus.
• Skills, knowledge required for this question? Worth enumerating in my opinion...
  
• I also believe strongly that our students should be tackling these kinds of problems on a regular basis to deepen their understanding of the relationship among the function, the coordinates of key points and the geometry. This used to be known as Analytic Geometry.