What goes up...Applying Quadratic Functions

This question was inspired by a released SAT question from a couple of years ago. Some of you may recall this question that appeared on the first released Sample Test for the 'new' SAT. Can you think of some reasons why this question was used by ETS as a Sample problem?

Because the height function given was not exactly in 'standard' form (using a,h, and k), even the strongest students I administered this problem to resorted to complicated algebra or used a physics formula (s = 0.5at²+...). They missed the point that when given the vertex of a parabola you're given more than just an ordered pair! I believe our students need more experience with this type of 'free-response' application. We see these in some textbooks but is enough time devoted to them or is that left to the physics teacher?

As usual, I've modified the question and developed it into an open-ended problem with several parts. Pls don't get exercised about the lack of reality of the physical model!

A model rocket is projected vertically upward from a point 877.5 ft above the ground and after 2.5 seconds reaches its maximum height of 1440 ft. We are given its height above the ground as a function of t:

h(t) = p(q-3t)² + r, where t is in sec, h is in ft; p, q, r are constants.


(a) Determine the values of p, q and r.

(b) Rewrite the given function in 'standard' form: h(t) = a(t-h)² + k.

(c) Determine, algebraically, all values of t for which h(t) = 1080. Explain, in terms of the motion of the rocket, why there are exactly 2 such values.

(d) After how many seconds did the object hit the ground? Use algebra.

(e) Verify your results by analyzing the function using graphing calculator technology.