Question

The function f(t) = 4t2 − 8t + 7 shows the height from the ground f(t), in meters, of a roller coaster car at different times t. Write f(t) in the vertex form a(x − h)2 + k, where a, h, and k are integers, and interpret the vertex of f(t).

(A) f(t) = 4(t − 1)2 + 3; the minimum height of the roller coaster is 3 meters from the ground
(B) f(t) = 4(t − 1)2 + 3; the minimum height of the roller coaster is 1 meter from the ground
(C) f(t) = 4(t − 1)2 + 2; the minimum height of the roller coaster is 2 meters from the ground
(D) f(t) = 4(t − 1)2 + 2; the minimum height of the roller coaster is 1 meter from the ground

Answer 1

T= 4 s (second option).

Answer 2

The function is a parabola, and the problem asks to transform the equation into f(t)=a(x-h)2 + k

Given f(t) = 4t2 -8t +7

= (4t2 - 8t + 4) + 7 - 4

=4 (t2 - 2t + 1) + 3

= 4 (t-1) 2 +3

This removes C and D from the viable choices.

Differentiating the f(t),

f’(t) = 8t – 8, the maximum/minimum value occurs at f’(t) = 0

0 = 8t – 8

t = 1

determining if maximum or minimum, f”(t) > 0 if minimum, f”(t) < 0 maximum

f”(t) = 8 > 0, therefore minimum

f(1) =4(1)^2 – 8(1) +7

= 3

Therefore, minimum height is 3.