Question

The function f(t) = 4t^2 − 8t + 6 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

C - f(t) = 4(t − 1)^2 + 2; the minimum height of the roller coaster is 2 meters from the ground.

Explanation:

Here we're asked to rewrite the given equation f(t) = 4t^2 − 8t + 6 in the form f(t) = a(t - h)^2 + k (which is known as the "vertex form of the equation of a parabola.") Here (h, k) is the vertex and a is a scale factor.

Let's begin by factoring 4 out of all three terms:

f(t) = 4 [ t^2 - 2t + 6/4 ]

Next, we must "complete the square" of t^2 - 2t + 6/4; in other words, we must re-write this expression in the form (t - h)^2 + k.

(To be continued)