Question

The function f(t) = 4t2 - 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

f(t) = $4 t^{2} -8+6$
The formula to use is: a(x-xv)^2+yv
where a is the coeficient of x^2, xv= -b/2a (b is the coeficient of x) and yv= -delta/4a or f(xv)

xv = 8/8 =1
yv = f(xv)=f(1) = 2

f(t) = 4(t-1)^2+2 the minimum value is 2 when t=1

The answer to your question is C. I hope this is the answer that you are looking for and it comes to your help.

Answer 2

The function f(t) can be written in the vertex form as: C. f(t) = 4(t - 1)² + 2; the minimum height of the roller coaster is 2 meters from the ground.

How to write this function in vertex form?

Since the leading coefficient in the given function f(t) is positive 4, the parabola will open upward and at the vertex. Also, the value of the function f(t) would be minimum.

Evaluating the function by applying completing the square, we have:

f(t) = 4t² - 8t + 6

f(t) = 4(t² - 2t) + 6

f(t) = 4(t² - (2 × t × 1) + 1² - 1²) + 6

f(t) = 4(t² - (2 × t × 1) + 1² - 1) + 6

f(t) = 4(t² - 2t + 1²) - 4 + 6

f(t) = 4(t² - 2t + 1²) + 2

f(t) = 4(t - 1)² + 2.

Therefore, the vertex of f(t) will be at (1, 2) and the minimum height of the roller coaster is 2 meters from the ground.

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