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).

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

Answer 1

Answer:

The correct option is 3.

Step-by-step explanation:

The vertex form of a parabola is

$f(x)=a(x-h)^2+k$            .... (1)

where a, h, and k are integers, and interpret the vertex of f(t). (h,k) is the vertex of the parabola.

The given function is

$f(x)=4t^2-8t+6$

It can be written as

$f(x)=4(t^2-2t)+6$

If an expression is defined as $x^2+bx$, then we need to add $(\frac{b}{2})^2$ to make it perfect square.

In the expression $t^2-2t$ the value of b is -2. So, we nned to add and subtract $(\frac{-2}{2})^2$ in the parenthesis.

$f(x)=4(t^2-2t+1^2-1^2)+6$

$f(x)=4(t^2-2t+1)+4(-1)+6$

$f(x)=4(t-1)^2-4+6$

$f(x)=4(t-1)^2+2$                .... (2)

The vertex form of the parabola is $f(x)=4(t-1)^2+2$.

From (1) and (2), we get h=1 and k=2. It means the vertex of the parabola is (1,2). Vertex of upward parabola is point of minima. So the  minimum height of the roller coaster is 2 meters from the ground.

Therefore the correct option is 3.