To solve the given problem, we need to interpret the function \( f(t) = 4t^2 - 8t + 6 \) in the context of its vertex form, which helps us identify the minimum height of the roller coaster.
First, let's rewrite the given quadratic function in its vertex form \( f(t) = a(t - h)^2 + k \).
To do this, we complete the square as follows:
1. Start with the function \( f(t) = 4t^2 - 8t + 6 \).
2. Factor out the coefficient of \( t^2 \) from the first two terms:
[tex]\[ f(t) = 4(t^2 - 2t) + 6 \][/tex]
3. To complete the square, take half of the coefficient of \( t \) (which is -2), square it, and add and subtract this value inside the parentheses:
[tex]\[ 4(t^2 - 2t + 1 - 1) + 6 \][/tex]
4. Simplify inside the parentheses:
[tex]\[ 4((t - 1)^2 - 1) + 6 \][/tex]
5. Distribute the 4 through the expression inside the parentheses:
[tex]\[ 4(t - 1)^2 - 4 + 6 \][/tex]
6. Combine the constants:
[tex]\[ f(t) = 4(t - 1)^2 + 2 \][/tex]
Now, we have the function in vertex form \( f(t) = 4(t - 1)^2 + 2 \), where \( a = 4 \), \( h = 1 \), and \( k = 2 \).
The vertex of the function is at \( (h, k) = (1, 2) \). This means that the minimum height of the roller coaster, which occurs at \( t = 1 \), is 2 meters from the ground.
Among the given choices, the correct interpretation is:
[tex]\[ f(t) = 4(t - 1)^2 + 2 \][/tex]; the minimum height of the roller coaster is 2 meters from the ground.
Thus, the correct answer is:
[tex]\( f(t) = 4(t - 1)^2 + 2 \)[/tex]; the minimum height of the roller coaster is 2 meters from the ground.
