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Question 10 (Multiple Choice Worth 1 point)

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 vertex form 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. [tex]\( f(t) = 4(t - 1)^2 + 2 \)[/tex]; the minimum height of the roller coaster is 1 meter from the ground

Sagot :

GinnyAnsweridn
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.
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