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The function f(t) = 4t2 - 8t + 7 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

Verified Answer :

InesWalston

Answer:

A. \(f(t)=4(t-1)^2+3\); the minimum height of the roller coaster is 3 meters from the ground.

Step-by-step explanation:

The given function is,

\(f(t) = 4t^2 - 8t + 7\)

As the leading coefficient is 4 which is positive, so the parabola will open upward and at the vertex the value of the function will be minimum.

\(f(t)=4t^2-8t+7\)

\(=4(t^2-2t)+7\)

\(=4(t^2-2\cdot t\cdot 1+1^2-1^2)+7\)

\(=4(t^2-2\cdot t\cdot 1+1^2-1)+7\)

\(=4(t^2-2\cdot t\cdot 1+1^2)-4+7\)

\(=4(t^2-2\cdot t\cdot 1+1^2)+3\)

\(=4(t-1)^2+3\)

So the vertex form will be,

\(f(t)=4(t-1)^2+3\)

Hence the vertex will be at \((1,3)\)

At vertex the value of the function ot f(t) is 3, so the minimum height of the roller coaster is 3 meters from the ground.

View image InesWalston

Answer:

Option A- \(f(t)=4(t-1)^2+3\) ; the minimum height of the roller coaster is 3 meters from the ground.

Step-by-step explanation:

Given : The function \(f(t) = 4t^2-8t+7\) shows the height from the ground f(t), in meters, of a roller coaster car at different times t.

To find : 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).

Solution :

We have given the function \(f(t) = 4t^2-8t+7\)

As the leading coefficient is 4 which is positive, so the parabola will open upward and at the vertex the value of the function will be minimum.

Now, we convert it into vertex form,

\(f(t)=4t^2-8t+7\)

\(f(t)=4(t^2-2t)+7\)

Making completing square,

\(f(t)=4(t^2-2t+1^2-1^2)+7\)

\(f(t)=4(t^2-2t+1^2)-4+7\)

\(f(t)=4(t-1)^2+3\)

So the vertex form will be, \(f(t)=4(t-1)^2+3\)

Where Vertex are (h,k)=(1,3) and a=4

At vertex the value of the function f(t) is 3.

So, the minimum height of the roller coaster is 3 meters from the ground.

Therefore, Option A is correct.

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