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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

Respuesta :

InesWalston

Answer:

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

Step-by-step explanation:

The given function is,

[tex]f(t) = 4t^2 - 8t + 7[/tex]

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.

[tex]f(t)=4t^2-8t+7[/tex]

[tex]=4(t^2-2t)+7[/tex]

[tex]=4(t^2-2\cdot t\cdot 1+1^2-1^2)+7[/tex]

[tex]=4(t^2-2\cdot t\cdot 1+1^2-1)+7[/tex]

[tex]=4(t^2-2\cdot t\cdot 1+1^2)-4+7[/tex]

[tex]=4(t^2-2\cdot t\cdot 1+1^2)+3[/tex]

[tex]=4(t-1)^2+3[/tex]

So the vertex form will be,

[tex]f(t)=4(t-1)^2+3[/tex]

Hence the vertex will be at [tex](1,3)[/tex]

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.

Ver imagen InesWalston

Answer:

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

Step-by-step explanation:

Given : The function [tex]f(t) = 4t^2-8t+7[/tex] 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 [tex]a(x - h)^2 + k[/tex], where a, h, and k are integers, and interpret the vertex of f(t).

Solution :

We have given the function [tex]f(t) = 4t^2-8t+7[/tex]

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,

[tex]f(t)=4t^2-8t+7[/tex]

[tex]f(t)=4(t^2-2t)+7[/tex]

Making completing square,

[tex]f(t)=4(t^2-2t+1^2-1^2)+7[/tex]

[tex]f(t)=4(t^2-2t+1^2)-4+7[/tex]

[tex]f(t)=4(t-1)^2+3[/tex]

So the vertex form will be, [tex]f(t)=4(t-1)^2+3[/tex]

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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