Question

Which function in vertex form is equivalent to f(x) = x2 + x +1?

O f(x) = (x + 3+
1
3
o f(x)
4
f(x) = (x +
f(x) = (x + +
f(x) = (x + 3D + 1
IN​

Answer 1

To find the vertex form of f(x) = x^2 + x + 1, we complete the square, resulting in f(x) = (x + 1/2)^2 + 3/4.

Step-by-step explanation:

To convert the quadratic function f(x) = x^2 + x +1 into vertex form, we need to complete the square. Vertex form is y = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. Let's complete the square for the given function:

• First, we take the coefficient of x, which is 1, divide it by 2, and square it, giving us (1/2)^2 = 1/4.
• Next, we add and subtract this value inside the square of the function: f(x) = x^2 + x + (1/4) - (1/4) + 1.
• Now, we can rewrite the function with a perfect square: f(x) = (x + 1/2)^2 + 3/4.

This process has transformed the original quadratic function into vertex form, which is f(x) = (x + 1/2)^2 + 3/4.

Answer 2

`\displaystyle f(x)=\left(x+(1)/(2)\right)^2+(3)/(4)`

Step-by-step explanation:

We are given the function:

`f(x)=x^2+x+1`.

And we want to turn this into vertex form.

Note that our given function is in the standard form:

`f(x)=ax^2+bx+c`

In other words, a = 1, b = 1, and c = 1.

To convert from standard form to vertex form, we can either: (1) complete the square, or (2) find the vertex manually.

In most cases, the second method is more time efficient.

Vertex form is given by:

`f(x)=a(x-h)^2+k`

Where a is the leading coefficient and (h, k) is the vertex.

We have already determined that a = 1.

Find the vertex. The x-coordinate of the vertex of a quadratic is given by:

`\displaystyle x=-(b)/(2a)`

Therefore, our point is:

`\displaystyle x=-((1))/(2(1))=-(1)/(2)`

To find the y-coordinate or k, substitute this value back into the function:

`\displaystyle f\left(-(1)/(2)\right)=\left(-(1)/(2)\right)^2+\left(-(1)/(2)\right)+1=(3)/(4)`

Thus, the vertex is (-1/2, 3/4). So, h = -1/2 and k = 3/4.

Hence, the vertex form is:

`\displaystyle f(x)=(1)\left(x-\left(-(1)/(2)\right)\right)^2+\left((3)/(4)\right)`

Simplify:

`\displaystyle f(x)=\left(x+(1)/(2)\right)^2+(3)/(4)`