Question

Choose the correct vertex of the function f(x) = x2 - X + 2.

Answer 1

`\large\boxed{\left((1)/(2),\ (7)/(4)\right)}`

Explanation:

The vertex form of an equation of a parabola f(x) = ax² + bx + c:

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

(h, k) - vertex

METHOD 1:

`h=(-b)/(2a),\ k=f(h)\\\\f(x)=x^2-x+2\to a=1,\ b=-1,\ c=2\\\\h=(-(-1))/((2)(1))=(1)/(2)\\\\k=f\bigg((1)/(2)\bigg)=\left((1)/(2)\right)^2-(1)/(2)+2=(1)/(4)-(1)/(2)+2=(1)/(4)-(1\cdot2)/(2\cdot2)+2\\\\=(1)/(4)-(2)/(4)+2=-(1)/(4)+2=1(3)/(4)=(7)/(4)\\\\\boxed{\left((1)/(2),\ (7)/(4)\right)}`

METHOD 2:

`\text{use}\ (a-b)^2=a^2-2ab+b^2\qquad(*)\\\\f(x)=x^2-x+2=\underbrace{x^2-2(x)\left((1)/(2)\right)+\left((1)/(2)\right)^2}_((*))-\left((1)/(2)\right)^2+2\\\\=\left(x-(1)/(2)\right)^2-(1)/(4)+2=\left(x-(1)/(2)\right)^2+1(3)/(4)=\left(x-(1)/(2)\right)^2+(7)/(4)\\\\h=(1)/(2),\ k=(7)/(4)\\\\\boxed{\left((1)/(2),\ (7)/(4)\right)}`