Question

1. Find the coordinates of the vertex
f(x) = x2 - 4x + 7

Answer 1

`\large\boxed{(2,\ 3)}`

Explanation:

`\bold{METHOD\ 1:}\\\\\text{For}\ f(x)=ax^2+bx+c\ \text{a vertex is}\ (h,\ k),\ \text{where}\ h=(-b)/(2a),\ \text{and}\ k=f(h).\\\\f(x)=x^2-4x+7\\\\a=1,\ b=-4,\ c=7\\\\\text{Substitute:}\\\\h=(-(-4))/(2(1))=(4)/(2)=2\\\\k=f(2)=2^2-4(2)+7=4-8+7=3\\\\\boxed{\boxed{vertex=(2,\ 3)}}`

`\bold{METHOD\ 2:}\\\\\text{The vrtex form of a quadratic function:}\\\\f(x)=a(x-h)^2+k,\ \text{where}\ (h,\ k)\ \text{is a vertex}.\\\\f(x)=x^2-4x+7\qquad\text{use}\ (*)\ \ (a-b)^2=a^2-2ab+b^2\\\\f(x)=\underbrace{x^2-2(x)(2)+2^2}_((*))-2^2+7=(x-2)^2-4+7\\\\f(x)=(x-2)^2+3\to h=2,\ k=3\\\\\boxed{\boxed{vertex=(2,\ 3)}}`