Question

How do you find the axis of symmetry, graph and find the maximum or minimum value of the function `y=x^2 - 2x - 1`?

Answer 1

Symmetry: `x=1` ; Maximum value is `oo` , Minimum value is `-2`

Explanation:

This is a parabola opening upwards `a>0`

`y= x^2-2x-1 (ax^2+bx+c); a=1 ; b = -2 ; c= -1`
Discriminant `D=b^2-4ac=4+4=8`
Vertex `(x,y) ; x= (-b)/(2a)= 2/2=1 ;` Putting `x=1` we can get `y= 1^2-2*1-1= -2 :.` Vertex is at `(1 , -2) :. x=1` is the axis of symmetry.

`a>0 :.` Maximum value is `oo` ; Minimum value is `y=-D/(4a) = -8/4=-2` at `x= -b/(2a) = (- (-2))/2 =1` i.e Vertex is the minimum point.
Symmetry: `x=1` ; Maximum value is `oo` , Minimum value is `-2` graph{x^2-2x-1 [-10, 10, -5, 5]} [Ans]