In this problem we have a quadratic equation
y=X²-4x-5
The domain of a quadratic equation is all real numbers
To find out the range we need to calculate the vertex of the parabola
Convert the quadratic equation into vertex form
so
(y-k)=(x-h)^2
y=X²-4x-5
y+5=x^2-4x
complete the square
y+5=(x^2-4x+4)-4
y+5+4=(x-2)^2
y+9=(x-2)^2
the vertex is the point (2,--9)
The quadratic equation represent a vertical parabola open upwards
so the range is the interval
{-9, infinite)
y-1=(x-2)^2 -------> is written as vertex form
The vertex of the parabola is the point (h,k)
(y-k)=(x-h)^2
using a graphing tool to better understand the problem
The range is the interval {-9, infinite)
The domain is the interva (-infinite, infinite)
the function is
y+9=(x-2)^2
f(x)=(x-2)^2-9
or
f(x)=x^2-4x-5