The vertex of the parabola below is at the point (4, -1). Which of the equations below could be the one for this parabola?

One possible equation for this quadratic would be 

y=(x-4)²-1.  This is vertex form:  y=a(x-h)²+k, where (h, k) is the vertex.

However, this is not the only possible equation.  There could be multiple values for a, in front of the parentheses, that we don't know about from the information we are given.

We can also write this equation in standard form (y=ax²+bx+c).  First write the squared binomial as the product of two binomials:
y=(x-4)(x-4)-1

Multiply the binomials:
y=x*x-4*x-4*x-4(-4)-1
 = x²-4x-4x--16-1
 = x²-8x+16-1
 = x²-8x+15

Again, this would change depending on what the value of a is in the functoin.