Question

The point P lies on the parabola y=x^2+x+ 1. Find the distance between P and the point (2,1) as a function of the x-coordinate of P.

Answer 1

Find the distance between two points.
One point is: (2, 1)
Second point is: (x, y) on the parabola and so (x, x^2+x+1) (in terms of x).

Find the distance between (2,1) and (x, x^2+x+1) using the distance formula.

Answer 2

\[
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} = \sqrt{(x-2)^2 + (x^2+x)^2} = \\
d = \sqrt{x^2-2x+4 + x^4 + 2x^3 + x^2} \\
d = \sqrt{x^4+2x^3+2x^2-2x+4}
\]

Answer 3

@aum ok cool thats the right answer, i just have a question about the (y2-y1)^2

how come its (x^2 + x)^2, isn't y2 the entire quadratic and then y1 just 1(from the 2,1)?

Answer 4

\[
y_2 = x^2 + x + 1 \\
y_1 = 1 \\
y_2-y_1 = x^2 + x + 1 - 1 = x^2 + x
\]

Answer 5

@aum oh man i feel dumb! thanks a ton for the help :)