Standard and Vertex Form of the Equation of a Parabola

The standard form of the equation of a parabola is:

y=Ax2+Bx+Cy=Ax2+Bx+C

while in vertex form, the equation of a parabola is:

y=p(x−h)2+ky=p(x−h)2+k

whereas:

pp = distance of vertex to focus

hh = x-coordinate of the vertex

kk = y-coordinate of the vertex

Answer and Explanation: 1

From the standard form, y=x2+4x−7y=x2+4x−7.

We transform it to vertex form to determine the hh and kk by completing the square.

First, we transpose 77 in the left side of the equation.

y+7=x2+4xy+7=x2+4x

Next, we add 44 to transform it into a perfect square.

y+7+4=x2+4x+4y+7+4=x2+4x+4

y+11=(x+2)2y+11=(x+2)2

Transposing again the 1111 to the other side of the equation into the form of vertex form, y=p(x−h)2+ky=p(x−h)2+k

y=(x+2)2−11y=(x+2)2−11

So, from its vertex form we can say that h=−2h=−2 and k=−11k=−11.

Therefore, the vertex of the parabola with equation y=x2+4x−7y=x2+4x−7 is at (−2,−11)(−2,−11).