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).
