What is the equation of a parabola?

Answer 1

Parabola's equations can be any second-order polynomial, but popular forms are:

Standard form for a parabola is y=Ax2+Bx+C

This is a very nice form for algebraic use, but physically, A B and C do not directly represent the shape of the curve

Vertex form: y=a(x-h)2+k

Often people use this because (h, k) would also be the location of the parabola's vertex. The a at the beginning represents the "wideness" (tight parabola's have high a values greater than one; wide parabola's have low a values less than one) and the direction of the parabola (negative a means it opens downward while a positive a mean it opens upwards)

Factored form: y=a(x-r)(y-s)

This form is very nice because it is factored, and it shows you the zeros of the function: r and s. a is the same as from the vertex form. This is less popular than other forms because some parabolas have imaginary numbers for r and s when in factored form, even if the parabola is made of completely real points.