A parabola has a vertex at (-1, 0) and opens down. What is the equation of the parabola?

Question

A parabola has a vertex at (-1, 0) and opens down. What is the equation of the parabola?

Answer 1

There is not enough information to get the exact equation! a parabola has the form

y = ax² + bx + c where you have 3 unknowns which must be given in some form.

One equation that satisfies your conditions is y = -(x+1)² + 0 (The negative sign says "open down")

but without another piece of information, you don't know just how the parabola "fans out"!

Answer 2

y-h= -a(x-k)² is the standard from for a parabola opening downward, with (k,h) as the vertex

but there's not enough information to determine the equation.

It could be a anywhere from a nearly flat line going through (-1,0) with y=0 as the limit to two nearly vertical lines going down from (-1,0) with x=-1 as the limit.

Without some information to show the value of a, how flat or vertical the parabola is, you can't find the equation.

But with (-1,0) as the vertex, the equation would be y-0 = -a(x+1)² or y=-a(x+1)² or y=-a(x²+2x+1)

taking the first derivative gives -a(2x+2) set = zero, then x=-1, meaning (-1,0) is the vertex.

Best you can do for an equation is y=-a(x+1)²

If a is very small, the equation approaches y=0

If a is very large, the equation approaches x=-1