Question

What is the equation of the parabola that has a vertex at `(34, -3)` and passes through point `(32,-43)`?

Answer 1

There are two parabolas that have the given vertex and can pass through the given point:

`y = -10(x-34)^2-3`
`x = -1/800(y--3)^2+34`

Explanation:

Using both vertex forms of the equation of a parabola:

`y = a(x-h)^2+k`
`x = a(y-k)^2+h`

Substitute the 34 for h and -3 for k:

`y = a(x-34)^2-3`
`x = a(y--3)^2+34`

Solve for both values of a by substituting 32 for x and 43 for y:

`-43 = a(32-34)^2-3`
`32 = a(-43--3)^2+34`

`-40 = a(-2)^2`
`-2 = a(40)^2`

`a = -10`
`a = -1/800`

`y = -10(x-34)^2-3`
`x = -1/800(y--3)^2+34`