Question

What is the equation of the parabola that has a vertex at `(-18, -12)` and passes through point `(-3,7)`?

Answer 1

`y=19/225(x+18)^2-12`

Explanation:

Use the general quadratic formula,

`y=a(x-b)^2+c`

Since the vertex is given `P(-18,-12)`, you know the value of `-b` and `c`,

`y=a(x--18)^2-12`
`y=a(x+18)^2-12`

The only unkown variable left is `a`, which can be solved for using `P(-3,7)` by subbing `y` and `x` into the equation,

`7=a(-3+18)^2-12`
`19=a(15)^2`
`19=225a`
`a=19/225`

Finally, the equation of the quadratic is,

`y=19/225(x+18)^2-12`

graph{19/225(x+18)^2-12 [-58.5, 58.53, -29.26, 29.25]}

Answer 2

There are two equations that represent two parabolas that have the same vertex and pass through the same point. The two equations are:

`y =19/225(x+18)^2-12` and `x = 15/361(y+12)^2-18`

Explanation:

Using the vertex forms:

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

Substitute `-18` for `h` and `-12` for `k` into both:

`y =a(x+18)^2-12` and `x = a(y+12)^2-18`

Substitute `-3` for `x` and 7 for `y` into both:

`7 =a(-3+18)^2-12` and `-3 = a(7+12)^2-18`

Solve for both values of `a`:

`19=a(-3+18)^2` and `15 = a(7+12)^2`

`19=a(15)^2` and `15 = a(19)^2`

`a = 19/225` and `a = 15/361`

The two equations are:

`y =19/225(x+18)^2-12` and `x = 15/361(y+12)^2-18`

Here is a graph of the two points and the two parabolas: