Question

The vertex and the focus of a parabola are shown. What is the equation of the parabola?

Answer 1

`x = -(1)/(8)(y - 4)^2 -2`

Explanation:

To find the equation for the parabola with the given focus and vertex, we can use the form:

`(y-b)^2 = 4p(x-a)`

where
`(a,b)` is the vertex and
`p` is the distance from the focus to the vertex, with respect to direction (right is a positive distance, left is a negative distance).

We are given that the vertex is:

`(-2, 4)`

Therefore, we can assign the following variable values:

• 
  `a=-2`
• 
  `b=4`

We can see that the focus is 2 units to the left of the vertex. Therefore, we can assign the following value for the variable
`p`:

• 
  `p = -2`

Using these values, we can form the equation:

`(y-4)^2 = 4(-2)(x-(-2))`

which can be simplified to solve for
`x`.

↓ rewriting subtraction of a negative as addition of a positive

`(y - 4)^2 = -8(x + 2)`

↓ applying the distributive property ...
`a(b + c) = ab + ac`

`(y - 4)^2 = -8x - 16`

↓ adding 16 to both sides

`(y - 4)^2 + 16 = -8x`

↓ dividing both sides by -8

`-((y - 4)^2 + 16)/(8) = x`

This can be rewritten with two terms on the side with
`y` as:

`\boxed{x = -(1)/(8)(y - 4)^2 -2}`