Question

The equation of a parabola is given. y=18x2+4x+20 What are the coordinates of the focus of the parabola?

Answer 1

(-16,-10)

Explanation:

I took the test. The other answer is wrong because you pasted it incorrectly.

Answer 2

The equation of a parabola is given. y=
`18x^2+4x+20`

Here the value of 'a' = 18 and it is positive, it means the parabola opens up.

For vertical parabola the focus point is (h , k+p)

Where (h,k) is the vertex and p =
`(1)/(4a)` (p is the distance between vertex and focus)

First we find vertex (h,k)

h =
`(-b)/(2a)`

a = 18 and b = 4

So h=
`(-4)/(2*18)` =
`(-4)/(36)` =
`(-1)/(9)`

Plug in -1/9 for x and find out k

k =
`y = 18x^2+4x+20 = 18((-1)/(9))^2 + 4((-1)/(9)) + 20`

=
`((2)/(9)) + ((-4)/(9)) + 20`

=
`(178)/(9)`

So vertex is (
`(-1)/(9)`,
`(178)/(9)`)

P =
`(1)/(4a)`, we know a= 18

So p =
`(1)/(4*18)` =
`(1)/(72)`

Focus is ( h, k +p)

h =
`(-1)/(9)` k =
`(178)/(9)`

and p =
`(1)/(72)`

k + P =
`(178)/(9)` +
`(1)/(72)`

Take common denominator 72

k + p =
`(1424)/(72)` +
`(1)/(72)` =
`(1425)/(72)` =
`(475)/(24)`

Focus point is (
`(-1)/(9)` ,
`(475)/(24)`)