Question

How do I find the focus of the parabola represented by `y=2x^2-8x+9`?

Answer 1

`(2,-135/8)`

Explanation:

First, we must convert the equation into the vertex-form:

`color(white)([1]XX)y=a(x-h)^2+k`

Solution:

`[1]color(white)(XX)y=2x^2-8x-9`

`[2]color(white)(XX)y+9=2x^2-8x`

`[3]color(white)(XX)y+9=2(x^2-4x)`

`[4]color(white)(XX)y+9+2(4)=2(x^2-4x)+2(4)`

`[5]color(white)(XX)y+9+8=2(x^2-4x+4)`

`[6]color(white)(XX)y+17=2(x-2)^2`

`[7]color(white)(XX)color(red)(y=2(x-2)^2-17)`

Now that we have the equation in vertex form, we can easily get the focus:

Focus: `color(white)([1]X)(h, k + 1/(4a))`

Solution:

`[1]color(white)(XX)(h, k+1/(4a))`

`[2]color(white)(XX)((2), (-17)+1/(4(2)))`

`[3]color(white)(XX)(2, -17+1/8)`

`[4]color(white)(XX)color(red)((2","-135/8))`