Question

What is the focus point of a parabola with this equation

Answer 1

The focus point of the parabola is (2, 0), which corresponds to option D.

To find the focus point of a parabola given by the equation
`\(y = (1)/(8)(x^2 - 4x - 12)\)`, we can rewrite the equation in the form
`\(y = a(x - h)^2 + k\)`, where
`\((h, k)\)` is the vertex of the parabola.

First, expand and simplify the given equation:

`\[ y = (1)/(8)(x^2 - 4x - 12) \]`

`\[ y = (1)/(8)x^2 - (1)/(2)x - (3)/(2) \]`

Now, factor out the leading coefficient from the
`\(x^2\)` and
`\(x\)` terms:

`\[ y = (1)/(8)(x^2 - 4x) - (3)/(2) \]`

Complete the square inside the parentheses. To complete the square for
`\(x^2 - 4x\)`, add
`\(\left((4)/(2)\right)^2 = 4\)` inside the parentheses:

`\[ y = (1)/(8)(x^2 - 4x + 4 - 4) - (3)/(2) \]`

`\[ y = (1)/(8)(x^2 - 4x + 4) - (1)/(2) - (3)/(2) \]`

`\[ y = (1)/(8)(x - 2)^2 - 2 \]`

Now, the equation is in the form
`\(y = a(x - h)^2 + k\)` with
`\((h, k) = (2, -2)\)`.

The focus point
`\((h, k + (1)/(4a))\)` for a parabola in this form is
`\((2, -2 + (1)/(4 * (1)/(8)))\)`.

`\[ k + (1)/(4a) = -2 + (1)/(4 * (1)/(8)) = -2 + 2 = 0 \]`

So, the focus point is
`\((2, 0)\)`.

Therefore, the correct answer is
`\((2, 0)\)`.

So, the focus point of the parabola is (2, 0), which corresponds to option D.