Find the Focus y=-4(x-9)^2-9

$y = - 4 {(x - 9)}^{2} - 9$

Step 1

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 4$

$h = 9$

$k = - 9$

Step 2

Find the vertex $(h, k)$ .

$(9, - 9)$

Step 3

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 4}$

Simplify.

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Move the negative in front of the fraction.

$- \frac{1}{16}$

Step 4

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(9, - \frac{145}{16})$

Step 5