Find the Axis of Symmetry f(x)=9-(x-4)^2

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

Step 1

Write $f (x) = 9 - {(x - 4)}^{2}$ as an equation.

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

Step 2

Reorder $9$ and $- {(x - 4)}^{2}$ .

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

Step 3

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

$a = - 1$

$h = 4$

$k = 9$

Step 4

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(4, 9)$

Step 6

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.

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

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 7

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.

$(4, \frac{35}{4})$

Step 8

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 4$

Step 9