Find the Axis of Symmetry f(x)=10x-x^2-16

$f (x) = 10 x - x^{2} - 16$

Step 1

Write $f (x) = 10 x - x^{2} - 16$ as an equation.

$y = 10 x - x^{2} - 16$

Step 2

Rewrite the equation in vertex form.

Reorder $10 x$ and $- x^{2}$ .

$y = - x^{2} + 10 x - 16$

Complete the square for $- x^{2} + 10 x - 16$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 10$

$c = - 16$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{10}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

$d = \frac{2 \cdot 5}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = - 16 - \frac{10^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $10$ to the power of $2$ .

$e = - 16 - \frac{100}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 16 - \frac{100}{- 4}$

Divide $100$ by $- 4$ .

$e = - 16 - - 25$

Multiply $- 1$ by $- 25$ .

$e = - 16 + 25$

Add $- 16$ and $25$ .

$e = 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 5)}^{2} + 9$ .

$- {(x - 5)}^{2} + 9$

Set $y$ equal to the new right side.

$y = - {(x - 5)}^{2} + 9$

Step 3

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

$a = - 1$

$h = 5$

$k = 9$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(5, 9)$

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.

$(5, \frac{35}{4})$

Step 8

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

$x = 5$

Step 9

Find the Axis of Symmetry f(x)=10x-x^2-16

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