Find the Axis of Symmetry y=-1/2x^2-8x-36

$y = - \frac{1}{2} x^{2} - 8 x - 36$

Step 1

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{2}$ .

$y = - \frac{x^{2}}{2} - 8 x - 36$

Complete the square for $- \frac{x^{2}}{2} - 8 x - 36$ .

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

$a = - \frac{1}{2}$

$b = - 8$

$c = - 36$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 (- \frac{1}{2})}$

Simplify the right side.

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 (- \frac{1}{2})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 (- \frac{1}{2})}$

Rewrite the expression.

$d = \frac{- 4}{- \frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - 4 (- 1 \cdot 2)$

Multiply $- 4 (- 1 \cdot 2)$ .

Multiply $- 1$ by $2$ .

$d = - 4 \cdot - 2$

Multiply $- 4$ by $- 2$ .

$d = 8$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 36 - \frac{{(- 8)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 8$ to the power of $2$ .

$e = - 36 - \frac{64}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 36 - \frac{64}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = - 36 - \frac{64}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - 36 - \frac{64}{- 2}$

Divide $64$ by $- 2$ .

$e = - 36 - - 32$

Multiply $- 1$ by $- 32$ .

$e = - 36 + 32$

Add $- 36$ and $32$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 8)}^{2} - 4$ .

$- \frac{1}{2} {(x + 8)}^{2} - 4$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + 8)}^{2} - 4$

Step 2

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

$a = - \frac{1}{2}$

$h = - 8$

$k = - 4$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- 8, - 4)$

Step 5

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.

$\frac{1}{4 \cdot (- \frac{1}{2})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{2})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Combine $4$ and $\frac{1}{2}$ .

$- \frac{1}{\frac{4}{2}}$

Divide $4$ by $2$ .

$- \frac{1}{2}$

Step 6

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.

$(- 8, - \frac{9}{2})$

Step 7

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

$x = - 8$

Step 8

Find the Axis of Symmetry y=-1/2x^2-8x-36

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