Find the Axis of Symmetry f(x)=-1/2x^2-3x-1/2

$f (x) = - \frac{1}{2} x^{2} - 3 x - \frac{1}{2}$

Step 1

Write $f (x) = - \frac{1}{2} x^{2} - 3 x - \frac{1}{2}$ as an equation.

$y = - \frac{1}{2} x^{2} - 3 x - \frac{1}{2}$

Step 2

Rewrite the equation in vertex form.

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

$y = - \frac{x^{2}}{2} - 3 x - \frac{1}{2}$

Complete the square for $- \frac{x^{2}}{2} - 3 x - \frac{1}{2}$ .

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

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

$b = - 3$

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

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

$d = \frac{- 3}{2 (- \frac{1}{2})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{- 3}{- 2 (\frac{1}{2})}$

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

$d = \frac{- 3}{\frac{- 2}{2}}$

Divide $- 2$ by $2$ .

$d = \frac{- 3}{- 1}$

Divide $- 3$ by $- 1$ .

$d = 3$

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

$e = - \frac{1}{2} - \frac{{(- 3)}^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $- 3$ to the power of $2$ .

$e = - \frac{1}{2} - \frac{9}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{1}{2} - \frac{9}{- 4 (\frac{1}{2})}$

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

$e = - \frac{1}{2} - \frac{9}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - \frac{1}{2} - \frac{9}{- 2}$

Move the negative in front of the fraction.

$e = - \frac{1}{2} - - \frac{9}{2}$

Multiply $- - \frac{9}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - \frac{1}{2} + 1 (\frac{9}{2})$

Multiply $\frac{9}{2}$ by $1$ .

$e = - \frac{1}{2} + \frac{9}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 1 + 9}{2}$

Add $- 1$ and $9$ .

$e = \frac{8}{2}$

Divide $8$ by $2$ .

$e = 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 3)}^{2} + 4$ .

$- \frac{1}{2} {(x + 3)}^{2} + 4$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + 3)}^{2} + 4$

Step 3

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

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

$h = - 3$

$k = 4$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 3, 4)$

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.

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

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 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.

$(- 3, \frac{7}{2})$

Step 8

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

$x = - 3$

Step 9

Find the Axis of Symmetry f(x)=-1/2x^2-3x-1/2

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