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

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

Step 1

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

$y = - 2 x^{2} - 2 x - 3$

Step 2

Rewrite the equation in vertex form.

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

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

$a = - 2$

$b = - 2$

$c = - 3$

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

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

Simplify the right side.

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

Factor $2$ out of $- 2$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of ${(- 2)}^{2}$ and $- 2$ .

Factor $- 2$ out of ${(- 2)}^{2}$ .

$e = - 3 - \frac{- 2 \cdot - 2}{4 \cdot - 2}$

Cancel the common factors.

Factor $- 2$ out of $4 \cdot - 2$ .

$e = - 3 - \frac{- 2 \cdot - 2}{- 2 (- 2 \cdot - 2)}$

Cancel the common factor.

$e = - 3 - \frac{- 2 \cdot - 2}{- 2 (- 2 \cdot - 2)}$

Rewrite the expression.

$e = - 3 - \frac{- 2}{- 2 \cdot - 2}$

Cancel the common factor of $- 2$ .

Cancel the common factor.

$e = - 3 - \frac{- 2}{- 2 \cdot - 2}$

Rewrite the expression.

$e = - 3 - \frac{1}{- 2}$

Move the negative in front of the fraction.

$e = - 3 - - \frac{1}{2}$

Multiply $- - \frac{1}{2}$ .

Multiply $- 1$ by $- 1$ .

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

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

$e = - 3 + \frac{1}{2}$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 3 \cdot \frac{2}{2} + \frac{1}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$e = \frac{- 3 \cdot 2}{2} + \frac{1}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 3 \cdot 2 + 1}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$e = \frac{- 6 + 1}{2}$

Add $- 6$ and $1$ .

$e = \frac{- 5}{2}$

Move the negative in front of the fraction.

$e = - \frac{5}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + \frac{1}{2})}^{2} - \frac{5}{2}$ .

$- 2 {(x + \frac{1}{2})}^{2} - \frac{5}{2}$

Set $y$ equal to the new right side.

$y = - 2 {(x + \frac{1}{2})}^{2} - \frac{5}{2}$

Step 3

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

$a = - 2$

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

$k = - \frac{5}{2}$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- \frac{1}{2}, - \frac{5}{2})$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- \frac{1}{2}, - \frac{21}{8})$

Step 8

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

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

Step 9

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

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