Find the Axis of Symmetry y=-2x^2-32x+15

$y = - 2 x^{2} - 32 x + 15$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} - 32 x + 15$ .

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

$a = - 2$

$b = - 32$

$c = 15$

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

$d = \frac{- 32}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $- 32$ and $2$ .

Factor $2$ out of $- 32$ .

$d = \frac{2 \cdot - 16}{2 \cdot - 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 16}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 16}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{- 16}{- 2}$

Cancel the common factor of $- 16$ and $- 2$ .

Factor $- 2$ out of $- 16$ .

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

Cancel the common factors.

Factor $- 2$ out of $- 2$ .

$d = \frac{- 2 \cdot 8}{- 2 \cdot 1}$

Cancel the common factor.

$d = \frac{- 2 \cdot 8}{- 2 \cdot 1}$

Rewrite the expression.

$d = \frac{8}{1}$

Divide $8$ by $1$ .

$d = 8$

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

$e = 15 - \frac{{(- 32)}^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $- 32$ to the power of $2$ .

$e = 15 - \frac{1024}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = 15 - \frac{1024}{- 8}$

Divide $1024$ by $- 8$ .

$e = 15 - - 128$

Multiply $- 1$ by $- 128$ .

$e = 15 + 128$

Add $15$ and $128$ .

$e = 143$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x + 8)}^{2} + 143$ .

$- 2 {(x + 8)}^{2} + 143$

Set $y$ equal to the new right side.

$y = - 2 {(x + 8)}^{2} + 143$

Step 2

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

$a = - 2$

$h = - 8$

$k = 143$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- 8, 143)$

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.

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

Simplify.

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Move the negative in front of the fraction.

$- \frac{1}{8}$

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.

$(- 8, \frac{1143}{8})$

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=-2x^2-32x+15

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