Find the Axis of Symmetry y=-x^2-14x-53

$y = - x^{2} - 14 x - 53$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- x^{2} - 14 x - 53$ .

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

$a = - 1$

$b = - 14$

$c = - 53$

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

$d = \frac{- 14}{2 \cdot - 1}$

Simplify the right side.

Cancel the common factor of $- 14$ and $2$ .

Factor $2$ out of $- 14$ .

$d = \frac{2 \cdot - 7}{2 \cdot - 1}$

Move the negative one from the denominator of $\frac{- 7}{- 1}$ .

$d = - 1 \cdot - 7$

Rewrite $- 1 \cdot - 7$ as $- - 7$ .

$d = - - 7$

Multiply $- 1$ by $- 7$ .

$d = 7$

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

$e = - 53 - \frac{{(- 14)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

Raise $- 14$ to the power of $2$ .

$e = - 53 - \frac{196}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 53 - \frac{196}{- 4}$

Divide $196$ by $- 4$ .

$e = - 53 - - 49$

Multiply $- 1$ by $- 49$ .

$e = - 53 + 49$

Add $- 53$ and $49$ .

$e = - 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 7)}^{2} - 4$ .

$- {(x + 7)}^{2} - 4$

Set $y$ equal to the new right side.

$y = - {(x + 7)}^{2} - 4$

Step 2

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

$a = - 1$

$h = - 7$

$k = - 4$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- 7, - 4)$

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.

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

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 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.

$(- 7, - \frac{17}{4})$

Step 7

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

$x = - 7$

Step 8

Find the Axis of Symmetry y=-x^2-14x-53

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