Find the Properties f(x)=-x^2-8x-7

$f (x) = - x^{2} - 8 x - 7$

Step 1

Write $f (x) = - x^{2} - 8 x - 7$ as an equation.

$y = - x^{2} - 8 x - 7$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- x^{2} - 8 x - 7$ .

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

$a = - 1$

$b = - 8$

$c = - 7$

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

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

Simplify the right side.

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

Factor $2$ out of $- 8$ .

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

Move the negative one from the denominator of $\frac{- 4}{- 1}$ .

$d = - 1 \cdot - 4$

Rewrite $- 1 \cdot - 4$ as $- - 4$ .

$d = - - 4$

Multiply $- 1$ by $- 4$ .

$d = 4$

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

$e = - 7 - \frac{{(- 8)}^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = - 7 - \frac{64}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = - 7 - \frac{64}{- 4}$

Divide $64$ by $- 4$ .

$e = - 7 - - 16$

Multiply $- 1$ by $- 16$ .

$e = - 7 + 16$

Add $- 7$ and $16$ .

$e = 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 4)}^{2} + 9$ .

$- {(x + 4)}^{2} + 9$

Set $y$ equal to the new right side.

$y = - {(x + 4)}^{2} + 9$

Step 3

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

$a = - 1$

$h = - 4$

$k = 9$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 4, 9)$

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.

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

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 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.

$(- 4, \frac{35}{4})$

Step 8

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

$x = - 4$

Step 9

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = \frac{37}{4}$

Step 10

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(- 4, 9)$

Focus: $(- 4, \frac{35}{4})$

Axis of Symmetry: $x = - 4$

Directrix: $y = \frac{37}{4}$

Step 11

Find the Properties f(x)=-x^2-8x-7

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