Find the Properties f(x)=-2x^2-12x-16

$f (x) = - 2 x^{2} - 12 x - 16$

Step 1

Write $f (x) = - 2 x^{2} - 12 x - 16$ as an equation.

$y = - 2 x^{2} - 12 x - 16$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 2 x^{2} - 12 x - 16$ .

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

$a = - 2$

$b = - 12$

$c = - 16$

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

$d = \frac{- 12}{2 \cdot - 2}$

Simplify the right side.

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot - 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 6}{2 (- 2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot - 2}$

Rewrite the expression.

$d = \frac{- 6}{- 2}$

Cancel the common factor of $- 6$ and $- 2$ .

Factor $- 2$ out of $- 6$ .

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

Cancel the common factors.

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

$d = \frac{- 2 \cdot 3}{- 2 \cdot 1}$

Cancel the common factor.

$d = \frac{- 2 \cdot 3}{- 2 \cdot 1}$

Rewrite the expression.

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

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

$e = - 16 - \frac{{(- 12)}^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = - 16 - \frac{144}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 16 - \frac{144}{- 8}$

Divide $144$ by $- 8$ .

$e = - 16 - - 18$

Multiply $- 1$ by $- 18$ .

$e = - 16 + 18$

Add $- 16$ and $18$ .

$e = 2$

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

$- 2 {(x + 3)}^{2} + 2$

Set $y$ equal to the new right side.

$y = - 2 {(x + 3)}^{2} + 2$

Step 3

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

$a = - 2$

$h = - 3$

$k = 2$

Step 4

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

Opens Down

Step 5

Find the vertex $(h, k)$ .

$(- 3, 2)$

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.

$(- 3, \frac{15}{8})$

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 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.

$y = \frac{17}{8}$

Step 10

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

Direction: Opens Down

Vertex: $(- 3, 2)$

Focus: $(- 3, \frac{15}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{17}{8}$

Step 11

Find the Properties f(x)=-2x^2-12x-16

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