Graph f(x)=-2x^2-12x-16

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

Step 1

Find the properties of the given parabola.

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.

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

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

$a = - 2$

$h = - 3$

$k = 2$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 3, 2)$

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.

$- \frac{1}{8}$

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.

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

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

$x = - 3$

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.

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

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

Direction: Opens Down

Vertex: $(- 3, 2)$

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

Axis of Symmetry: $x = - 3$

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

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

Replace the variable $x$ with $- 4$ in the expression.

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

Simplify the result.

Simplify each term.

Raise $- 4$ to the power of $2$ .

$f (- 4) = - 2 \cdot 16 - 12 \cdot - 4 - 16$

Multiply $- 2$ by $16$ .

$f (- 4) = - 32 - 12 \cdot - 4 - 16$

Multiply $- 12$ by $- 4$ .

$f (- 4) = - 32 + 48 - 16$

Simplify by adding and subtracting.

Add $- 32$ and $48$ .

$f (- 4) = 16 - 16$

Subtract $16$ from $16$ .

$f (- 4) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = - 2 {(- 5)}^{2} - 12 \cdot - 5 - 16$

Simplify the result.

Simplify each term.

Raise $- 5$ to the power of $2$ .

$f (- 5) = - 2 \cdot 25 - 12 \cdot - 5 - 16$

Multiply $- 2$ by $25$ .

$f (- 5) = - 50 - 12 \cdot - 5 - 16$

Multiply $- 12$ by $- 5$ .

$f (- 5) = - 50 + 60 - 16$

Simplify by adding and subtracting.

Add $- 50$ and $60$ .

$f (- 5) = 10 - 16$

Subtract $16$ from $10$ .

$f (- 5) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = - 5$ is $- 6$ .

$y = - 6$

Replace the variable $x$ with $- 2$ in the expression.

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

Simplify the result.

Simplify each term.

Multiply $- 2$ by ${(- 2)}^{2}$ by adding the exponents.

Multiply $- 2$ by ${(- 2)}^{2}$ .

Raise $- 2$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 2) = {(- 2)}^{1 + 2} - 12 \cdot - 2 - 16$

Add $1$ and $2$ .

$f (- 2) = {(- 2)}^{3} - 12 \cdot - 2 - 16$

Raise $- 2$ to the power of $3$ .

$f (- 2) = - 8 - 12 \cdot - 2 - 16$

Multiply $- 12$ by $- 2$ .

$f (- 2) = - 8 + 24 - 16$

Simplify by adding and subtracting.

Add $- 8$ and $24$ .

$f (- 2) = 16 - 16$

Subtract $16$ from $16$ .

$f (- 2) = 0$

The final answer is $0$ .

$0$

The $y$ value at $x = - 2$ is $0$ .

$y = 0$

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - 2 {(- 1)}^{2} - 12 \cdot - 1 - 16$

Simplify the result.

Simplify each term.

Raise $- 1$ to the power of $2$ .

$f (- 1) = - 2 \cdot 1 - 12 \cdot - 1 - 16$

Multiply $- 2$ by $1$ .

$f (- 1) = - 2 - 12 \cdot - 1 - 16$

Multiply $- 12$ by $- 1$ .

$f (- 1) = - 2 + 12 - 16$

Simplify by adding and subtracting.

Add $- 2$ and $12$ .

$f (- 1) = 10 - 16$

Subtract $16$ from $10$ .

$f (- 1) = - 6$

The final answer is $- 6$ .

$- 6$

The $y$ value at $x = - 1$ is $- 6$ .

$y = - 6$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & - 6 \\ - 4 & 0 \\ - 3 & 2 \\ - 2 & 0 \\ - 1 & - 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 3, 2)$

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

Axis of Symmetry: $x = - 3$

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

$\begin{array}{c} x & y \\ - 5 & - 6 \\ - 4 & 0 \\ - 3 & 2 \\ - 2 & 0 \\ - 1 & - 6 \end{array}$

Step 4

Graph f(x)=-2x^2-12x-16

Related Questions

Support LCMGCF.com by disabling your adblocker.