Graph y=-6x^2-12x-1

$y = - 6 x^{2} - 12 x - 1$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $- 6 x^{2} - 12 x - 1$ .

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

$a = - 6$

$b = - 12$

$c = - 1$

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

$d = \frac{- 12}{2 \cdot - 6}$

Simplify the right side.

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

Factor $2$ out of $- 12$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = - 1 - \frac{{(- 12)}^{2}}{4 \cdot - 6}$

Simplify the right side.

Simplify each term.

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

$e = - 1 - \frac{144}{4 \cdot - 6}$

Multiply $4$ by $- 6$ .

$e = - 1 - \frac{144}{- 24}$

Divide $144$ by $- 24$ .

$e = - 1 - - 6$

Multiply $- 1$ by $- 6$ .

$e = - 1 + 6$

Add $- 1$ and $6$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 6 {(x + 1)}^{2} + 5$ .

$- 6 {(x + 1)}^{2} + 5$

Set $y$ equal to the new right side.

$y = - 6 {(x + 1)}^{2} + 5$

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

$a = - 6$

$h = - 1$

$k = 5$

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

Opens Down

Find the vertex $(h, k)$ .

$(- 1, 5)$

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.

$\frac{1}{4 \cdot - 6}$

Simplify.

Multiply $4$ by $- 6$ .

$\frac{1}{- 24}$

Move the negative in front of the fraction.

$- \frac{1}{24}$

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.

$(- 1, \frac{119}{24})$

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

$x = - 1$

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.

$y = \frac{121}{24}$

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

Direction: Opens Down

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{119}{24})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{121}{24}$

Direction: Opens Down

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{119}{24})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{121}{24}$

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 $- 2$ in the expression.

$f (- 2) = - 6 {(- 2)}^{2} - 12 \cdot - 2 - 1$

Simplify the result.

Simplify each term.

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

$f (- 2) = - 6 \cdot 4 - 12 \cdot - 2 - 1$

Multiply $- 6$ by $4$ .

$f (- 2) = - 24 - 12 \cdot - 2 - 1$

Multiply $- 12$ by $- 2$ .

$f (- 2) = - 24 + 24 - 1$

Simplify by adding and subtracting.

Add $- 24$ and $24$ .

$f (- 2) = 0 - 1$

Subtract $1$ from $0$ .

$f (- 2) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = - 2$ is $- 1$ .

$y = - 1$

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = - 6 {(- 3)}^{2} - 12 \cdot - 3 - 1$

Simplify the result.

Simplify each term.

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

$f (- 3) = - 6 \cdot 9 - 12 \cdot - 3 - 1$

Multiply $- 6$ by $9$ .

$f (- 3) = - 54 - 12 \cdot - 3 - 1$

Multiply $- 12$ by $- 3$ .

$f (- 3) = - 54 + 36 - 1$

Simplify by adding and subtracting.

Add $- 54$ and $36$ .

$f (- 3) = - 18 - 1$

Subtract $1$ from $- 18$ .

$f (- 3) = - 19$

The final answer is $- 19$ .

$- 19$

The $y$ value at $x = - 3$ is $- 19$ .

$y = - 19$

Replace the variable $x$ with $0$ in the expression.

$f (0) = - 6 {(0)}^{2} - 12 \cdot 0 - 1$

Simplify the result.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$f (0) = - 6 \cdot 0 - 12 \cdot 0 - 1$

Multiply $- 6$ by $0$ .

$f (0) = 0 - 12 \cdot 0 - 1$

Multiply $- 12$ by $0$ .

$f (0) = 0 + 0 - 1$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$f (0) = 0 - 1$

Subtract $1$ from $0$ .

$f (0) = - 1$

The final answer is $- 1$ .

$- 1$

The $y$ value at $x = 0$ is $- 1$ .

$y = - 1$

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

$f (1) = - 6 {(1)}^{2} - 12 \cdot 1 - 1$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = - 6 \cdot 1 - 12 \cdot 1 - 1$

Multiply $- 6$ by $1$ .

$f (1) = - 6 - 12 \cdot 1 - 1$

Multiply $- 12$ by $1$ .

$f (1) = - 6 - 12 - 1$

Simplify by subtracting numbers.

Subtract $12$ from $- 6$ .

$f (1) = - 18 - 1$

Subtract $1$ from $- 18$ .

$f (1) = - 19$

The final answer is $- 19$ .

$- 19$

The $y$ value at $x = 1$ is $- 19$ .

$y = - 19$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - 19 \\ - 2 & - 1 \\ - 1 & 5 \\ 0 & - 1 \\ 1 & - 19 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 1, 5)$

Focus: $(- 1, \frac{119}{24})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{121}{24}$

$\begin{array}{c} x & y \\ - 3 & - 19 \\ - 2 & - 1 \\ - 1 & 5 \\ 0 & - 1 \\ 1 & - 19 \end{array}$

Step 4

Graph y=-6x^2-12x-1

Related Questions

Support LCMGCF.com by disabling your adblocker.