Graph x=3y^2+6y+2

$x = 3 y^{2} + 6 y + 2$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $3 y^{2} + 6 y + 2$ .

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

$a = 3$

$b = 6$

$c = 2$

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

$d = \frac{6}{2 \cdot 3}$

Simplify the right side.

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

Factor $2$ out of $6$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

$e = 2 - \frac{6^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 2 - \frac{36}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 2 - \frac{36}{12}$

Divide $36$ by $12$ .

$e = 2 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$e = 2 - 3$

Subtract $3$ from $2$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y + 1)}^{2} - 1$ .

$3 {(y + 1)}^{2} - 1$

Set $x$ equal to the new right side.

$x = 3 {(y + 1)}^{2} - 1$

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

$a = 3$

$h = - 1$

$k = - 1$

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Find the vertex $(h, k)$ .

$(- 1, - 1)$

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.

$\frac{1}{4 \cdot 3}$

Multiply $4$ by $3$ .

$\frac{1}{12}$

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(- \frac{11}{12}, - 1)$

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

$y = - 1$

Find the directrix.

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

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

$x = - \frac{13}{12}$

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

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(- \frac{11}{12}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{13}{12}$

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(- \frac{11}{12}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{13}{12}$

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.

Substitute the $x$ value $0$ into $f (x) = - \frac{3 - \sqrt{3 (1 + x)}}{3}$ . In this case, the point is $(0, - 0.42264973)$ .

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

$f (0) = - \frac{3 - \sqrt{3 (1 + 0)}}{3}$

Simplify the result.

Remove parentheses.

$f (0) = - \frac{3 - \sqrt{3 (1 + 0)}}{3}$

Simplify the numerator.

Add $1$ and $0$ .

$f (0) = - \frac{3 - \sqrt{3 \cdot 1}}{3}$

Multiply $3$ by $1$ .

$f (0) = - \frac{3 - \sqrt{3}}{3}$

The final answer is $- \frac{3 - \sqrt{3}}{3}$ .

$y = - \frac{3 - \sqrt{3}}{3}$

Convert $- \frac{3 - \sqrt{3}}{3}$ to decimal.

$= - 0.42264973$

Substitute the $x$ value $0$ into $f (x) = - \frac{3 + \sqrt{3 (x + 1)}}{3}$ . In this case, the point is $(0, - 1.57735026)$ .

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

$f (0) = - \frac{3 + \sqrt{3 ((0) + 1)}}{3}$

Simplify the result.

Simplify the numerator.

Add $0$ and $1$ .

$f (0) = - \frac{3 + \sqrt{3 \cdot 1}}{3}$

Multiply $3$ by $1$ .

$f (0) = - \frac{3 + \sqrt{3}}{3}$

The final answer is $- \frac{3 + \sqrt{3}}{3}$ .

$y = - \frac{3 + \sqrt{3}}{3}$

Convert $- \frac{3 + \sqrt{3}}{3}$ to decimal.

$= - 1.57735026$

Substitute the $x$ value $1$ into $f (x) = - \frac{3 - \sqrt{3 (1 + x)}}{3}$ . In this case, the point is $(1, - 0.18350341)$ .

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

$f (1) = - \frac{3 - \sqrt{3 (1 + 1)}}{3}$

Simplify the result.

Remove parentheses.

$f (1) = - \frac{3 - \sqrt{3 (1 + 1)}}{3}$

Simplify the numerator.

Add $1$ and $1$ .

$f (1) = - \frac{3 - \sqrt{3 \cdot 2}}{3}$

Multiply $3$ by $2$ .

$f (1) = - \frac{3 - \sqrt{6}}{3}$

The final answer is $- \frac{3 - \sqrt{6}}{3}$ .

$y = - \frac{3 - \sqrt{6}}{3}$

Convert $- \frac{3 - \sqrt{6}}{3}$ to decimal.

$= - 0.18350341$

Substitute the $x$ value $1$ into $f (x) = - \frac{3 + \sqrt{3 (x + 1)}}{3}$ . In this case, the point is $(1, - 1.81649658)$ .

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

$f (1) = - \frac{3 + \sqrt{3 ((1) + 1)}}{3}$

Simplify the result.

Simplify the numerator.

Add $1$ and $1$ .

$f (1) = - \frac{3 + \sqrt{3 \cdot 2}}{3}$

Multiply $3$ by $2$ .

$f (1) = - \frac{3 + \sqrt{6}}{3}$

The final answer is $- \frac{3 + \sqrt{6}}{3}$ .

$y = - \frac{3 + \sqrt{6}}{3}$

Convert $- \frac{3 + \sqrt{6}}{3}$ to decimal.

$= - 1.81649658$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & - 0.42 \\ 0 & - 1.58 \\ 1 & - 0.18 \\ 1 & - 1.82 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(- \frac{11}{12}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{13}{12}$

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & - 0.42 \\ 0 & - 1.58 \\ 1 & - 0.18 \\ 1 & - 1.82 \end{array}$

Step 4

Graph x=3y^2+6y+2

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