Graph x+2=6(y-3)^2

$x + 2 = 6 {(y - 3)}^{2}$

Step 1

Simplify $6 {(y - 3)}^{2}$ .

Rewrite.

$x + 2 = 0 + 0 + 6 {(y - 3)}^{2}$

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$x + 2 = 6 ((y - 3) (y - 3))$

Expand $(y - 3) (y - 3)$ using the FOIL Method.

Apply the distributive property.

$x + 2 = 6 (y (y - 3) - 3 (y - 3))$

Apply the distributive property.

$x + 2 = 6 (y \cdot y + y \cdot - 3 - 3 (y - 3))$

Apply the distributive property.

$x + 2 = 6 (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3)$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$x + 2 = 6 (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3)$

Move $- 3$ to the left of $y$ .

$x + 2 = 6 (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3)$

Multiply $- 3$ by $- 3$ .

$x + 2 = 6 (y^{2} - 3 y - 3 y + 9)$

Subtract $3 y$ from $- 3 y$ .

$x + 2 = 6 (y^{2} - 6 y + 9)$

Apply the distributive property.

$x + 2 = 6 y^{2} + 6 (- 6 y) + 6 \cdot 9$

Simplify.

Multiply $- 6$ by $6$ .

$x + 2 = 6 y^{2} - 36 y + 6 \cdot 9$

Multiply $6$ by $9$ .

$x + 2 = 6 y^{2} - 36 y + 54$

Step 2

Move all terms not containing $x$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$x = 6 y^{2} - 36 y + 54 - 2$

Subtract $2$ from $54$ .

$x = 6 y^{2} - 36 y + 52$

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for $6 y^{2} - 36 y + 52$ .

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

$a = 6$

$b = - 36$

$c = 52$

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

$d = \frac{- 36}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $- 36$ and $2$ .

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot 6}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 18}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{- 18}{6}$

Cancel the common factor of $- 18$ and $6$ .

Factor $6$ out of $- 18$ .

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

Cancel the common factors.

Factor $6$ out of $6$ .

$d = \frac{6 \cdot - 3}{6 (1)}$

Cancel the common factor.

$d = \frac{6 \cdot - 3}{6 \cdot 1}$

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

$e = 52 - \frac{{(- 36)}^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $- 36$ to the power of $2$ .

$e = 52 - \frac{1296}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = 52 - \frac{1296}{24}$

Divide $1296$ by $24$ .

$e = 52 - 1 \cdot 54$

Multiply $- 1$ by $54$ .

$e = 52 - 54$

Subtract $54$ from $52$ .

$e = - 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(y - 3)}^{2} - 2$ .

$6 {(y - 3)}^{2} - 2$

Set $x$ equal to the new right side.

$x = 6 {(y - 3)}^{2} - 2$

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

$a = 6$

$h = - 2$

$k = 3$

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

Opens Right

Find the vertex $(h, k)$ .

$(- 2, 3)$

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.

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

Multiply $4$ by $6$ .

$\frac{1}{24}$

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.

$(- \frac{47}{24}, 3)$

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

$y = 3$

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.

$x = - \frac{49}{24}$

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

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(- \frac{47}{24}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{49}{24}$

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(- \frac{47}{24}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{49}{24}$

Step 4

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 $- 1$ into $f (x) = \frac{\sqrt{6 (x + 2)}}{6} + 3$ . In this case, the point is $(- 1, 3.40824829)$ .

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

$f (- 1) = \frac{\sqrt{6 ((- 1) + 2)}}{6} + 3$

Simplify the result.

Add $- 1$ and $2$ .

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

Multiply $6$ by $1$ .

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

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

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

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

$= 3.40824829$

Substitute the $x$ value $- 1$ into $f (x) = - \frac{\sqrt{6 (x + 2)}}{6} + 3$ . In this case, the point is $(- 1, 2.5917517)$ .

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

$f (- 1) = - \frac{\sqrt{6 ((- 1) + 2)}}{6} + 3$

Simplify the result.

Simplify the numerator.

Add $- 1$ and $2$ .

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

Multiply $6$ by $1$ .

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

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

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

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

$= 2.5917517$

Substitute the $x$ value $0$ into $f (x) = \frac{\sqrt{6 (x + 2)}}{6} + 3$ . In this case, the point is $(0, 3.57735026)$ .

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

$f (0) = \frac{\sqrt{6 ((0) + 2)}}{6} + 3$

Simplify the result.

Simplify the expression.

Add $0$ and $2$ .

$f (0) = \frac{\sqrt{6 \cdot 2}}{6} + 3$

Multiply $6$ by $2$ .

$f (0) = \frac{\sqrt{12}}{6} + 3$

Simplify each term.

Simplify the numerator.

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$f (0) = \frac{\sqrt{4 (3)}}{6} + 3$

Rewrite $4$ as $2^{2}$ .

$f (0) = \frac{\sqrt{2^{2} \cdot 3}}{6} + 3$

Pull terms out from under the radical.

$f (0) = \frac{2 \sqrt{3}}{6} + 3$

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

Factor $2$ out of $2 \sqrt{3}$ .

$f (0) = \frac{2 (\sqrt{3})}{6} + 3$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (0) = \frac{2 \sqrt{3}}{2 \cdot 3} + 3$

Cancel the common factor.

$f (0) = \frac{2 \sqrt{3}}{2 \cdot 3} + 3$

Rewrite the expression.

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

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

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

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

$= 3.57735026$

Substitute the $x$ value $0$ into $f (x) = - \frac{\sqrt{6 (x + 2)}}{6} + 3$ . In this case, the point is $(0, 2.42264973)$ .

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

$f (0) = - \frac{\sqrt{6 ((0) + 2)}}{6} + 3$

Simplify the result.

Simplify each term.

Simplify the numerator.

Add $0$ and $2$ .

$f (0) = - \frac{\sqrt{6 \cdot 2}}{6} + 3$

Multiply $6$ by $2$ .

$f (0) = - \frac{\sqrt{12}}{6} + 3$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$f (0) = - \frac{\sqrt{4 (3)}}{6} + 3$

Rewrite $4$ as $2^{2}$ .

$f (0) = - \frac{\sqrt{2^{2} \cdot 3}}{6} + 3$

Pull terms out from under the radical.

$f (0) = - \frac{2 \sqrt{3}}{6} + 3$

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

Factor $2$ out of $2 \sqrt{3}$ .

$f (0) = - \frac{2 (\sqrt{3})}{6} + 3$

Cancel the common factors.

Factor $2$ out of $6$ .

$f (0) = - \frac{2 \sqrt{3}}{2 \cdot 3} + 3$

Cancel the common factor.

$f (0) = - \frac{2 \sqrt{3}}{2 \cdot 3} + 3$

Rewrite the expression.

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

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

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

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

$= 2.42264973$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 3.41 \\ - 1 & 2.59 \\ 0 & 3.58 \\ 0 & 2.42 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 2, 3)$

Focus: $(- \frac{47}{24}, 3)$

Axis of Symmetry: $y = 3$

Directrix: $x = - \frac{49}{24}$

$\begin{array}{c} x & y \\ - 2 & 3 \\ - 1 & 3.41 \\ - 1 & 2.59 \\ 0 & 3.58 \\ 0 & 2.42 \end{array}$

Step 6

Graph x+2=6(y-3)^2

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