Graph (y+1)^2=12(x+1)

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Trigonometry

${(y + 1)}^{2} = 12 (x + 1)$

Rewrite the equation as $12 (x + 1) = {(y + 1)}^{2}$ .

$12 (x + 1) = {(y + 1)}^{2}$

Divide each term by $12$ and simplify.

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Divide each term in $12 (x + 1) = {(y + 1)}^{2}$ by $12$ .

$\frac{12 (x + 1)}{12} = \frac{{(y + 1)}^{2}}{12}$

Cancel the common factor of $12$ .

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Cancel the common factor.

$\frac{12 (x + 1)}{12} = \frac{{(y + 1)}^{2}}{12}$

Divide $x + 1$ by $1$ .

$x + 1 = \frac{{(y + 1)}^{2}}{12}$

Subtract $1$ from both sides of the equation.

$x = \frac{{(y + 1)}^{2}}{12} - 1$

Find the properties of the given parabola.

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Rewrite the equation in vertex form.

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Reorder terms.

$x = \frac{1}{12} \cdot {(y + 1)}^{2} - 1$

Complete the square for $\frac{1}{12} \cdot {(y + 1)}^{2} - 1$ .

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Simplify each term.

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Rewrite ${(y + 1)}^{2}$ as $(y + 1) (y + 1)$ .

$\frac{1}{12} \cdot ((y + 1) (y + 1)) - 1$

Expand $(y + 1) (y + 1)$ using the FOIL Method.

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Apply the distributive property.

$\frac{1}{12} \cdot (y (y + 1) + 1 (y + 1)) - 1$

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot 1 + 1 (y + 1)) - 1$

Apply the distributive property.

$\frac{1}{12} \cdot (y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1) - 1$

Simplify and combine like terms.

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Simplify each term.

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Multiply $y$ by $y$ .

$\frac{1}{12} \cdot (y^{2} + y \cdot 1 + 1 y + 1 \cdot 1) - 1$

Multiply $y$ by $1$ .

$\frac{1}{12} \cdot (y^{2} + y + 1 y + 1 \cdot 1) - 1$

Multiply $y$ by $1$ .

$\frac{1}{12} \cdot (y^{2} + y + y + 1 \cdot 1) - 1$

Multiply $1$ by $1$ .

$\frac{1}{12} \cdot (y^{2} + y + y + 1) - 1$

Add $y$ and $y$ .

$\frac{1}{12} \cdot (y^{2} + 2 y + 1) - 1$

Apply the distributive property.

$\frac{1}{12} y^{2} + \frac{1}{12} (2 y) + \frac{1}{12} \cdot 1 - 1$

Simplify.

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Combine $\frac{1}{12}$ and $y^{2}$ .

$\frac{y^{2}}{12} + \frac{1}{12} (2 y) + \frac{1}{12} \cdot 1 - 1$

Cancel the common factor of $2$ .

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Factor $2$ out of $12$ .

$\frac{y^{2}}{12} + \frac{1}{2 (6)} (2 y) + \frac{1}{12} \cdot 1 - 1$

Factor $2$ out of $2 y$ .

$\frac{y^{2}}{12} + \frac{1}{2 (6)} (2 (y)) + \frac{1}{12} \cdot 1 - 1$

Cancel the common factor.

$\frac{y^{2}}{12} + \frac{1}{2 \cdot 6} (2 y) + \frac{1}{12} \cdot 1 - 1$

Rewrite the expression.

$\frac{y^{2}}{12} + \frac{1}{6} y + \frac{1}{12} \cdot 1 - 1$

Combine $\frac{1}{6}$ and $y$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1}{12} \cdot 1 - 1$

Multiply $\frac{1}{12}$ by $1$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1}{12} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1}{12} - 1 \cdot \frac{12}{12}$

Combine $- 1$ and $\frac{12}{12}$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1}{12} + \frac{- 1 \cdot 12}{12}$

Combine the numerators over the common denominator.

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1 - 1 \cdot 12}{12}$

Simplify the numerator.

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Multiply $- 1$ by $12$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{1 - 12}{12}$

Subtract $12$ from $1$ .

$\frac{y^{2}}{12} + \frac{y}{6} + \frac{- 11}{12}$

Move the negative in front of the fraction.

$\frac{y^{2}}{12} + \frac{y}{6} - \frac{11}{12}$

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

$a = \frac{1}{12}, b = \frac{1}{6}, c = - \frac{11}{12}$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{6}}{2 (\frac{1}{12})}$

Simplify the right side.

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Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{6} \cdot \frac{1}{2 (\frac{1}{12})}$

Combine $2$ and $\frac{1}{12}$ .

$d = \frac{1}{6} \cdot \frac{1}{\frac{2}{12}}$

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

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Factor $2$ out of $2$ .

$d = \frac{1}{6} \cdot \frac{1}{\frac{2 (1)}{12}}$

Cancel the common factors.

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Factor $2$ out of $12$ .

$d = \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Cancel the common factor.

$d = \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $6$ .

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Factor $6$ out of $1 \cdot 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Simplify the numerator.

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Apply the product rule to $\frac{1}{6}$ .

$e = - \frac{11}{12} - \frac{\frac{1^{2}}{6^{2}}}{4 (\frac{1}{12})}$

One to any power is one.

$e = - \frac{11}{12} - \frac{\frac{1}{6^{2}}}{4 (\frac{1}{12})}$

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

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{4 (\frac{1}{12})}$

Combine $4$ and $\frac{1}{12}$ .

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

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Factor $4$ out of $4$ .

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{\frac{4 (1)}{12}}$

Cancel the common factors.

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Factor $4$ out of $12$ .

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Cancel the common factor.

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Rewrite the expression.

$e = - \frac{11}{12} - \frac{\frac{1}{36}}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{11}{12} - (\frac{1}{36} \cdot 3)$

Cancel the common factor of $3$ .

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Factor $3$ out of $36$ .

$e = - \frac{11}{12} - (\frac{1}{3 (12)} \cdot 3)$

Cancel the common factor.

$e = - \frac{11}{12} - (\frac{1}{3 \cdot 12} \cdot 3)$

Rewrite the expression.

$e = - \frac{11}{12} - \frac{1}{12}$

Combine fractions.

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Combine the numerators over the common denominator.

$e = \frac{- 11 - 1}{12}$

Simplify the expression.

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Subtract $1$ from $- 11$ .

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

Divide $- 12$ by $12$ .

$e = - 1$

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

$\frac{1}{12} \cdot {(y + 1)}^{2} - 1$

Set $x$ equal to the new right side.

$x = \frac{1}{12} \cdot {(y + 1)}^{2} - 1$

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

$a = \frac{1}{12}$

$h = - 1$

$k = - 1$

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.

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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.

$\frac{1}{4 \cdot \frac{1}{12}}$

Simplify.

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Combine $4$ and $\frac{1}{12}$ .

$\frac{1}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

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Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{12}}$

Cancel the common factors.

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Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 3$

Multiply $3$ by $1$ .

$3$

Find the focus.

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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.

$(2, - 1)$

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

$y = - 1$

Find the directrix.

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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.

$x = - 4$

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

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(2, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 4$

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(2, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 4$

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.

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Substitute the $x$ value $0$ into $f (x) = 2 \sqrt{3 (x + 1)} - 1$ . In this case, the point is $(0, 2.46410161)$ .

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

$f (0) = 2 \sqrt{3 ((0) + 1)} - 1$

Simplify the result.

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Simplify each term.

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Add $0$ and $1$ .

$f (0) = 2 \sqrt{3 \cdot 1} - 1$

Multiply $3$ by $1$ .

$f (0) = 2 \sqrt{3} - 1$

The final answer is $2 \sqrt{3} - 1$ .

$y = 2 \sqrt{3} - 1$

Convert $2 \sqrt{3} - 1$ to decimal.

$= 2.46410161$

Substitute the $x$ value $0$ into $f (x) = - 2 \sqrt{3 (x + 1)} - 1$ . In this case, the point is $(0, - 4.46410161)$ .

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

$f (0) = - 2 \sqrt{3 ((0) + 1)} - 1$

Simplify the result.

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Simplify each term.

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Add $0$ and $1$ .

$f (0) = - 2 \sqrt{3 \cdot 1} - 1$

Multiply $3$ by $1$ .

$f (0) = - 2 \sqrt{3} - 1$

The final answer is $- 2 \sqrt{3} - 1$ .

$y = - 2 \sqrt{3} - 1$

Convert $- 2 \sqrt{3} - 1$ to decimal.

$= - 4.46410161$

Substitute the $x$ value $1$ into $f (x) = 2 \sqrt{3 (x + 1)} - 1$ . In this case, the point is $(1, 3.89897948)$ .

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

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

Simplify the result.

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Simplify each term.

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Add $1$ and $1$ .

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

Multiply $3$ by $2$ .

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

The final answer is $2 \sqrt{6} - 1$ .

$y = 2 \sqrt{6} - 1$

Convert $2 \sqrt{6} - 1$ to decimal.

$= 3.89897948$

Substitute the $x$ value $1$ into $f (x) = - 2 \sqrt{3 (x + 1)} - 1$ . In this case, the point is $(1, - 5.89897948)$ .

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

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

Simplify the result.

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Simplify each term.

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Add $1$ and $1$ .

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

Multiply $3$ by $2$ .

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

The final answer is $- 2 \sqrt{6} - 1$ .

$y = - 2 \sqrt{6} - 1$

Convert $- 2 \sqrt{6} - 1$ to decimal.

$= - 5.89897948$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & 2.46 \\ 0 & - 4.46 \\ 1 & 3.9 \\ 1 & - 5.9 \end{array}$

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 1, - 1)$

Focus: $(2, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - 4$

$\begin{array}{c} x & y \\ - 1 & - 1 \\ 0 & 2.46 \\ 0 & - 4.46 \\ 1 & 3.9 \\ 1 & - 5.9 \end{array}$

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