Precalculus Examples

$y^{2} - 4 y + 6 x - 8 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

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

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Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$- 4 y + 6 x - 8 = - y^{2}$

Step 1.1.1.2

Add $4 y$ to both sides of the equation.

$6 x - 8 = - y^{2} + 4 y$

Step 1.1.1.3

Add $8$ to both sides of the equation.

$6 x = - y^{2} + 4 y + 8$

Step 1.1.2

Divide each term in $6 x = - y^{2} + 4 y + 8$ by $6$ and simplify.

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Step 1.1.2.1

Divide each term in $6 x = - y^{2} + 4 y + 8$ by $6$ .

$\frac{6 x}{6} = \frac{- y^{2}}{6} + \frac{4 y}{6} + \frac{8}{6}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $6$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{- y^{2}}{6} + \frac{4 y}{6} + \frac{8}{6}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{6} + \frac{4 y}{6} + \frac{8}{6}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{y^{2}}{6} + \frac{4 y}{6} + \frac{8}{6}$

Step 1.1.2.3.1.2

Cancel the common factor of $4$ and $6$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $4 y$ .

$x = - \frac{y^{2}}{6} + \frac{2 (2 y)}{6} + \frac{8}{6}$

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $6$ .

$x = - \frac{y^{2}}{6} + \frac{2 (2 y)}{2 (3)} + \frac{8}{6}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{6} + \frac{2 (2 y)}{2 \cdot 3} + \frac{8}{6}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{8}{6}$

Step 1.1.2.3.1.3

Cancel the common factor of $8$ and $6$ .

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Step 1.1.2.3.1.3.1

Factor $2$ out of $8$ .

$x = - \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{2 (4)}{6}$

Step 1.1.2.3.1.3.2

Cancel the common factors.

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Step 1.1.2.3.1.3.2.1

Factor $2$ out of $6$ .

$x = - \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{2 \cdot 4}{2 \cdot 3}$

Step 1.1.2.3.1.3.2.2

Cancel the common factor.

$x = - \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{2 \cdot 4}{2 \cdot 3}$

Step 1.1.2.3.1.3.2.3

Rewrite the expression.

$x = - \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{4}{3}$

Step 1.2

Complete the square for $- \frac{y^{2}}{6} + \frac{2 y}{3} + \frac{4}{3}$ .

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Step 1.2.1

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

$a = - \frac{1}{6}$

$b = \frac{2}{3}$

$c = \frac{4}{3}$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

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

$d = \frac{\frac{2}{3}}{2 (- \frac{1}{6})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{2}{3} \cdot \frac{1}{2 (- \frac{1}{6})}$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Cancel the common factor.

$d = \frac{2}{3} \cdot \frac{1}{2 (- \frac{1}{6})}$

Step 1.2.3.2.2.2

Rewrite the expression.

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

Step 1.2.3.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{- \frac{1}{6}}$ .

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

Step 1.2.3.2.4

Multiply $- 1$ by $3$ .

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

Step 1.2.3.2.5

Combine $- 3$ and $\frac{1}{6}$ .

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

Step 1.2.3.2.6

Cancel the common factor of $- 3$ and $6$ .

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Step 1.2.3.2.6.1

Factor $3$ out of $- 3$ .

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

Step 1.2.3.2.6.2

Cancel the common factors.

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Step 1.2.3.2.6.2.1

Factor $3$ out of $6$ .

$d = \frac{1}{\frac{3 \cdot - 1}{3 \cdot 2}}$

Step 1.2.3.2.6.2.2

Cancel the common factor.

$d = \frac{1}{\frac{3 \cdot - 1}{3 \cdot 2}}$

Step 1.2.3.2.6.2.3

Rewrite the expression.

$d = \frac{1}{\frac{- 1}{2}}$

Step 1.2.3.2.7

Multiply the numerator by the reciprocal of the denominator.

$d = 1 (\frac{2}{- 1})$

Step 1.2.3.2.8

Multiply $\frac{2}{- 1}$ by $1$ .

$d = \frac{2}{- 1}$

Step 1.2.3.2.9

Divide $2$ by $- 1$ .

$d = - 2$

Step 1.2.4

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

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{4}{3} - \frac{{(\frac{2}{3})}^{2}}{4 (- \frac{1}{6})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{2}{3}$ .

$e = \frac{4}{3} - \frac{\frac{2^{2}}{3^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.2

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

$e = \frac{4}{3} - \frac{\frac{4}{3^{2}}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.1.3

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

$e = \frac{4}{3} - \frac{\frac{4}{9}}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{4}{3} - \frac{\frac{4}{9}}{- 4 (\frac{1}{6})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{6}$ .

$e = \frac{4}{3} - \frac{\frac{4}{9}}{\frac{- 4}{6}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

Cancel the common factor of $- 4$ and $6$ .

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Step 1.2.4.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = \frac{4}{3} - \frac{\frac{4}{9}}{\frac{2 (- 2)}{6}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = \frac{4}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = \frac{4}{3} - \frac{\frac{4}{9}}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = \frac{4}{3} - \frac{\frac{4}{9}}{\frac{- 2}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = \frac{4}{3} - \frac{\frac{4}{9}}{- \frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{4}{3} - (\frac{4}{9} (- \frac{3}{2}))$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$e = \frac{4}{3} - (\frac{4}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.2

Factor $2$ out of $4$ .

$e = \frac{4}{3} - (\frac{2 (2)}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = \frac{4}{3} - (\frac{2 \cdot 2}{9} \cdot \frac{- 3}{2})$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = \frac{4}{3} - (\frac{2}{9} \cdot - 3)$

Step 1.2.4.2.1.6

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.6.1

Factor $3$ out of $9$ .

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

Step 1.2.4.2.1.6.2

Factor $3$ out of $- 3$ .

$e = \frac{4}{3} - (\frac{2}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.3

Cancel the common factor.

$e = \frac{4}{3} - (\frac{2}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 1.2.4.2.1.6.4

Rewrite the expression.

$e = \frac{4}{3} - (\frac{2}{3} \cdot - 1)$

Step 1.2.4.2.1.7

Combine $\frac{2}{3}$ and $- 1$ .

$e = \frac{4}{3} - \frac{2 \cdot - 1}{3}$

Step 1.2.4.2.1.8

Multiply $2$ by $- 1$ .

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

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

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

Step 1.2.4.2.1.10

Multiply $- - \frac{2}{3}$ .

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Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = \frac{4}{3} + 1 (\frac{2}{3})$

Step 1.2.4.2.1.10.2

Multiply $\frac{2}{3}$ by $1$ .

$e = \frac{4}{3} + \frac{2}{3}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{4 + 2}{3}$

Step 1.2.4.2.3

Add $4$ and $2$ .

$e = \frac{6}{3}$

Step 1.2.4.2.4

Divide $6$ by $3$ .

$e = 2$

Step 1.2.5

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{6}$

$h = 2$

$k = 2$

Step 3

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

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(2, 2)$

Step 5

Find $p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

Simplify.

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Step 5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{6})}$

Step 5.3.2

Combine $4$ and $\frac{1}{6}$ .

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

Step 5.3.3

Cancel the common factor of $4$ and $6$ .

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Step 5.3.3.1

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{6}}$

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $2$ out of $6$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 5.3.3.2.2

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 5.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{2}{3}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{2}))$

Step 5.3.5

Multiply $\frac{3}{2}$ by $1$ .

$- \frac{3}{2}$

Step 6

Find the focus.

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Step 6.1

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

$(h + p, k)$

Step 6.2

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

$(\frac{1}{2}, 2)$

Step 7

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

$y = 2$

Step 8

Find the directrix.

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Step 8.1

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.

$x = h - p$

Step 8.2

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

$x = \frac{7}{2}$

Step 9

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

Direction: Opens Left

Vertex: $(2, 2)$

Focus: $(\frac{1}{2}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = \frac{7}{2}$

Step 10