Precalculus Examples

$y^{2} - 2 y - x = 2$

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.

$- 2 y - x = 2 - y^{2}$

Step 1.1.1.2

Add $2 y$ to both sides of the equation.

$- x = 2 - y^{2} + 2 y$

Step 1.1.2

Divide each term in $- x = 2 - y^{2} + 2 y$ by $- 1$ and simplify.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.2.2

Divide $x$ by $1$ .

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

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

Divide $2$ by $- 1$ .

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

Step 1.1.2.3.1.2

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.3

Divide $y^{2}$ by $1$ .

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

Step 1.1.2.3.1.4

Move the negative one from the denominator of $\frac{2 y}{- 1}$ .

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

Step 1.1.2.3.1.5

Rewrite $- 1 \cdot (2 y)$ as $- (2 y)$ .

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

Step 1.1.2.3.1.6

Multiply $2$ by $- 1$ .

$x = - 2 + y^{2} - 2 y$

Step 1.2

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

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Step 1.2.1

Move $- 2$ .

$y^{2} - 2 y - 2$

Step 1.2.2

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

$a = 1$

$b = - 2$

$c = - 2$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

Factor $2$ out of $- 2$ .

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

Step 1.2.4.2.2

Cancel the common factors.

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Step 1.2.4.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 1.2.4.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.2.3

Rewrite the expression.

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

Step 1.2.4.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

Step 1.2.5

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

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Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

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

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

Step 1.2.5.2.1.2

Multiply $4$ by $1$ .

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

Step 1.2.5.2.1.3

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.3.1

Cancel the common factor.

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

Step 1.2.5.2.1.3.2

Rewrite the expression.

$e = - 2 - 1 \cdot 1$

Step 1.2.5.2.1.4

Multiply $- 1$ by $1$ .

$e = - 2 - 1$

Step 1.2.5.2.2

Subtract $1$ from $- 2$ .

$e = - 3$

Step 1.2.6

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = 1$

$h = - 3$

$k = 1$

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(- 3, 1)$

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.

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

Step 5.3

Cancel the common factor of $1$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

$\frac{1}{4}$

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.

$(- \frac{11}{4}, 1)$

Step 7

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

$y = 1$

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.

$x = - \frac{13}{4}$

Step 9

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

Direction: Opens Right

Vertex: $(- 3, 1)$

Focus: $(- \frac{11}{4}, 1)$

Axis of Symmetry: $y = 1$

Directrix: $x = - \frac{13}{4}$

Step 10