Precalculus Examples

$y^{2} - 4 y - 4 x = 0$

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.

$- 4 y - 4 x = - y^{2}$

Step 1.1.1.2

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

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

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

Dividing two negative values results in a positive value.

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

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

Factor $4$ out of $4 y$ .

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

Step 1.1.2.3.1.2.2

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

$x = \frac{y^{2}}{4} - 1 \cdot y$

Step 1.1.2.3.1.3

Rewrite $- 1 \cdot y$ as $- y$ .

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

Step 1.2

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

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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 $c$ .

$a = \frac{1}{4}$

$b = - 1$

$c = 0$

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

$d = \frac{- 1}{2 (\frac{1}{4})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Combine $2$ and $\frac{1}{4}$ .

$d = \frac{- 1}{\frac{2}{4}}$

Step 1.2.3.2.2

Cancel the common factor of $2$ and $4$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $2$ .

$d = \frac{- 1}{\frac{2 (1)}{4}}$

Step 1.2.3.2.2.2

Cancel the common factors.

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Step 1.2.3.2.2.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Step 1.2.3.2.4

Multiply $- 1$ by $2$ .

$d = - 2$

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

$e = 0 - \frac{{(- 1)}^{2}}{4 (\frac{1}{4})}$

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

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1}{4 (\frac{1}{4})}$

Step 1.2.4.2.1.2

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

$e = 0 - \frac{1}{\frac{4}{4}}$

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

$e = 0 - \frac{1}{1}$

Step 1.2.4.2.1.4

Cancel the common factor of $1$ .

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Step 1.2.4.2.1.4.1

Cancel the common factor.

$e = 0 - \frac{1}{1}$

Step 1.2.4.2.1.4.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 1.2.4.2.1.5

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 1.2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(y - 2)}^{2} - 1$ .

$\frac{1}{4} {(y - 2)}^{2} - 1$

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = \frac{1}{4}$

$h = - 1$

$k = 2$

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(- 1, 2)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

Simplify by dividing numbers.

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Step 5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 5.3.2.2

Divide $1$ by $1$ .

$1$

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.

$(0, 2)$

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.

$x = - 2$

Step 9

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

Direction: Opens Right

Vertex: $(- 1, 2)$

Focus: $(0, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - 2$

Step 10