Precalculus Examples

$- 2 y^{2} + x - 4 y + 1 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Add $2 y^{2}$ to both sides of the equation.

$x - 4 y + 1 = 2 y^{2}$

Step 1.1.2

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

$x + 1 = 2 y^{2} + 4 y$

Step 1.1.3

Subtract $1$ from both sides of the equation.

$x = 2 y^{2} + 4 y - 1$

Step 1.2

Complete the square for $2 y^{2} + 4 y - 1$ .

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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 $c$ .

$a = 2$

$b = 4$

$c = - 1$

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

$d = \frac{4}{2 \cdot 2}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

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

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

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.

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

Step 1.2.3.2.2.2

Rewrite the expression.

$d = 1$

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

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

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 1.2.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 1.2.4.2.1.1.2

Cancel the common factors.

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Step 1.2.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 2$ .

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

Step 1.2.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.1.2.3

Rewrite the expression.

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

Step 1.2.4.2.1.2

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

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Step 1.2.4.2.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.4.2.1.2.2

Cancel the common factors.

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Step 1.2.4.2.1.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.1.2.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.2.2.3

Rewrite the expression.

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

Step 1.2.4.2.1.2.2.4

Divide $2$ by $1$ .

$e = - 1 - 1 \cdot 2$

Step 1.2.4.2.1.3

Multiply $- 1$ by $2$ .

$e = - 1 - 2$

Step 1.2.4.2.2

Subtract $2$ from $- 1$ .

$e = - 3$

Step 1.2.5

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = 2$

$h = - 3$

$k = - 1$

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.

$\frac{1}{4 \cdot 2}$

Step 5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(- \frac{23}{8}, - 1)$

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.

$x = - \frac{25}{8}$

Step 9

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

Direction: Opens Right

Vertex: $(- 3, - 1)$

Focus: $(- \frac{23}{8}, - 1)$

Axis of Symmetry: $y = - 1$

Directrix: $x = - \frac{25}{8}$

Step 10