Precalculus Examples

$3 y^{2} - 2 y + x + 1 = 0$

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

Subtract $3 y^{2}$ from both sides of the equation.

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

Step 1.1.2

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

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

Step 1.1.3

Subtract $1$ from both sides of the equation.

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

Step 1.2

Complete the square for $- 3 y^{2} + 2 y - 1$ .

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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 $c$ .

$a = - 3$

$b = 2$

$c = - 1$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Rewrite the expression.

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

Step 1.2.3.2.2

Move the negative in front of the fraction.

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

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

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

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 $2$ to the power of $2$ .

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

Step 1.2.4.2.1.2

Multiply $4$ by $- 3$ .

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

Step 1.2.4.2.1.3

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

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

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

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $- 12$ .

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

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = - 1 - \frac{1}{- 3}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = - 1 - - \frac{1}{3}$

Step 1.2.4.2.1.5

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

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Step 1.2.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4.2.1.5.2

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

$e = - 1 + \frac{1}{3}$

Step 1.2.4.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e = - 1 \cdot \frac{3}{3} + \frac{1}{3}$

Step 1.2.4.2.3

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

$e = \frac{- 1 \cdot 3}{3} + \frac{1}{3}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 1 \cdot 3 + 1}{3}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $- 1$ by $3$ .

$e = \frac{- 3 + 1}{3}$

Step 1.2.4.2.5.2

Add $- 3$ and $1$ .

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

Step 1.2.4.2.6

Move the negative in front of the fraction.

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

Step 1.2.5

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

$a = - 3$

$h = - \frac{2}{3}$

$k = \frac{1}{3}$

Step 3

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

Opens Left

Step 4

Find the vertex $(h, k)$ .

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

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

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.

$(- \frac{3}{4}, \frac{1}{3})$

Step 7

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

$y = \frac{1}{3}$

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.

$x = - \frac{7}{12}$

Step 9

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

Direction: Opens Left

Vertex: $(- \frac{2}{3}, \frac{1}{3})$

Focus: $(- \frac{3}{4}, \frac{1}{3})$

Axis of Symmetry: $y = \frac{1}{3}$

Directrix: $x = - \frac{7}{12}$

Step 10