Precalculus Examples

$x^{2} + 6 x + 4 y + 5 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

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Step 1.1.1.1

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

$6 x + 4 y + 5 = - x^{2}$

Step 1.1.1.2

Subtract $6 x$ from both sides of the equation.

$4 y + 5 = - x^{2} - 6 x$

Step 1.1.1.3

Subtract $5$ from both sides of the equation.

$4 y = - x^{2} - 6 x - 5$

Step 1.1.2

Divide each term in $4 y = - x^{2} - 6 x - 5$ by $4$ and simplify.

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Step 1.1.2.1

Divide each term in $4 y = - x^{2} - 6 x - 5$ by $4$ .

$\frac{4 y}{4} = \frac{- x^{2}}{4} + \frac{- 6 x}{4} + \frac{- 5}{4}$

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.

$\frac{4 y}{4} = \frac{- x^{2}}{4} + \frac{- 6 x}{4} + \frac{- 5}{4}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{4} + \frac{- 6 x}{4} + \frac{- 5}{4}$

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

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} + \frac{- 6 x}{4} + \frac{- 5}{4}$

Step 1.1.2.3.1.2

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

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Step 1.1.2.3.1.2.1

Factor $2$ out of $- 6 x$ .

$y = - \frac{x^{2}}{4} + \frac{2 (- 3 x)}{4} + \frac{- 5}{4}$

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 $4$ .

$y = - \frac{x^{2}}{4} + \frac{2 (- 3 x)}{2 (2)} + \frac{- 5}{4}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{4} + \frac{2 (- 3 x)}{2 \cdot 2} + \frac{- 5}{4}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{4} + \frac{- 3 x}{2} + \frac{- 5}{4}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} - \frac{(3) x}{2} + \frac{- 5}{4}$

Step 1.1.2.3.1.4

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{4} - \frac{3 x}{2} - \frac{5}{4}$

Step 1.2

Complete the square for $- \frac{x^{2}}{4} - \frac{3 x}{2} - \frac{5}{4}$ .

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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 $c$ .

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

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

$c = - \frac{5}{4}$

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.3

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

$d = \frac{3}{2} \cdot \frac{1}{\frac{2}{4}}$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

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

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $4$ .

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

Step 1.2.3.2.4.2.2

Cancel the common factor.

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

Step 1.2.3.2.4.2.3

Rewrite the expression.

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

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.6

Cancel the common factor of $2$ .

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Step 1.2.3.2.6.1

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

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

Step 1.2.3.2.6.2

Cancel the common factor.

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

Step 1.2.3.2.6.3

Rewrite the expression.

$d = 3$

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

$e = - \frac{5}{4} - \frac{{(- \frac{3}{2})}^{2}}{4 (- \frac{1}{4})}$

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

Simplify the numerator.

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Step 1.2.4.2.1.1.1

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

$e = - \frac{5}{4} - \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.2

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

$e = - \frac{5}{4} - \frac{1 {(\frac{3}{2})}^{2}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.3

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

$e = - \frac{5}{4} - \frac{1 \frac{3^{2}}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.4

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

$e = - \frac{5}{4} - \frac{1 \frac{9}{2^{2}}}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.5

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

$e = - \frac{5}{4} - \frac{1 (\frac{9}{4})}{4 (- \frac{1}{4})}$

Step 1.2.4.2.1.1.6

Multiply $\frac{9}{4}$ by $1$ .

$e = - \frac{5}{4} - \frac{\frac{9}{4}}{4 (- \frac{1}{4})}$

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 $4$ .

$e = - \frac{5}{4} - \frac{\frac{9}{4}}{- 4 (\frac{1}{4})}$

Step 1.2.4.2.1.2.2

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

$e = - \frac{5}{4} - \frac{\frac{9}{4}}{\frac{- 4}{4}}$

Step 1.2.4.2.1.3

Divide $- 4$ by $4$ .

$e = - \frac{5}{4} - \frac{\frac{9}{4}}{- 1}$

Step 1.2.4.2.1.4

Move the negative one from the denominator of $\frac{\frac{9}{4}}{- 1}$ .

$e = - \frac{5}{4} - (- 1 \cdot \frac{9}{4})$

Step 1.2.4.2.1.5

Rewrite $- 1 \cdot \frac{9}{4}$ as $- \frac{9}{4}$ .

$e = - \frac{5}{4} - - \frac{9}{4}$

Step 1.2.4.2.1.6

Multiply $- - \frac{9}{4}$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{5}{4} + 1 (\frac{9}{4})$

Step 1.2.4.2.1.6.2

Multiply $\frac{9}{4}$ by $1$ .

$e = - \frac{5}{4} + \frac{9}{4}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 5 + 9}{4}$

Step 1.2.4.2.3

Add $- 5$ and $9$ .

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

Step 1.2.4.2.4

Divide $4$ by $4$ .

$e = 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{4} {(x + 3)}^{2} + 1$ .

$- \frac{1}{4} {(x + 3)}^{2} + 1$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{4} \cdot {(x + 3)}^{2} + 1$

Step 2

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

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

$h = - 3$

$k = 1$

Step 3

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

Opens Down

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.

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

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 $- 1 (- 1)$ .

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

Step 5.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.2

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

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

Step 5.3.3

Divide $4$ by $4$ .

$- \frac{1}{1}$

Step 5.3.4

Cancel the common factor of $1$ .

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Step 5.3.4.1

Cancel the common factor.

$- \frac{1}{1}$

Step 5.3.4.2

Rewrite the expression.

$- 1 \cdot 1$

Step 5.3.5

Multiply $- 1$ 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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(- 3, 0)$

Step 7

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

$x = - 3$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 8.2

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

$y = 2$

Step 9

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

Direction: Opens Down

Vertex: $(- 3, 1)$

Focus: $(- 3, 0)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 2$

Step 10