Precalculus Examples

$\frac{{(x + 4)}^{2}}{36} - \frac{y + 4}{81} = 1$

Step 1

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

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Step 1.1

Simplify $\frac{{(x + 4)}^{2}}{36} - \frac{y + 4}{81}$ .

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Step 1.1.1

To write $\frac{{(x + 4)}^{2}}{36}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{{(x + 4)}^{2}}{36} \cdot \frac{9}{9} - \frac{y + 4}{81} = 1$

Step 1.1.2

To write $- \frac{y + 4}{81}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{{(x + 4)}^{2}}{36} \cdot \frac{9}{9} - \frac{y + 4}{81} \cdot \frac{4}{4} = 1$

Step 1.1.3

Write each expression with a common denominator of $324$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.3.1

Multiply $\frac{{(x + 4)}^{2}}{36}$ by $\frac{9}{9}$ .

$\frac{{(x + 4)}^{2} \cdot 9}{36 \cdot 9} - \frac{y + 4}{81} \cdot \frac{4}{4} = 1$

Step 1.1.3.2

Multiply $36$ by $9$ .

$\frac{{(x + 4)}^{2} \cdot 9}{324} - \frac{y + 4}{81} \cdot \frac{4}{4} = 1$

Step 1.1.3.3

Multiply $\frac{y + 4}{81}$ by $\frac{4}{4}$ .

$\frac{{(x + 4)}^{2} \cdot 9}{324} - \frac{(y + 4) \cdot 4}{81 \cdot 4} = 1$

Step 1.1.3.4

Multiply $81$ by $4$ .

$\frac{{(x + 4)}^{2} \cdot 9}{324} - \frac{(y + 4) \cdot 4}{324} = 1$

Step 1.1.4

Combine the numerators over the common denominator.

$\frac{{(x + 4)}^{2} \cdot 9 - (y + 4) \cdot 4}{324} = 1$

Step 1.1.5

Simplify the numerator.

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Step 1.1.5.1

Move $9$ to the left of ${(x + 4)}^{2}$ .

$\frac{9 \cdot {(x + 4)}^{2} - (y + 4) \cdot 4}{324} = 1$

Step 1.1.5.2

Apply the distributive property.

$\frac{9 {(x + 4)}^{2} + (- y - 1 \cdot 4) \cdot 4}{324} = 1$

Step 1.1.5.3

Multiply $- 1$ by $4$ .

$\frac{9 {(x + 4)}^{2} + (- y - 4) \cdot 4}{324} = 1$

Step 1.1.5.4

Apply the distributive property.

$\frac{9 {(x + 4)}^{2} - y \cdot 4 - 4 \cdot 4}{324} = 1$

Step 1.1.5.5

Multiply $4$ by $- 1$ .

$\frac{9 {(x + 4)}^{2} - 4 y - 4 \cdot 4}{324} = 1$

Step 1.1.5.6

Multiply $- 4$ by $4$ .

$\frac{9 {(x + 4)}^{2} - 4 y - 16}{324} = 1$

Step 1.2

Multiply both sides by $324$ .

$\frac{9 {(x + 4)}^{2} - 4 y - 16}{324} \cdot 324 = 1 \cdot 324$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Cancel the common factor of $324$ .

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Step 1.3.1.1.1

Cancel the common factor.

$\frac{9 {(x + 4)}^{2} - 4 y - 16}{324} \cdot 324 = 1 \cdot 324$

Step 1.3.1.1.2

Rewrite the expression.

$9 {(x + 4)}^{2} - 4 y - 16 = 1 \cdot 324$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Multiply $324$ by $1$ .

$9 {(x + 4)}^{2} - 4 y - 16 = 324$

Step 1.4

Solve for $y$ .

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Step 1.4.1

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

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Step 1.4.1.1

Subtract $9 {(x + 4)}^{2}$ from both sides of the equation.

$- 4 y - 16 = 324 - 9 {(x + 4)}^{2}$

Step 1.4.1.2

Add $16$ to both sides of the equation.

$- 4 y = 324 - 9 {(x + 4)}^{2} + 16$

Step 1.4.1.3

Add $324$ and $16$ .

$- 4 y = - 9 {(x + 4)}^{2} + 340$

Step 1.4.2

Divide each term in $- 4 y = - 9 {(x + 4)}^{2} + 340$ by $- 4$ and simplify.

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Step 1.4.2.1

Divide each term in $- 4 y = - 9 {(x + 4)}^{2} + 340$ by $- 4$ .

$\frac{- 4 y}{- 4} = \frac{- 9 {(x + 4)}^{2}}{- 4} + \frac{340}{- 4}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 y}{- 4} = \frac{- 9 {(x + 4)}^{2}}{- 4} + \frac{340}{- 4}$

Step 1.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9 {(x + 4)}^{2}}{- 4} + \frac{340}{- 4}$

Step 1.4.2.3

Simplify the right side.

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Step 1.4.2.3.1

Simplify each term.

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Step 1.4.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{9 {(x + 4)}^{2}}{4} + \frac{340}{- 4}$

Step 1.4.2.3.1.2

Divide $340$ by $- 4$ .

$y = \frac{9 {(x + 4)}^{2}}{4} - 85$

Step 1.5

Reorder terms.

$y = \frac{9}{4} \cdot {(x + 4)}^{2} - 85$

Step 2

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

$a = \frac{9}{4}$

$h = - 4$

$k = - 85$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(- 4, - 85)$

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.

$\frac{1}{4 \cdot \frac{9}{4}}$

Step 5.3

Simplify.

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Step 5.3.1

Combine $4$ and $\frac{9}{4}$ .

$\frac{1}{\frac{4 \cdot 9}{4}}$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Multiply $4$ by $9$ .

$\frac{1}{\frac{36}{4}}$

Step 5.3.2.2

Divide $36$ by $4$ .

$\frac{1}{9}$

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.

$(- 4, - \frac{764}{9})$

Step 7

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

$x = - 4$

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.

$y = - \frac{766}{9}$

Step 9

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

Direction: Opens Up

Vertex: $(- 4, - 85)$

Focus: $(- 4, - \frac{764}{9})$

Axis of Symmetry: $x = - 4$

Directrix: $y = - \frac{766}{9}$

Step 10