Precalculus Examples

$\frac{{(x + 3)}^{2}}{36} + \frac{y - 4^{2}}{32} = 1$

Step 1

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

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Step 1.1

Simplify $\frac{{(x + 3)}^{2}}{36} + \frac{y - 4^{2}}{32}$ .

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Step 1.1.1

Simplify the numerator.

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Step 1.1.1.1

Raise $4$ to the power of $2$ .

$\frac{{(x + 3)}^{2}}{36} + \frac{y - 1 \cdot 16}{32} = 1$

Step 1.1.1.2

Multiply $- 1$ by $16$ .

$\frac{{(x + 3)}^{2}}{36} + \frac{y - 16}{32} = 1$

Step 1.1.2

To write $\frac{{(x + 3)}^{2}}{36}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{{(x + 3)}^{2}}{36} \cdot \frac{8}{8} + \frac{y - 16}{32} = 1$

Step 1.1.3

To write $\frac{y - 16}{32}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{{(x + 3)}^{2}}{36} \cdot \frac{8}{8} + \frac{y - 16}{32} \cdot \frac{9}{9} = 1$

Step 1.1.4

Write each expression with a common denominator of $288$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.4.1

Multiply $\frac{{(x + 3)}^{2}}{36}$ by $\frac{8}{8}$ .

$\frac{{(x + 3)}^{2} \cdot 8}{36 \cdot 8} + \frac{y - 16}{32} \cdot \frac{9}{9} = 1$

Step 1.1.4.2

Multiply $36$ by $8$ .

$\frac{{(x + 3)}^{2} \cdot 8}{288} + \frac{y - 16}{32} \cdot \frac{9}{9} = 1$

Step 1.1.4.3

Multiply $\frac{y - 16}{32}$ by $\frac{9}{9}$ .

$\frac{{(x + 3)}^{2} \cdot 8}{288} + \frac{(y - 16) \cdot 9}{32 \cdot 9} = 1$

Step 1.1.4.4

Multiply $32$ by $9$ .

$\frac{{(x + 3)}^{2} \cdot 8}{288} + \frac{(y - 16) \cdot 9}{288} = 1$

Step 1.1.5

Combine the numerators over the common denominator.

$\frac{{(x + 3)}^{2} \cdot 8 + (y - 16) \cdot 9}{288} = 1$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Move $8$ to the left of ${(x + 3)}^{2}$ .

$\frac{8 \cdot {(x + 3)}^{2} + (y - 16) \cdot 9}{288} = 1$

Step 1.1.6.2

Apply the distributive property.

$\frac{8 {(x + 3)}^{2} + y \cdot 9 - 16 \cdot 9}{288} = 1$

Step 1.1.6.3

Move $9$ to the left of $y$ .

$\frac{8 {(x + 3)}^{2} + 9 \cdot y - 16 \cdot 9}{288} = 1$

Step 1.1.6.4

Multiply $- 16$ by $9$ .

$\frac{8 {(x + 3)}^{2} + 9 y - 144}{288} = 1$

Step 1.2

Multiply both sides by $288$ .

$\frac{8 {(x + 3)}^{2} + 9 y - 144}{288} \cdot 288 = 1 \cdot 288$

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 $288$ .

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Step 1.3.1.1.1

Cancel the common factor.

$\frac{8 {(x + 3)}^{2} + 9 y - 144}{288} \cdot 288 = 1 \cdot 288$

Step 1.3.1.1.2

Rewrite the expression.

$8 {(x + 3)}^{2} + 9 y - 144 = 1 \cdot 288$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Multiply $288$ by $1$ .

$8 {(x + 3)}^{2} + 9 y - 144 = 288$

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 $8 {(x + 3)}^{2}$ from both sides of the equation.

$9 y - 144 = 288 - 8 {(x + 3)}^{2}$

Step 1.4.1.2

Add $144$ to both sides of the equation.

$9 y = 288 - 8 {(x + 3)}^{2} + 144$

Step 1.4.1.3

Add $288$ and $144$ .

$9 y = - 8 {(x + 3)}^{2} + 432$

Step 1.4.2

Divide each term in $9 y = - 8 {(x + 3)}^{2} + 432$ by $9$ and simplify.

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Step 1.4.2.1

Divide each term in $9 y = - 8 {(x + 3)}^{2} + 432$ by $9$ .

$\frac{9 y}{9} = \frac{- 8 {(x + 3)}^{2}}{9} + \frac{432}{9}$

Step 1.4.2.2

Simplify the left side.

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Step 1.4.2.2.1

Cancel the common factor of $9$ .

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Step 1.4.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{- 8 {(x + 3)}^{2}}{9} + \frac{432}{9}$

Step 1.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 8 {(x + 3)}^{2}}{9} + \frac{432}{9}$

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

Move the negative in front of the fraction.

$y = - \frac{8 {(x + 3)}^{2}}{9} + \frac{432}{9}$

Step 1.4.2.3.1.2

Divide $432$ by $9$ .

$y = - \frac{8 {(x + 3)}^{2}}{9} + 48$

Step 1.5

Reorder terms.

$y = - \frac{8}{9} \cdot {(x + 3)}^{2} + 48$

Step 2

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

$a = - \frac{8}{9}$

$h = - 3$

$k = 48$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(- 3, 48)$

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.

$\frac{1}{4 \cdot (- \frac{8}{9})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{8}{9})}$

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{8}{9})}$

Step 5.3.2

Combine $4$ and $\frac{8}{9}$ .

$- \frac{1}{\frac{4 \cdot 8}{9}}$

Step 5.3.3

Multiply $4$ by $8$ .

$- \frac{1}{\frac{32}{9}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{9}{32}))$

Step 5.3.5

Multiply $\frac{9}{32}$ by $1$ .

$- \frac{9}{32}$

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.

$(- 3, \frac{1527}{32})$

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.

$y = \frac{1545}{32}$

Step 9

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

Direction: Opens Down

Vertex: $(- 3, 48)$

Focus: $(- 3, \frac{1527}{32})$

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{1545}{32}$

Step 10