Precalculus Examples

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

Step 1

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

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Step 1.1

Rewrite the equation as $- 8 (y + 3) = {(x - 2)}^{2}$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y + 3$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Move the negative in front of the fraction.

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

Step 1.3

Subtract $3$ from both sides of the equation.

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

Step 1.4

Reorder terms.

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

Step 2

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

$a = - \frac{1}{8}$

$h = 2$

$k = - 3$

Step 3

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

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(2, - 3)$

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.

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

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 $- 1 (- 1)$ .

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

Step 5.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.2

Combine $4$ and $\frac{1}{8}$ .

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

Step 5.3.3

Cancel the common factor of $4$ and $8$ .

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Step 5.3.3.1

Factor $4$ out of $4$ .

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

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $4$ out of $8$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 5.3.5

Multiply $- (1 \cdot 2)$ .

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Step 5.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 5.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

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.

$(2, - 5)$

Step 7

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

$x = 2$

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.

$y = - 1$

Step 9

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

Direction: Opens Down

Vertex: $(2, - 3)$

Focus: $(2, - 5)$

Axis of Symmetry: $x = 2$

Directrix: $y = - 1$

Step 10