Precalculus Examples

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

Step 1

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

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Step 1.1

Rewrite the equation as $- 2 (y - 1) = {(x + 3)}^{2}$ .

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

Step 1.2

Divide each term in $- 2 (y - 1) = {(x + 3)}^{2}$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 (y - 1) = {(x + 3)}^{2}$ by $- 2$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y - 1$ by $1$ .

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

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.

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

Step 1.3

Add $1$ to both sides of the equation.

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

Step 1.4

Reorder terms.

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

Step 2

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

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

$h = - 3$

$k = 1$

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.

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

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 $- 1 (- 1)$ .

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

Step 5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{2})}$

Step 5.3.2

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

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

Step 5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

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

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.

$y = \frac{3}{2}$

Step 9

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

Direction: Opens Down

Vertex: $(- 3, 1)$

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

Axis of Symmetry: $x = - 3$

Directrix: $y = \frac{3}{2}$

Step 10