Precalculus Examples

$(x + 1) + {(y + 4)}^{2} = 0$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Subtract $1$ from both sides of the equation.

$x + {(y + 4)}^{2} = - 1$

Step 1.1.2

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

$x = - 1 - {(y + 4)}^{2}$

Step 1.2

Reorder terms.

$x = - {(y + 4)}^{2} - 1$

Step 2

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

$a = - 1$

$h = - 1$

$k = - 4$

Step 3

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

Opens Left

Step 4

Find the vertex $(h, k)$ .

$(- 1, - 4)$

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.

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

Step 5.3

Cancel the common factor of $1$ and $- 1$ .

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Step 5.3.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.3.2

Move the negative in front of the fraction.

$- \frac{1}{4}$

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 x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

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

$(- \frac{5}{4}, - 4)$

Step 7

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

$y = - 4$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 8.2

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

$x = - \frac{3}{4}$

Step 9

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

Direction: Opens Left

Vertex: $(- 1, - 4)$

Focus: $(- \frac{5}{4}, - 4)$

Axis of Symmetry: $y = - 4$

Directrix: $x = - \frac{3}{4}$

Step 10