Algebra Examples

$x = - 4 y^{2}$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $- 4 y^{2}$ .

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Step 1.1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 4$

$b = 0$

$c = 0$

Step 1.1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 \cdot - 4}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 4}$

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot - 4$ .

$d = \frac{2 (0)}{2 (- 4)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot - 4}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{0}{- 4}$

Step 1.1.1.3.2.2

Cancel the common factor of $0$ and $- 4$ .

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Step 1.1.1.3.2.2.1

Factor $4$ out of $0$ .

$d = \frac{4 (0)}{- 4}$

Step 1.1.1.3.2.2.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 1.1.1.3.2.3

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 1.1.1.3.2.4

Multiply $- 1$ by $0$ .

$d = 0$

Step 1.1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{0^{2}}{4 \cdot - 4}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 \cdot - 4}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{0}{- 16}$

Step 1.1.1.4.2.1.3

Divide $0$ by $- 16$ .

$e = 0 - 0$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 y^{2}$ .

$- 4 y^{2}$

Step 1.1.2

Set $x$ equal to the new right side.

$x = - 4 y^{2}$

Step 1.2

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

$a = - 4$

$h = 0$

$k = 0$

Step 1.3

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

Opens Left

Step 1.4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 1.5

Find $p$ , the distance from the vertex to the focus.

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Step 1.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $4$ by $- 4$ .

$\frac{1}{- 16}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{16}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(- \frac{1}{16}, 0)$

Step 1.7

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

$y = 0$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$x = \frac{1}{16}$

Step 1.9

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{1}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{16}$

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{1}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{16}$

Step 2

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 2.1

Substitute the $x$ value $- 2$ into $f (x) = \frac{\sqrt{- x}}{2}$ . In this case, the point is $(- 2, 0.70710678)$ .

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Step 2.1.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{\sqrt{- (- 2)}}{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = \frac{\sqrt{2}}{2}$

Step 2.1.2.2

The final answer is $\frac{\sqrt{2}}{2}$ .

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

Step 2.1.3

Convert $\frac{\sqrt{2}}{2}$ to decimal.

$= 0.70710678$

Step 2.2

Substitute the $x$ value $- 2$ into $f (x) = - \frac{\sqrt{- x}}{2}$ . In this case, the point is $(- 2, - 0.70710678)$ .

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Step 2.2.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - \frac{\sqrt{- (- 2)}}{2}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = - \frac{\sqrt{2}}{2}$

Step 2.2.2.2

The final answer is $- \frac{\sqrt{2}}{2}$ .

$y = - \frac{\sqrt{2}}{2}$

Step 2.2.3

Convert $- \frac{\sqrt{2}}{2}$ to decimal.

$= - 0.70710678$

Step 2.3

Substitute the $x$ value $- 1$ into $f (x) = \frac{\sqrt{- x}}{2}$ . In this case, the point is $(- 1, 0.5)$ .

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Step 2.3.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \frac{\sqrt{- (- 1)}}{2}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify the numerator.

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Step 2.3.2.1.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = \frac{\sqrt{1}}{2}$

Step 2.3.2.1.2

Any root of $1$ is $1$ .

$f (- 1) = \frac{1}{2}$

Step 2.3.2.2

The final answer is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 2.3.3

Convert $\frac{1}{2}$ to decimal.

$= 0.5$

Step 2.4

Substitute the $x$ value $- 1$ into $f (x) = - \frac{\sqrt{- x}}{2}$ . In this case, the point is $(- 1, - 0.5)$ .

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Step 2.4.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = - \frac{\sqrt{- (- 1)}}{2}$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

Simplify the numerator.

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Step 2.4.2.1.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = - \frac{\sqrt{1}}{2}$

Step 2.4.2.1.2

Any root of $1$ is $1$ .

$f (- 1) = - \frac{1}{2}$

Step 2.4.2.2

The final answer is $- \frac{1}{2}$ .

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

Step 2.4.3

Convert $- \frac{1}{2}$ to decimal.

$= - 0.5$

Step 2.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 0.71 \\ - 2 & - 0.71 \\ - 1 & 0.5 \\ - 1 & - 0.5 \\ 0 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{1}{16}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{16}$

$\begin{array}{c} x & y \\ - 2 & 0.71 \\ - 2 & - 0.71 \\ - 1 & 0.5 \\ - 1 & - 0.5 \\ 0 & 0 \end{array}$

Step 4