Precalculus Examples

$y^{2} + 3 x = 0$

Step 1

Subtract $y^{2}$ from both sides of the equation.

$3 x = - y^{2}$

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $x$ by $1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

Step 3

Find the properties of the given parabola.

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Step 3.1

Rewrite the equation in vertex form.

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Step 3.1.1

Complete the square for $- \frac{y^{2}}{3}$ .

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Step 3.1.1.1

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

$a = - \frac{1}{3}$

$b = 0$

$c = 0$

Step 3.1.1.2

Consider the vertex form of a parabola.

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

Step 3.1.1.3

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

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Step 3.1.1.3.1

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

$d = \frac{0}{2 (- \frac{1}{3})}$

Step 3.1.1.3.2

Simplify the right side.

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Step 3.1.1.3.2.1

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

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Step 3.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{3})}$

Step 3.1.1.3.2.1.2

Cancel the common factors.

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Step 3.1.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{3})}$

Step 3.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{1}{3}}$

Step 3.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 3)$

Step 3.1.1.3.2.3

Multiply $0 (- 1 \cdot 3)$ .

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Step 3.1.1.3.2.3.1

Multiply $- 1$ by $3$ .

$d = 0 \cdot - 3$

Step 3.1.1.3.2.3.2

Multiply $0$ by $- 3$ .

$d = 0$

Step 3.1.1.4

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

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Step 3.1.1.4.1

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

$e = 0 - \frac{0^{2}}{4 (- \frac{1}{3})}$

Step 3.1.1.4.2

Simplify the right side.

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Step 3.1.1.4.2.1

Simplify each term.

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Step 3.1.1.4.2.1.1

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

$e = 0 - \frac{0}{4 (- \frac{1}{3})}$

Step 3.1.1.4.2.1.2

Simplify the denominator.

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Step 3.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{1}{3})}$

Step 3.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{3}$ .

$e = 0 - \frac{0}{\frac{- 4}{3}}$

Step 3.1.1.4.2.1.3

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{4}{3}}$

Step 3.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- \frac{3}{4}))$

Step 3.1.1.4.2.1.5

Multiply $0 (- \frac{3}{4})$ .

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Step 3.1.1.4.2.1.5.1

Multiply $- 1$ by $0$ .

$e = 0 - (0 (\frac{3}{4}))$

Step 3.1.1.4.2.1.5.2

Multiply $0$ by $\frac{3}{4}$ .

$e = 0 - 0$

Step 3.1.1.4.2.1.6

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 3.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 3.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{3} y^{2}$ .

$- \frac{1}{3} y^{2}$

Step 3.1.2

Set $x$ equal to the new right side.

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

Step 3.2

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

$a = - \frac{1}{3}$

$h = 0$

$k = 0$

Step 3.3

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

Opens Left

Step 3.4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 3.5

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

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Step 3.5.1

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

$\frac{1}{4 a}$

Step 3.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{3})}$

Step 3.5.3

Simplify.

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Step 3.5.3.1

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

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Step 3.5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{3})}$

Step 3.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{3})}$

Step 3.5.3.2

Combine $4$ and $\frac{1}{3}$ .

$- \frac{1}{\frac{4}{3}}$

Step 3.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{4}))$

Step 3.5.3.4

Multiply $\frac{3}{4}$ by $1$ .

$- \frac{3}{4}$

Step 3.6

Find the focus.

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Step 3.6.1

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 3.6.2

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

$(- \frac{3}{4}, 0)$

Step 3.7

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

$y = 0$

Step 3.8

Find the directrix.

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Step 3.8.1

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 3.8.2

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

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

Step 3.9

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

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

Step 4

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 4.1

Substitute the $x$ value $- 2$ into $f (x) = \sqrt{- 3 x}$ . In this case, the point is $(- 2, 2.44948974)$ .

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Step 4.1.1

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

$f (- 2) = \sqrt{- 3 \cdot - 2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Multiply $- 3$ by $- 2$ .

$f (- 2) = \sqrt{6}$

Step 4.1.2.2

The final answer is $\sqrt{6}$ .

$y = \sqrt{6}$

Step 4.1.3

Convert $\sqrt{6}$ to decimal.

$= 2.44948974$

Step 4.2

Substitute the $x$ value $- 2$ into $f (x) = - \sqrt{- 3 x}$ . In this case, the point is $(- 2, - 2.44948974)$ .

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Step 4.2.1

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

$f (- 2) = - \sqrt{- 3 \cdot - 2}$

Step 4.2.2

Simplify the result.

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Step 4.2.2.1

Multiply $- 3$ by $- 2$ .

$f (- 2) = - \sqrt{6}$

Step 4.2.2.2

The final answer is $- \sqrt{6}$ .

$y = - \sqrt{6}$

Step 4.2.3

Convert $- \sqrt{6}$ to decimal.

$= - 2.44948974$

Step 4.3

Substitute the $x$ value $- 1$ into $f (x) = \sqrt{- 3 x}$ . In this case, the point is $(- 1, 1.7320508)$ .

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Step 4.3.1

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

$f (- 1) = \sqrt{- 3 \cdot - 1}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Multiply $- 3$ by $- 1$ .

$f (- 1) = \sqrt{3}$

Step 4.3.2.2

The final answer is $\sqrt{3}$ .

$y = \sqrt{3}$

Step 4.3.3

Convert $\sqrt{3}$ to decimal.

$= 1.7320508$

Step 4.4

Substitute the $x$ value $- 1$ into $f (x) = - \sqrt{- 3 x}$ . In this case, the point is $(- 1, - 1.7320508)$ .

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Step 4.4.1

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

$f (- 1) = - \sqrt{- 3 \cdot - 1}$

Step 4.4.2

Simplify the result.

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Step 4.4.2.1

Multiply $- 3$ by $- 1$ .

$f (- 1) = - \sqrt{3}$

Step 4.4.2.2

The final answer is $- \sqrt{3}$ .

$y = - \sqrt{3}$

Step 4.4.3

Convert $- \sqrt{3}$ to decimal.

$= - 1.7320508$

Step 4.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 2.45 \\ - 2 & - 2.45 \\ - 1 & 1.73 \\ - 1 & - 1.73 \\ 0 & 0 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

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

$\begin{array}{c} x & y \\ - 2 & 2.45 \\ - 2 & - 2.45 \\ - 1 & 1.73 \\ - 1 & - 1.73 \\ 0 & 0 \end{array}$

Step 6