Calculus Examples

$x = 3 y^{2} - 4 y$

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 $3 y^{2} - 4 y$ .

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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 $c$ .

$a = 3$

$b = - 4$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot 3}$

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 $- 4$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 3}$

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 $2 \cdot 3$ .

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

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 3}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 2}{3}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{2}{3}$

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 $e = c - \frac{b^{2}}{4 a}$ .

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

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 1.1.1.4.2.1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 1.1.1.4.2.1.2

Apply the product rule to $- 1 (4)$ .

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

Step 1.1.1.4.2.1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.1.1.4.2.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 3}$

Step 1.1.1.4.2.1.5

Factor $4$ out of $4^{2}$ .

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

Step 1.1.1.4.2.1.6

Cancel the common factors.

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Step 1.1.1.4.2.1.6.1

Factor $4$ out of $4 \cdot 3$ .

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

Step 1.1.1.4.2.1.6.2

Cancel the common factor.

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

Step 1.1.1.4.2.1.6.3

Rewrite the expression.

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

Step 1.1.1.4.2.2

Subtract $\frac{4}{3}$ from $0$ .

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

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(y - \frac{2}{3})}^{2} - \frac{4}{3}$ .

$3 {(y - \frac{2}{3})}^{2} - \frac{4}{3}$

Step 1.1.2

Set $x$ equal to the new right side.

$x = 3 {(y - \frac{2}{3})}^{2} - \frac{4}{3}$

Step 1.2

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

$a = 3$

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

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

Step 1.3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{4}{3}, \frac{2}{3})$

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.

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

Step 1.5.3

Multiply $4$ by $3$ .

$\frac{1}{12}$

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.

$(- \frac{5}{4}, \frac{2}{3})$

Step 1.7

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

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

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.

$x = - \frac{17}{12}$

Step 1.9

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

Direction: Opens Right

Vertex: $(- \frac{4}{3}, \frac{2}{3})$

Focus: $(- \frac{5}{4}, \frac{2}{3})$

Axis of Symmetry: $y = \frac{2}{3}$

Directrix: $x = - \frac{17}{12}$

Direction: Opens Right

Vertex: $(- \frac{4}{3}, \frac{2}{3})$

Focus: $(- \frac{5}{4}, \frac{2}{3})$

Axis of Symmetry: $y = \frac{2}{3}$

Directrix: $x = - \frac{17}{12}$

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 $- 0.\overline{3}$ into $f (x) = \frac{2 + \sqrt{- 1 (- 4 - 3 x)}}{3}$ . In this case, the point is $(- 0.33333333, \frac{2 + \sqrt{3}}{3})$ .

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Step 2.1.1

Replace the variable $x$ with $- 0.\overline{3}$ in the expression.

$f (- 0.\overline{3}) = \frac{2 + \sqrt{- 1 (- 4 - 3 \cdot - 0.\overline{3})}}{3}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Multiply $- 3$ by $- 0.\overline{3}$ .

$f (- 0.\overline{3}) = \frac{2 + \sqrt{- 1 (- 4 + 1)}}{3}$

Step 2.1.2.1.2

Add $- 4$ and $1$ .

$f (- 0.\overline{3}) = \frac{2 + \sqrt{- 1 \cdot - 3}}{3}$

Step 2.1.2.1.3

Multiply $- 1$ by $- 3$ .

$f (- 0.\overline{3}) = \frac{2 + \sqrt{3}}{3}$

Step 2.1.2.2

The final answer is $\frac{2 + \sqrt{3}}{3}$ .

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

Step 2.2

Substitute the $x$ value $- 0.\overline{3}$ into $f (x) = \frac{2 - \sqrt{- 1 (- 4 - 3 x)}}{3}$ . In this case, the point is $(- 0.33333333, \frac{2 - \sqrt{3}}{3})$ .

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Step 2.2.1

Replace the variable $x$ with $- 0.\overline{3}$ in the expression.

$f (- 0.\overline{3}) = \frac{2 - \sqrt{- 1 (- 4 - 3 \cdot - 0.\overline{3})}}{3}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

Multiply $- 3$ by $- 0.\overline{3}$ .

$f (- 0.\overline{3}) = \frac{2 - \sqrt{- 1 (- 4 + 1)}}{3}$

Step 2.2.2.1.2

Add $- 4$ and $1$ .

$f (- 0.\overline{3}) = \frac{2 - \sqrt{- 1 \cdot - 3}}{3}$

Step 2.2.2.1.3

Multiply $- 1$ by $- 3$ .

$f (- 0.\overline{3}) = \frac{2 - \sqrt{3}}{3}$

Step 2.2.2.2

The final answer is $\frac{2 - \sqrt{3}}{3}$ .

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

Step 2.3

Substitute the $x$ value $0.\overline{6}$ into $f (x) = \frac{2 + \sqrt{- 1 (- 4 - 3 x)}}{3}$ . In this case, the point is $(0.\overline{6}, \frac{2 + \sqrt{6}}{3})$ .

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Step 2.3.1

Replace the variable $x$ with $0.\overline{6}$ in the expression.

$f (0.\overline{6}) = \frac{2 + \sqrt{- 1 (- 4 - 3 \cdot 0.\overline{6})}}{3}$

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 $- 3$ by $0.\overline{6}$ .

$f (0.\overline{6}) = \frac{2 + \sqrt{- 1 (- 4 - 2)}}{3}$

Step 2.3.2.1.2

Subtract $2$ from $- 4$ .

$f (0.\overline{6}) = \frac{2 + \sqrt{- 1 \cdot - 6}}{3}$

Step 2.3.2.1.3

Multiply $- 1$ by $- 6$ .

$f (0.\overline{6}) = \frac{2 + \sqrt{6}}{3}$

Step 2.3.2.2

The final answer is $\frac{2 + \sqrt{6}}{3}$ .

$y = \frac{2 + \sqrt{6}}{3}$

Step 2.4

Substitute the $x$ value $0.\overline{6}$ into $f (x) = \frac{2 - \sqrt{- 1 (- 4 - 3 x)}}{3}$ . In this case, the point is $(0.\overline{6}, \frac{2 - \sqrt{6}}{3})$ .

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Step 2.4.1

Replace the variable $x$ with $0.\overline{6}$ in the expression.

$f (0.\overline{6}) = \frac{2 - \sqrt{- 1 (- 4 - 3 \cdot 0.\overline{6})}}{3}$

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 $- 3$ by $0.\overline{6}$ .

$f (0.\overline{6}) = \frac{2 - \sqrt{- 1 (- 4 - 2)}}{3}$

Step 2.4.2.1.2

Subtract $2$ from $- 4$ .

$f (0.\overline{6}) = \frac{2 - \sqrt{- 1 \cdot - 6}}{3}$

Step 2.4.2.1.3

Multiply $- 1$ by $- 6$ .

$f (0.\overline{6}) = \frac{2 - \sqrt{6}}{3}$

Step 2.4.2.2

The final answer is $\frac{2 - \sqrt{6}}{3}$ .

$y = \frac{2 - \sqrt{6}}{3}$

Step 2.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - \frac{4}{3} & \frac{2}{3} \\ - 0.333 & 1.24 \\ - 0.333 & 0.09 \\ 0.667 & 1.48 \\ 0.667 & - 0.15 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- \frac{4}{3}, \frac{2}{3})$

Focus: $(- \frac{5}{4}, \frac{2}{3})$

Axis of Symmetry: $y = \frac{2}{3}$

Directrix: $x = - \frac{17}{12}$

$\begin{array}{c} x & y \\ - \frac{4}{3} & \frac{2}{3} \\ - 0.333 & 1.24 \\ - 0.333 & 0.09 \\ 0.667 & 1.48 \\ 0.667 & - 0.15 \end{array}$

Step 4