Calculus Examples

$x = y^{2} - 5 y$

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 $y^{2} - 5 y$ .

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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 $c$ .

$a = 1$

$b = - 5$

$c = 0$

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

$d = \frac{- 5}{2 \cdot 1}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Multiply $2$ by $1$ .

$d = \frac{- 5}{2}$

Step 1.1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

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

$e = 0 - \frac{{(- 5)}^{2}}{4 \cdot 1}$

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

Raise $- 5$ to the power of $2$ .

$e = 0 - \frac{25}{4 \cdot 1}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{25}{4}$

Step 1.1.1.4.2.2

Subtract $\frac{25}{4}$ from $0$ .

$e = - \frac{25}{4}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{5}{2})}^{2} - \frac{25}{4}$ .

${(y - \frac{5}{2})}^{2} - \frac{25}{4}$

Step 1.1.2

Set $x$ equal to the new right side.

$x = {(y - \frac{5}{2})}^{2} - \frac{25}{4}$

Step 1.2

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

$a = 1$

$h = - \frac{25}{4}$

$k = \frac{5}{2}$

Step 1.3

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

Opens Right

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{25}{4}, \frac{5}{2})$

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.

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

Step 1.5.3

Cancel the common factor of $1$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4}$

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.

$(- 6, \frac{5}{2})$

Step 1.7

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

$y = \frac{5}{2}$

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.

$x = - \frac{13}{2}$

Step 1.9

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

Direction: Opens Right

Vertex: $(- \frac{25}{4}, \frac{5}{2})$

Focus: $(- 6, \frac{5}{2})$

Axis of Symmetry: $y = \frac{5}{2}$

Directrix: $x = - \frac{13}{2}$

Direction: Opens Right

Vertex: $(- \frac{25}{4}, \frac{5}{2})$

Focus: $(- 6, \frac{5}{2})$

Axis of Symmetry: $y = \frac{5}{2}$

Directrix: $x = - \frac{13}{2}$

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 $- 5.25$ into $f (x) = \frac{5 + \sqrt{25 + 4 x}}{2}$ . In this case, the point is $(- 5.25, 3.5)$ .

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Step 2.1.1

Replace the variable $x$ with $- 5.25$ in the expression.

$f (- 5.25) = \frac{5 + \sqrt{25 + 4 (- 5.25)}}{2}$

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 $4$ by $- 5.25$ .

$f (- 5.25) = \frac{5 + \sqrt{25 - 21}}{2}$

Step 2.1.2.1.2

Subtract $21$ from $25$ .

$f (- 5.25) = \frac{5 + \sqrt{4}}{2}$

Step 2.1.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (- 5.25) = \frac{5 + \sqrt{2^{2}}}{2}$

Step 2.1.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (- 5.25) = \frac{5 + 2}{2}$

Step 2.1.2.1.5

Add $5$ and $2$ .

$f (- 5.25) = \frac{7}{2}$

Step 2.1.2.2

Cancel the common factor of $7$ and $2$ .

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Step 2.1.2.2.1

Rewrite $7$ as $1 (7)$ .

$f (- 5.25) = \frac{1 (7)}{2}$

Step 2.1.2.2.2

Cancel the common factors.

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Step 2.1.2.2.2.1

Rewrite $2$ as $1 (2)$ .

$f (- 5.25) = \frac{1 \cdot 7}{1 \cdot 2}$

Step 2.1.2.2.2.2

Cancel the common factor.

$f (- 5.25) = \frac{1 \cdot 7}{1 \cdot 2}$

Step 2.1.2.2.2.3

Rewrite the expression.

$f (- 5.25) = \frac{7}{2}$

Step 2.1.2.3

The final answer is $\frac{7}{2}$ .

$y = \frac{7}{2}$

Step 2.1.3

Convert $\frac{7}{2}$ to decimal.

$= 3.5$

Step 2.2

Substitute the $x$ value $- 5.25$ into $f (x) = \frac{5 - \sqrt{25 + 4 x}}{2}$ . In this case, the point is $(- 5.25, 1.5)$ .

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Step 2.2.1

Replace the variable $x$ with $- 5.25$ in the expression.

$f (- 5.25) = \frac{5 - \sqrt{25 + 4 (- 5.25)}}{2}$

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 $4$ by $- 5.25$ .

$f (- 5.25) = \frac{5 - \sqrt{25 - 21}}{2}$

Step 2.2.2.1.2

Subtract $21$ from $25$ .

$f (- 5.25) = \frac{5 - \sqrt{4}}{2}$

Step 2.2.2.1.3

Rewrite $4$ as $2^{2}$ .

$f (- 5.25) = \frac{5 - \sqrt{2^{2}}}{2}$

Step 2.2.2.1.4

Pull terms out from under the radical, assuming positive real numbers.

$f (- 5.25) = \frac{5 - 1 \cdot 2}{2}$

Step 2.2.2.1.5

Multiply $- 1$ by $2$ .

$f (- 5.25) = \frac{5 - 2}{2}$

Step 2.2.2.1.6

Subtract $2$ from $5$ .

$f (- 5.25) = \frac{3}{2}$

Step 2.2.2.2

Cancel the common factor of $3$ and $2$ .

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Step 2.2.2.2.1

Rewrite $3$ as $1 (3)$ .

$f (- 5.25) = \frac{1 (3)}{2}$

Step 2.2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.2.1

Rewrite $2$ as $1 (2)$ .

$f (- 5.25) = \frac{1 \cdot 3}{1 \cdot 2}$

Step 2.2.2.2.2.2

Cancel the common factor.

$f (- 5.25) = \frac{1 \cdot 3}{1 \cdot 2}$

Step 2.2.2.2.2.3

Rewrite the expression.

$f (- 5.25) = \frac{3}{2}$

Step 2.2.2.3

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

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

Step 2.2.3

Convert $\frac{3}{2}$ to decimal.

$= 1.5$

Step 2.3

Substitute the $x$ value $- 4.25$ into $f (x) = \frac{5 + \sqrt{25 + 4 x}}{2}$ . In this case, the point is $(- 4.25, \frac{5 + \sqrt{8}}{2})$ .

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Step 2.3.1

Replace the variable $x$ with $- 4.25$ in the expression.

$f (- 4.25) = \frac{5 + \sqrt{25 + 4 (- 4.25)}}{2}$

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 $4$ by $- 4.25$ .

$f (- 4.25) = \frac{5 + \sqrt{25 - 17}}{2}$

Step 2.3.2.1.2

Subtract $17$ from $25$ .

$f (- 4.25) = \frac{5 + \sqrt{8}}{2}$

Step 2.3.2.2

The final answer is $\frac{5 + \sqrt{8}}{2}$ .

$y = \frac{5 + \sqrt{8}}{2}$

Step 2.4

Substitute the $x$ value $- 4.25$ into $f (x) = \frac{5 - \sqrt{25 + 4 x}}{2}$ . In this case, the point is $(- 4.25, \frac{5 - \sqrt{8}}{2})$ .

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Step 2.4.1

Replace the variable $x$ with $- 4.25$ in the expression.

$f (- 4.25) = \frac{5 - \sqrt{25 + 4 (- 4.25)}}{2}$

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 $4$ by $- 4.25$ .

$f (- 4.25) = \frac{5 - \sqrt{25 - 17}}{2}$

Step 2.4.2.1.2

Subtract $17$ from $25$ .

$f (- 4.25) = \frac{5 - \sqrt{8}}{2}$

Step 2.4.2.2

The final answer is $\frac{5 - \sqrt{8}}{2}$ .

$y = \frac{5 - \sqrt{8}}{2}$

Step 2.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - \frac{25}{4} & \frac{5}{2} \\ - 5.25 & 3.5 \\ - 5.25 & 1.5 \\ - 4.25 & 3.91 \\ - 4.25 & 1.09 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- \frac{25}{4}, \frac{5}{2})$

Focus: $(- 6, \frac{5}{2})$

Axis of Symmetry: $y = \frac{5}{2}$

Directrix: $x = - \frac{13}{2}$

$\begin{array}{c} x & y \\ - \frac{25}{4} & \frac{5}{2} \\ - 5.25 & 3.5 \\ - 5.25 & 1.5 \\ - 4.25 & 3.91 \\ - 4.25 & 1.09 \end{array}$

Step 4