Calculus Examples

$\frac{x^{3} - 8}{x - 2}$

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 $x^{2} + 2 x + 4$ .

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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 $c$ .

$a = 1$

$b = 2$

$c = 4$

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

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

Step 1.1.1.3.2

Cancel the common factor of $2$ .

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Step 1.1.1.3.2.1

Cancel the common factor.

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

Step 1.1.1.3.2.2

Rewrite the expression.

$d = 1$

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

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

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 $2$ to the power of $2$ .

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

Step 1.1.1.4.2.1.2

Multiply $4$ by $1$ .

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

Step 1.1.1.4.2.1.3

Cancel the common factor of $4$ .

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Step 1.1.1.4.2.1.3.1

Cancel the common factor.

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

Step 1.1.1.4.2.1.3.2

Rewrite the expression.

$e = 4 - 1 \cdot 1$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 4 - 1$

Step 1.1.1.4.2.2

Subtract $1$ from $4$ .

$e = 3$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 1)}^{2} + 3$ .

${(x + 1)}^{2} + 3$

Step 1.1.2

Set $y$ equal to the new right side.

$y = {(x + 1)}^{2} + 3$

Step 1.2

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

$a = 1$

$h = - 1$

$k = 3$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 1, 3)$

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.

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

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

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

$(- 1, \frac{13}{4})$

Step 1.7

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

$x = - 1$

Step 1.8

Find the directrix.

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Step 1.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 1.8.2

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

$y = \frac{11}{4}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{13}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{11}{4}$

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{13}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{11}{4}$

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

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

$f (- 2) = {(- 2)}^{2} + 2 (- 2) + 4$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 2) = 4 + 2 (- 2) + 4$

Step 2.2.1.2

Multiply $2$ by $- 2$ .

$f (- 2) = 4 - 4 + 4$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $4$ from $4$ .

$f (- 2) = 0 + 4$

Step 2.2.2.2

Add $0$ and $4$ .

$f (- 2) = 4$

Step 2.2.3

The final answer is $4$ .

$4$

Step 2.3

The $y$ value at $x = - 2$ is $4$ .

$y = 4$

Step 2.4

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{2} + 2 (- 3) + 4$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = 9 + 2 (- 3) + 4$

Step 2.5.1.2

Multiply $2$ by $- 3$ .

$f (- 3) = 9 - 6 + 4$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $6$ from $9$ .

$f (- 3) = 3 + 4$

Step 2.5.2.2

Add $3$ and $4$ .

$f (- 3) = 7$

Step 2.5.3

The final answer is $7$ .

$7$

Step 2.6

The $y$ value at $x = - 3$ is $7$ .

$y = 7$

Step 2.7

Replace the variable $x$ with $0$ in the expression.

$f (0) = {(0)}^{2} + 2 (0) + 4$

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

$f (0) = 0 + 2 (0) + 4$

Step 2.8.1.2

Multiply $2$ by $0$ .

$f (0) = 0 + 0 + 4$

Step 2.8.2

Simplify by adding numbers.

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Step 2.8.2.1

Add $0$ and $0$ .

$f (0) = 0 + 4$

Step 2.8.2.2

Add $0$ and $4$ .

$f (0) = 4$

Step 2.8.3

The final answer is $4$ .

$4$

Step 2.9

The $y$ value at $x = 0$ is $4$ .

$y = 4$

Step 2.10

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

$f (1) = {(1)}^{2} + 2 (1) + 4$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

One to any power is one.

$f (1) = 1 + 2 (1) + 4$

Step 2.11.1.2

Multiply $2$ by $1$ .

$f (1) = 1 + 2 + 4$

Step 2.11.2

Simplify by adding numbers.

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Step 2.11.2.1

Add $1$ and $2$ .

$f (1) = 3 + 4$

Step 2.11.2.2

Add $3$ and $4$ .

$f (1) = 7$

Step 2.11.3

The final answer is $7$ .

$7$

Step 2.12

The $y$ value at $x = 1$ is $7$ .

$y = 7$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & 7 \\ - 2 & 4 \\ - 1 & 3 \\ 0 & 4 \\ 1 & 7 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 1, 3)$

Focus: $(- 1, \frac{13}{4})$

Axis of Symmetry: $x = - 1$

Directrix: $y = \frac{11}{4}$

$\begin{array}{c} x & y \\ - 3 & 7 \\ - 2 & 4 \\ - 1 & 3 \\ 0 & 4 \\ 1 & 7 \end{array}$

Step 4