Calculus Examples

$f (x) = 2 x^{2} + 12 x - 8$

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 $2 x^{2} + 12 x - 8$ .

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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 $c$ .

$a = 2$

$b = 12$

$c = - 8$

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

$d = \frac{12}{2 \cdot 2}$

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 $12$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot 2}$

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 $2 \cdot 2$ .

$d = \frac{2 \cdot 6}{2 (2)}$

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot 2}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{6}{2}$

Step 1.1.1.3.2.2

Cancel the common factor of $6$ and $2$ .

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Step 1.1.1.3.2.2.1

Factor $2$ out of $6$ .

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

Step 1.1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 1.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot 1}$

Step 1.1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.1.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

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

$e = - 8 - \frac{12^{2}}{4 \cdot 2}$

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 $12$ to the power of $2$ .

$e = - 8 - \frac{144}{4 \cdot 2}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = - 8 - \frac{144}{8}$

Step 1.1.1.4.2.1.3

Divide $144$ by $8$ .

$e = - 8 - 1 \cdot 18$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $18$ .

$e = - 8 - 18$

Step 1.1.1.4.2.2

Subtract $18$ from $- 8$ .

$e = - 26$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 3)}^{2} - 26$ .

$2 {(x + 3)}^{2} - 26$

Step 1.1.2

Set $y$ equal to the new right side.

$y = 2 {(x + 3)}^{2} - 26$

Step 1.2

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

$a = 2$

$h = - 3$

$k = - 26$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 3, - 26)$

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.

$\frac{1}{4 \cdot 2}$

Step 1.5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

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.

$(- 3, - \frac{207}{8})$

Step 1.7

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

$x = - 3$

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.

$y = - \frac{209}{8}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 3, - 26)$

Focus: $(- 3, - \frac{207}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{209}{8}$

Direction: Opens Up

Vertex: $(- 3, - 26)$

Focus: $(- 3, - \frac{207}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{209}{8}$

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 $- 4$ in the expression.

$f (- 4) = 2 {(- 4)}^{2} + 12 (- 4) - 8$

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 $- 4$ to the power of $2$ .

$f (- 4) = 2 \cdot 16 + 12 (- 4) - 8$

Step 2.2.1.2

Multiply $2$ by $16$ .

$f (- 4) = 32 + 12 (- 4) - 8$

Step 2.2.1.3

Multiply $12$ by $- 4$ .

$f (- 4) = 32 - 48 - 8$

Step 2.2.2

Simplify by subtracting numbers.

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Step 2.2.2.1

Subtract $48$ from $32$ .

$f (- 4) = - 16 - 8$

Step 2.2.2.2

Subtract $8$ from $- 16$ .

$f (- 4) = - 24$

Step 2.2.3

The final answer is $- 24$ .

$- 24$

Step 2.3

The $y$ value at $x = - 4$ is $- 24$ .

$y = - 24$

Step 2.4

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = 2 {(- 5)}^{2} + 12 (- 5) - 8$

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

$f (- 5) = 2 \cdot 25 + 12 (- 5) - 8$

Step 2.5.1.2

Multiply $2$ by $25$ .

$f (- 5) = 50 + 12 (- 5) - 8$

Step 2.5.1.3

Multiply $12$ by $- 5$ .

$f (- 5) = 50 - 60 - 8$

Step 2.5.2

Simplify by subtracting numbers.

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Step 2.5.2.1

Subtract $60$ from $50$ .

$f (- 5) = - 10 - 8$

Step 2.5.2.2

Subtract $8$ from $- 10$ .

$f (- 5) = - 18$

Step 2.5.3

The final answer is $- 18$ .

$- 18$

Step 2.6

The $y$ value at $x = - 5$ is $- 18$ .

$y = - 18$

Step 2.7

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

$f (- 2) = 2 {(- 2)}^{2} + 12 (- 2) - 8$

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

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

$f (- 2) = 2 \cdot 4 + 12 (- 2) - 8$

Step 2.8.1.2

Multiply $2$ by $4$ .

$f (- 2) = 8 + 12 (- 2) - 8$

Step 2.8.1.3

Multiply $12$ by $- 2$ .

$f (- 2) = 8 - 24 - 8$

Step 2.8.2

Simplify by subtracting numbers.

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Step 2.8.2.1

Subtract $24$ from $8$ .

$f (- 2) = - 16 - 8$

Step 2.8.2.2

Subtract $8$ from $- 16$ .

$f (- 2) = - 24$

Step 2.8.3

The final answer is $- 24$ .

$- 24$

Step 2.9

The $y$ value at $x = - 2$ is $- 24$ .

$y = - 24$

Step 2.10

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

$f (- 1) = 2 {(- 1)}^{2} + 12 (- 1) - 8$

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

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

$f (- 1) = 2 \cdot 1 + 12 (- 1) - 8$

Step 2.11.1.2

Multiply $2$ by $1$ .

$f (- 1) = 2 + 12 (- 1) - 8$

Step 2.11.1.3

Multiply $12$ by $- 1$ .

$f (- 1) = 2 - 12 - 8$

Step 2.11.2

Simplify by subtracting numbers.

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Step 2.11.2.1

Subtract $12$ from $2$ .

$f (- 1) = - 10 - 8$

Step 2.11.2.2

Subtract $8$ from $- 10$ .

$f (- 1) = - 18$

Step 2.11.3

The final answer is $- 18$ .

$- 18$

Step 2.12

The $y$ value at $x = - 1$ is $- 18$ .

$y = - 18$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & - 18 \\ - 4 & - 24 \\ - 3 & - 26 \\ - 2 & - 24 \\ - 1 & - 18 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, - 26)$

Focus: $(- 3, - \frac{207}{8})$

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{209}{8}$

$\begin{array}{c} x & y \\ - 5 & - 18 \\ - 4 & - 24 \\ - 3 & - 26 \\ - 2 & - 24 \\ - 1 & - 18 \end{array}$

Step 4