Trigonometry Examples

${(x + 8)}^{2} + 2$

Step 1

Find the properties of the given parabola.

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Step 1.1

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

$a = 1$

$h = - 8$

$k = 2$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(- 8, 2)$

Step 1.4

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

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Step 1.4.1

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

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of $a$ into the formula.

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

Step 1.4.3

Cancel the common factor of $1$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 1.5

Find the focus.

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Step 1.5.1

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 1.5.2

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

$(- 8, \frac{9}{4})$

Step 1.6

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

$x = - 8$

Step 1.7

Find the directrix.

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Step 1.7.1

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 1.7.2

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

$y = \frac{7}{4}$

Step 1.8

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

Direction: Opens Up

Vertex: $(- 8, 2)$

Focus: $(- 8, \frac{9}{4})$

Axis of Symmetry: $x = - 8$

Directrix: $y = \frac{7}{4}$

Direction: Opens Up

Vertex: $(- 8, 2)$

Focus: $(- 8, \frac{9}{4})$

Axis of Symmetry: $x = - 8$

Directrix: $y = \frac{7}{4}$

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 $- 9$ in the expression.

$f (- 9) = {(- 9)}^{2} + 16 (- 9) + 66$

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 $- 9$ to the power of $2$ .

$f (- 9) = 81 + 16 (- 9) + 66$

Step 2.2.1.2

Multiply $16$ by $- 9$ .

$f (- 9) = 81 - 144 + 66$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $144$ from $81$ .

$f (- 9) = - 63 + 66$

Step 2.2.2.2

Add $- 63$ and $66$ .

$f (- 9) = 3$

Step 2.2.3

The final answer is $3$ .

$3$

Step 2.3

The $y$ value at $x = - 9$ is $3$ .

$y = 3$

Step 2.4

Replace the variable $x$ with $- 10$ in the expression.

$f (- 10) = {(- 10)}^{2} + 16 (- 10) + 66$

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 $- 10$ to the power of $2$ .

$f (- 10) = 100 + 16 (- 10) + 66$

Step 2.5.1.2

Multiply $16$ by $- 10$ .

$f (- 10) = 100 - 160 + 66$

Step 2.5.2

Simplify by adding and subtracting.

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Step 2.5.2.1

Subtract $160$ from $100$ .

$f (- 10) = - 60 + 66$

Step 2.5.2.2

Add $- 60$ and $66$ .

$f (- 10) = 6$

Step 2.5.3

The final answer is $6$ .

$6$

Step 2.6

The $y$ value at $x = - 10$ is $6$ .

$y = 6$

Step 2.7

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = {(- 7)}^{2} + 16 (- 7) + 66$

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 $- 7$ to the power of $2$ .

$f (- 7) = 49 + 16 (- 7) + 66$

Step 2.8.1.2

Multiply $16$ by $- 7$ .

$f (- 7) = 49 - 112 + 66$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $112$ from $49$ .

$f (- 7) = - 63 + 66$

Step 2.8.2.2

Add $- 63$ and $66$ .

$f (- 7) = 3$

Step 2.8.3

The final answer is $3$ .

$3$

Step 2.9

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

$y = 3$

Step 2.10

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = {(- 6)}^{2} + 16 (- 6) + 66$

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 $- 6$ to the power of $2$ .

$f (- 6) = 36 + 16 (- 6) + 66$

Step 2.11.1.2

Multiply $16$ by $- 6$ .

$f (- 6) = 36 - 96 + 66$

Step 2.11.2

Simplify by adding and subtracting.

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Step 2.11.2.1

Subtract $96$ from $36$ .

$f (- 6) = - 60 + 66$

Step 2.11.2.2

Add $- 60$ and $66$ .

$f (- 6) = 6$

Step 2.11.3

The final answer is $6$ .

$6$

Step 2.12

The $y$ value at $x = - 6$ is $6$ .

$y = 6$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 10 & 6 \\ - 9 & 3 \\ - 8 & 2 \\ - 7 & 3 \\ - 6 & 6 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 8, 2)$

Focus: $(- 8, \frac{9}{4})$

Axis of Symmetry: $x = - 8$

Directrix: $y = \frac{7}{4}$

$\begin{array}{c} x & y \\ - 10 & 6 \\ - 9 & 3 \\ - 8 & 2 \\ - 7 & 3 \\ - 6 & 6 \end{array}$

Step 4