Trigonometry Examples

$y = \frac{1}{6} x^{2} + 2 x + 11$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Combine $\frac{1}{6}$ and $x^{2}$ .

$y = \frac{x^{2}}{6} + 2 x + 11$

Step 1.2

Complete the square for $\frac{x^{2}}{6} + 2 x + 11$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{1}{6}$

$b = 2$

$c = 11$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 (\frac{1}{6})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$d = \frac{2}{2 (\frac{1}{6})}$

Step 1.2.3.2.1.2

Rewrite the expression.

$d = \frac{1}{\frac{1}{6}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 6$

Step 1.2.3.2.3

Multiply $6$ by $1$ .

$d = 6$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 11 - \frac{2^{2}}{4 (\frac{1}{6})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

$e = 11 - \frac{4}{4 (\frac{1}{6})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{6}$ .

$e = 11 - \frac{4}{\frac{4}{6}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $6$ .

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Step 1.2.4.2.1.3.1

Factor $2$ out of $4$ .

$e = 11 - \frac{4}{\frac{2 (2)}{6}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $2$ out of $6$ .

$e = 11 - \frac{4}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = 11 - \frac{4}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = 11 - \frac{4}{\frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 11 - (4 (\frac{3}{2}))$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Factor $2$ out of $4$ .

$e = 11 - (2 (2) \frac{3}{2})$

Step 1.2.4.2.1.5.2

Cancel the common factor.

$e = 11 - (2 \cdot 2 \frac{3}{2})$

Step 1.2.4.2.1.5.3

Rewrite the expression.

$e = 11 - (2 \cdot 3)$

Step 1.2.4.2.1.6

Multiply $2$ by $3$ .

$e = 11 - 1 \cdot 6$

Step 1.2.4.2.1.7

Multiply $- 1$ by $6$ .

$e = 11 - 6$

Step 1.2.4.2.2

Subtract $6$ from $11$ .

$e = 5$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{6} {(x + 6)}^{2} + 5$ .

$\frac{1}{6} {(x + 6)}^{2} + 5$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{6} \cdot {(x + 6)}^{2} + 5$

Step 2

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

$a = \frac{1}{6}$

$h = - 6$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(- 6, 5)$

Step 4