Trigonometry Examples

$y = - \frac{1}{6} x^{2} + 7 x - 80$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

$y = - \frac{x^{2}}{6} + 7 x - 80$

Step 1.2

Complete the square for $- \frac{x^{2}}{6} + 7 x - 80$ .

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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 $c$ .

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

$b = 7$

$c = - 80$

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

$d = \frac{7}{2 (- \frac{1}{6})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Simplify the denominator.

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Step 1.2.3.2.1.1

Multiply $- 1$ by $2$ .

$d = \frac{7}{- 2 (\frac{1}{6})}$

Step 1.2.3.2.1.2

Combine $- 2$ and $\frac{1}{6}$ .

$d = \frac{7}{\frac{- 2}{6}}$

Step 1.2.3.2.2

Reduce the expression by cancelling the common factors.

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Step 1.2.3.2.2.1

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

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Step 1.2.3.2.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{7}{\frac{2 (- 1)}{6}}$

Step 1.2.3.2.2.1.2

Cancel the common factors.

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Step 1.2.3.2.2.1.2.1

Factor $2$ out of $6$ .

$d = \frac{7}{\frac{2 \cdot - 1}{2 \cdot 3}}$

Step 1.2.3.2.2.1.2.2

Cancel the common factor.

$d = \frac{7}{\frac{2 \cdot - 1}{2 \cdot 3}}$

Step 1.2.3.2.2.1.2.3

Rewrite the expression.

$d = \frac{7}{\frac{- 1}{3}}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{7}{- \frac{1}{3}}$

Step 1.2.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 7 (- 1 \cdot 3)$

Step 1.2.3.2.4

Multiply $7 (- 1 \cdot 3)$ .

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Step 1.2.3.2.4.1

Multiply $- 1$ by $3$ .

$d = 7 \cdot - 3$

Step 1.2.3.2.4.2

Multiply $7$ by $- 3$ .

$d = - 21$

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

$e = - 80 - \frac{7^{2}}{4 (- \frac{1}{6})}$

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 $7$ to the power of $2$ .

$e = - 80 - \frac{49}{4 (- \frac{1}{6})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - 80 - \frac{49}{- 4 (\frac{1}{6})}$

Step 1.2.4.2.1.2.2

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

$e = - 80 - \frac{49}{\frac{- 4}{6}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

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

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Step 1.2.4.2.1.3.1.1

Factor $2$ out of $- 4$ .

$e = - 80 - \frac{49}{\frac{2 (- 2)}{6}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $2$ out of $6$ .

$e = - 80 - \frac{49}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = - 80 - \frac{49}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = - 80 - \frac{49}{\frac{- 2}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = - 80 - \frac{49}{- \frac{2}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 80 - (49 (- \frac{3}{2}))$

Step 1.2.4.2.1.5

Multiply $49 (- \frac{3}{2})$ .

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Step 1.2.4.2.1.5.1

Multiply $- 1$ by $49$ .

$e = - 80 - (- 49 (\frac{3}{2}))$

Step 1.2.4.2.1.5.2

Combine $- 49$ and $\frac{3}{2}$ .

$e = - 80 - \frac{- 49 \cdot 3}{2}$

Step 1.2.4.2.1.5.3

Multiply $- 49$ by $3$ .

$e = - 80 - \frac{- 147}{2}$

Step 1.2.4.2.1.6

Move the negative in front of the fraction.

$e = - 80 - - \frac{147}{2}$

Step 1.2.4.2.1.7

Multiply $- - \frac{147}{2}$ .

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Step 1.2.4.2.1.7.1

Multiply $- 1$ by $- 1$ .

$e = - 80 + 1 (\frac{147}{2})$

Step 1.2.4.2.1.7.2

Multiply $\frac{147}{2}$ by $1$ .

$e = - 80 + \frac{147}{2}$

Step 1.2.4.2.2

To write $- 80$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 80 \cdot \frac{2}{2} + \frac{147}{2}$

Step 1.2.4.2.3

Combine $- 80$ and $\frac{2}{2}$ .

$e = \frac{- 80 \cdot 2}{2} + \frac{147}{2}$

Step 1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 80 \cdot 2 + 147}{2}$

Step 1.2.4.2.5

Simplify the numerator.

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Step 1.2.4.2.5.1

Multiply $- 80$ by $2$ .

$e = \frac{- 160 + 147}{2}$

Step 1.2.4.2.5.2

Add $- 160$ and $147$ .

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

Step 1.2.4.2.6

Move the negative in front of the fraction.

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

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{6} {(x - 21)}^{2} - \frac{13}{2}$ .

$- \frac{1}{6} {(x - 21)}^{2} - \frac{13}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{1}{6} \cdot {(x - 21)}^{2} - \frac{13}{2}$

Step 2

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

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

$h = 21$

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

Step 3

Find the vertex $(h, k)$ .

$(21, - \frac{13}{2})$

Step 4

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

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Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{6})}$

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{6})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{6})}$

Step 4.3.2

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

$- \frac{1}{\frac{4}{6}}$

Step 4.3.3

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

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Step 4.3.3.1

Factor $2$ out of $4$ .

$- \frac{1}{\frac{2 (2)}{6}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $2$ out of $6$ .

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 4.3.3.2.2

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 2}{2 \cdot 3}}$

Step 4.3.3.2.3

Rewrite the expression.

$- \frac{1}{\frac{2}{3}}$

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{2}))$

Step 4.3.5

Multiply $\frac{3}{2}$ by $1$ .

$- \frac{3}{2}$

Step 5

Find the directrix.

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Step 5.1

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 5.2

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

$y = - 5$

Step 6