Trigonometry Examples

${(y - 3)}^{2} = 20 (x + 1)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as $20 (x + 1) = {(y - 3)}^{2}$ .

$20 (x + 1) = {(y - 3)}^{2}$

Step 1.1.2

Divide each term in $20 (x + 1) = {(y - 3)}^{2}$ by $20$ and simplify.

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Step 1.1.2.1

Divide each term in $20 (x + 1) = {(y - 3)}^{2}$ by $20$ .

$\frac{20 (x + 1)}{20} = \frac{{(y - 3)}^{2}}{20}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $20$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{20 (x + 1)}{20} = \frac{{(y - 3)}^{2}}{20}$

Step 1.1.2.2.1.2

Divide $x + 1$ by $1$ .

$x + 1 = \frac{{(y - 3)}^{2}}{20}$

Step 1.1.3

Subtract $1$ from both sides of the equation.

$x = \frac{{(y - 3)}^{2}}{20} - 1$

Step 1.1.4

Reorder terms.

$x = \frac{1}{20} \cdot {(y - 3)}^{2} - 1$

Step 1.2

Complete the square for $\frac{1}{20} \cdot {(y - 3)}^{2} - 1$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$\frac{1}{20} \cdot ((y - 3) (y - 3)) - 1$

Step 1.2.1.1.2

Expand $(y - 3) (y - 3)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{20} \cdot (y (y - 3) - 3 (y - 3)) - 1$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 3 - 3 (y - 3)) - 1$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3) - 1$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

$\frac{1}{20} \cdot (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3) - 1$

Step 1.2.1.1.3.1.2

Move $- 3$ to the left of $y$ .

$\frac{1}{20} \cdot (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3) - 1$

Step 1.2.1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$\frac{1}{20} \cdot (y^{2} - 3 y - 3 y + 9) - 1$

Step 1.2.1.1.3.2

Subtract $3 y$ from $- 3 y$ .

$\frac{1}{20} \cdot (y^{2} - 6 y + 9) - 1$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{20} y^{2} + \frac{1}{20} (- 6 y) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Combine $\frac{1}{20}$ and $y^{2}$ .

$\frac{y^{2}}{20} + \frac{1}{20} (- 6 y) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.2

Cancel the common factor of $2$ .

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Step 1.2.1.1.5.2.1

Factor $2$ out of $20$ .

$\frac{y^{2}}{20} + \frac{1}{2 (10)} (- 6 y) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.2.2

Factor $2$ out of $- 6 y$ .

$\frac{y^{2}}{20} + \frac{1}{2 (10)} (2 (- 3 y)) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{y^{2}}{20} + \frac{1}{2 \cdot 10} (2 (- 3 y)) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{y^{2}}{20} + \frac{1}{10} (- 3 y) + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.3

Combine $- 3$ and $\frac{1}{10}$ .

$\frac{y^{2}}{20} + \frac{- 3}{10} y + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.4

Combine $\frac{- 3}{10}$ and $y$ .

$\frac{y^{2}}{20} + \frac{- 3 y}{10} + \frac{1}{20} \cdot 9 - 1$

Step 1.2.1.1.5.5

Combine $\frac{1}{20}$ and $9$ .

$\frac{y^{2}}{20} + \frac{- 3 y}{10} + \frac{9}{20} - 1$

Step 1.2.1.1.6

Move the negative in front of the fraction.

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{9}{20} - 1$

Step 1.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{9}{20} - 1 \cdot \frac{20}{20}$

Step 1.2.1.3

Combine $- 1$ and $\frac{20}{20}$ .

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{9}{20} + \frac{- 1 \cdot 20}{20}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{9 - 1 \cdot 20}{20}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 1$ by $20$ .

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{9 - 20}{20}$

Step 1.2.1.5.2

Subtract $20$ from $9$ .

$\frac{y^{2}}{20} - \frac{3 y}{10} + \frac{- 11}{20}$

Step 1.2.1.6

Move the negative in front of the fraction.

$\frac{y^{2}}{20} - \frac{3 y}{10} - \frac{11}{20}$

Step 1.2.2

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

$a = \frac{1}{20}$

$b = - \frac{3}{10}$

$c = - \frac{11}{20}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

$d = \frac{- \frac{3}{10}}{2 (\frac{1}{20})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{10} \cdot \frac{1}{2 (\frac{1}{20})}$

Step 1.2.4.2.2

Combine $2$ and $\frac{1}{20}$ .

$d = - \frac{3}{10} \cdot \frac{1}{\frac{2}{20}}$

Step 1.2.4.2.3

Cancel the common factor of $2$ and $20$ .

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Step 1.2.4.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{3}{10} \cdot \frac{1}{\frac{2 (1)}{20}}$

Step 1.2.4.2.3.2

Cancel the common factors.

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Step 1.2.4.2.3.2.1

Factor $2$ out of $20$ .

$d = - \frac{3}{10} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}}$

Step 1.2.4.2.3.2.2

Cancel the common factor.

$d = - \frac{3}{10} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}}$

Step 1.2.4.2.3.2.3

Rewrite the expression.

$d = - \frac{3}{10} \cdot \frac{1}{\frac{1}{10}}$

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{3}{10} (1 \cdot 10)$

Step 1.2.4.2.5

Cancel the common factor of $10$ .

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Step 1.2.4.2.5.1

Move the leading negative in $- \frac{3}{10}$ into the numerator.

$d = \frac{- 3}{10} (1 \cdot 10)$

Step 1.2.4.2.5.2

Factor $10$ out of $1 \cdot 10$ .

$d = \frac{- 3}{10} (10 \cdot 1)$

Step 1.2.4.2.5.3

Cancel the common factor.

$d = \frac{- 3}{10} (10 \cdot 1)$

Step 1.2.4.2.5.4

Rewrite the expression.

$d = - 3$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{11}{20} - \frac{{(- \frac{3}{10})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $- \frac{3}{10}$ .

$e = - \frac{11}{20} - \frac{{(- 1)}^{2} {(\frac{3}{10})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.2

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

$e = - \frac{11}{20} - \frac{1 {(\frac{3}{10})}^{2}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.3

Apply the product rule to $\frac{3}{10}$ .

$e = - \frac{11}{20} - \frac{1 \frac{3^{2}}{10^{2}}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.4

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

$e = - \frac{11}{20} - \frac{1 \frac{9}{10^{2}}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.5

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

$e = - \frac{11}{20} - \frac{1 (\frac{9}{100})}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.1.6

Multiply $\frac{9}{100}$ by $1$ .

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{4 (\frac{1}{20})}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{1}{20}$ .

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{\frac{4}{20}}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ and $20$ .

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{\frac{4 (1)}{20}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $4$ out of $20$ .

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = - \frac{11}{20} - \frac{\frac{9}{100}}{\frac{1}{5}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{11}{20} - (\frac{9}{100} \cdot 5)$

Step 1.2.5.2.1.5

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.5.1

Factor $5$ out of $100$ .

$e = - \frac{11}{20} - (\frac{9}{5 (20)} \cdot 5)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = - \frac{11}{20} - (\frac{9}{5 \cdot 20} \cdot 5)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = - \frac{11}{20} - \frac{9}{20}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 11 - 9}{20}$

Step 1.2.5.2.3

Subtract $9$ from $- 11$ .

$e = \frac{- 20}{20}$

Step 1.2.5.2.4

Divide $- 20$ by $20$ .

$e = - 1$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{20} {(y - 3)}^{2} - 1$ .

$\frac{1}{20} {(y - 3)}^{2} - 1$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{20} \cdot {(y - 3)}^{2} - 1$

Step 2

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

$a = \frac{1}{20}$

$h = - 1$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 3)$

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.

$\frac{1}{4 \cdot \frac{1}{20}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine $4$ and $\frac{1}{20}$ .

$\frac{1}{\frac{4}{20}}$

Step 4.3.2

Cancel the common factor of $4$ and $20$ .

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Step 4.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{20}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor $4$ out of $20$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 5}}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{5}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 5$

Step 4.3.4

Multiply $5$ by $1$ .

$5$

Step 5

Find the directrix.

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Step 5.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 5.2

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

$x = - 6$

Step 6