Trigonometry Examples

$y^{2} = - \frac{9}{8} x$

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 $- \frac{9}{8} x = y^{2}$ .

$- \frac{9}{8} x = y^{2}$

Step 1.1.2

Multiply both sides of the equation by $- \frac{8}{9}$ .

$- \frac{8}{9} (- \frac{9}{8} x) = - \frac{8}{9} y^{2}$

Step 1.1.3

Simplify both sides of the equation.

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Step 1.1.3.1

Simplify the left side.

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Step 1.1.3.1.1

Simplify $- \frac{8}{9} (- \frac{9}{8} x)$ .

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Step 1.1.3.1.1.1

Combine $x$ and $\frac{9}{8}$ .

$- \frac{8}{9} (- \frac{x \cdot 9}{8}) = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.2

Cancel the common factor of $8$ .

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Step 1.1.3.1.1.2.1

Move the leading negative in $- \frac{8}{9}$ into the numerator.

$\frac{- 8}{9} (- \frac{x \cdot 9}{8}) = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.2.2

Move the leading negative in $- \frac{x \cdot 9}{8}$ into the numerator.

$\frac{- 8}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.2.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.2.4

Cancel the common factor.

$\frac{8 \cdot - 1}{9} \cdot \frac{- x \cdot 9}{8} = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.2.5

Rewrite the expression.

$\frac{- 1}{9} (- x \cdot 9) = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.3

Cancel the common factor of $9$ .

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Step 1.1.3.1.1.3.1

Factor $9$ out of $- x \cdot 9$ .

$\frac{- 1}{9} (9 \cdot (- x)) = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.3.2

Cancel the common factor.

$\frac{- 1}{9} (9 \cdot (- x)) = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.3.3

Rewrite the expression.

$- - x = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.4

Multiply.

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Step 1.1.3.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{8}{9} y^{2}$

Step 1.1.3.1.1.4.2

Multiply $x$ by $1$ .

$x = - \frac{8}{9} y^{2}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Simplify $- \frac{8}{9} y^{2}$ .

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Step 1.1.3.2.1.1

Combine $y^{2}$ and $\frac{8}{9}$ .

$x = - \frac{y^{2} \cdot 8}{9}$

Step 1.1.3.2.1.2

Move $8$ to the left of $y^{2}$ .

$x = - \frac{8 y^{2}}{9}$

Step 1.2

Complete the square for $- \frac{8 y^{2}}{9}$ .

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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 $c$ .

$a = - \frac{8}{9}$

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (- \frac{8}{9})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{8}{9})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{8}{9})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{8}{9}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- \frac{9}{8})$

Step 1.2.3.2.3

Multiply $0 (- \frac{9}{8})$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $0$ .

$d = 0 (\frac{9}{8})$

Step 1.2.3.2.3.2

Multiply $0$ by $\frac{9}{8}$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (- \frac{8}{9})}$

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

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 (- \frac{8}{9})}$

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 $4$ .

$e = 0 - \frac{0}{- 4 (\frac{8}{9})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{8}{9}$ .

$e = 0 - \frac{0}{\frac{- 4 \cdot 8}{9}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $8$ .

$e = 0 - \frac{0}{\frac{- 32}{9}}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{32}{9}}$

Step 1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- \frac{9}{32}))$

Step 1.2.4.2.1.6

Multiply $0 (- \frac{9}{32})$ .

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Step 1.2.4.2.1.6.1

Multiply $- 1$ by $0$ .

$e = 0 - (0 (\frac{9}{32}))$

Step 1.2.4.2.1.6.2

Multiply $0$ by $\frac{9}{32}$ .

$e = 0 - 0$

Step 1.2.4.2.1.7

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{8}{9} y^{2}$ .

$- \frac{8}{9} y^{2}$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{8}{9} y^{2}$

Step 2

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

$a = - \frac{8}{9}$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

Step 4