Trigonometry Examples

y^{2} - 4 y - 4 x^{2} + 8 x = 4

$y^{2} - 4 y - 4 x^{2} + 8 x = 4$

Step 1

Complete the square for

y^{2} - 4 y

$y^{2} - 4 y$ .

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Step 1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = - 4

$b = - 4$

c = 0

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

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

Step 1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 4}{2 \cdot 1}

$d = \frac{- 4}{2 \cdot 1}$

Step 1.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 1.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

d = \frac{2 \cdot - 2}{2 \cdot 1}

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

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot - 2}{2 (1)}

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

Step 1.3.2.2.2

Cancel the common factor.

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

Step 1.3.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 1.3.2.2.4

Divide

$- 2$ by

$1$ .

$d = - 2$

Step 1.4

Find the value of

$e$ using the formula

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

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Step 1.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Cancel the common factor of

${(- 4)}^{2}$ and

$4$ .

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Step 1.4.2.1.1.1

Rewrite

$- 4$ as

$- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 1.4.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 1.4.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 1.4.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 1}$

Step 1.4.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.4.2.1.1.6

Cancel the common factors.

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Step 1.4.2.1.1.6.1

Factor

$4$ out of

$4 \cdot 1$ .

$e = 0 - \frac{4 \cdot 4}{4 (1)}$

Step 1.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{1}$

Step 1.4.2.1.1.6.4

Divide

$4$ by

$1$ .

$e = 0 - 1 \cdot 4$

Step 1.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = 0 - 4$

Step 1.4.2.2

Subtract

$4$ from

$0$ .

$e = - 4$

Step 1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(y - 2)}^{2} - 4$ .

${(y - 2)}^{2} - 4$

Step 2

Substitute

${(y - 2)}^{2} - 4$ for

$y^{2} - 4 y$ in the equation

$y^{2} - 4 y - 4 x^{2} + 8 x = 4$ .

${(y - 2)}^{2} - 4 - 4 x^{2} + 8 x = 4$

Step 3

Move

$- 4$ to the right side of the equation by adding

$4$ to both sides.

${(y - 2)}^{2} - 4 x^{2} + 8 x = 4 + 4$

Step 4

Complete the square for

$- 4 x^{2} + 8 x$ .

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Step 4.1

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = - 4$

$b = 8$

$c = 0$

Step 4.2

Consider the vertex form of a parabola.

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

Step 4.3

Find the value of

$d$ using the formula

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

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Step 4.3.1

Substitute the values of

$a$ and

$b$ into the formula

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

$d = \frac{8}{2 \cdot - 4}$

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

Cancel the common factor of

$8$ and

$2$ .

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Step 4.3.2.1.1

Factor

$2$ out of

$8$ .

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Step 4.3.2.1.2

Cancel the common factors.

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Step 4.3.2.1.2.1

Factor

$2$ out of

$2 \cdot - 4$ .

$d = \frac{2 \cdot 4}{2 (- 4)}$

Step 4.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot - 4}$

Step 4.3.2.1.2.3

Rewrite the expression.

$d = \frac{4}{- 4}$

Step 4.3.2.2

Cancel the common factor of

$4$ and

$- 4$ .

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Step 4.3.2.2.1

Factor

$4$ out of

$4$ .

$d = \frac{4 (1)}{- 4}$

Step 4.3.2.2.2

Move the negative one from the denominator of

$\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Step 4.3.2.3

Multiply

$- 1$ by

$1$ .

$d = - 1$

Step 4.4

Find the value of

$e$ using the formula

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

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Step 4.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

$e = 0 - \frac{8^{2}}{4 \cdot - 4}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Raise

$8$ to the power of

$2$ .

$e = 0 - \frac{64}{4 \cdot - 4}$

Step 4.4.2.1.2

Multiply

$4$ by

$- 4$ .

$e = 0 - \frac{64}{- 16}$

Step 4.4.2.1.3

Divide

$64$ by

$- 16$ .

$e = 0 - - 4$

Step 4.4.2.1.4

Multiply

$- 1$ by

$- 4$ .

$e = 0 + 4$

Step 4.4.2.2

Add

$0$ and

$4$ .

$e = 4$

Step 4.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$- 4 {(x - 1)}^{2} + 4$ .

$- 4 {(x - 1)}^{2} + 4$

Step 5

Substitute

$- 4 {(x - 1)}^{2} + 4$ for

$- 4 x^{2} + 8 x$ in the equation

$y^{2} - 4 y - 4 x^{2} + 8 x = 4$ .

${(y - 2)}^{2} - 4 {(x - 1)}^{2} + 4 = 4 + 4$

Step 6

Move

$4$ to the right side of the equation by adding

$4$ to both sides.

${(y - 2)}^{2} - 4 {(x - 1)}^{2} = 4 + 4 - 4$

Step 7

Simplify

$4 + 4 - 4$ .

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Step 7.1

Add

$4$ and

$4$ .

${(y - 2)}^{2} - 4 {(x - 1)}^{2} = 8 - 4$

Step 7.2

Subtract

$4$ from

$8$ .

${(y - 2)}^{2} - 4 {(x - 1)}^{2} = 4$

Step 8

Divide each term by

$4$ to make the right side equal to one.

$\frac{{(y - 2)}^{2}}{4} - \frac{4 {(x - 1)}^{2}}{4} = \frac{4}{4}$

Step 9

Simplify each term in the equation in order to set the right side equal to

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

$1$ .

$\frac{{(y - 2)}^{2}}{4} - \frac{{(x - 1)}^{2}}{1} = 1$