Trigonometry Examples

x^{2} + y^{2} = 2 y

$x^{2} + y^{2} = 2 y$

Step 1

Subtract

2 y

$2 y$ from both sides of the equation.

x^{2} + y^{2} - 2 y = 0

$x^{2} + y^{2} - 2 y = 0$

Step 2

Complete the square for

y^{2} - 2 y

$y^{2} - 2 y$ .

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Step 2.1

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 = - 2

$b = - 2$

c = 0

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

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

Step 2.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

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

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

Step 2.3.2

Cancel the common factor of

- 2

$- 2$ and

2

$2$ .

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Step 2.3.2.1

Factor

2

$2$ out of

- 2

$- 2$ .

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

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide

$- 1$ by

$1$ .

$d = - 1$

Step 2.4

Find the value of

$e$ using the formula

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

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Step 2.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise

$- 2$ to the power of

$2$ .

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

Step 2.4.2.1.2

Multiply

$4$ by

$1$ .

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

Step 2.4.2.1.3

Cancel the common factor of

$4$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

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

Step 2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.4

Multiply

$- 1$ by

$1$ .

$e = 0 - 1$

Step 2.4.2.2

Subtract

$1$ from

$0$ .

$e = - 1$

Step 2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(y - 1)}^{2} - 1$ .

${(y - 1)}^{2} - 1$

Step 3

Substitute

${(y - 1)}^{2} - 1$ for

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

$x^{2} + y^{2} - 2 y = 0$ .

$x^{2} + {(y - 1)}^{2} - 1 = 0$

Step 4

Move

$- 1$ to the right side of the equation by adding

$1$ to both sides.

$x^{2} + {(y - 1)}^{2} = 0 + 1$

Step 5

Add

$0$ and

$1$ .

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

Step 6

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 7

Match the values in this circle to those of the standard form. The variable

$r$ represents the radius of the circle,

$h$ represents the x-offset from the origin, and

$k$ represents the y-offset from origin.

$r = 1$

$h = 0$

$k = 1$

Step 8

The center of the circle is found at

$(h, k)$ .

Center:

$(0, 1)$

Step 9

These values represent the important values for graphing and analyzing a circle.

Center:

$(0, 1)$

Radius:

$1$

Step 10