Precalculus Examples

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

Step 1

Complete the square for $y^{2} - \frac{1}{2} y$ .

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Step 1.1

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

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

Step 1.2

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

$a = 1$

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

$c = 0$

Step 1.3

Consider the vertex form of a parabola.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

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

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Step 1.4.2.1.1

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

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

Step 1.4.2.1.2

Cancel the common factor.

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

Step 1.4.2.1.3

Rewrite the expression.

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

Step 1.4.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4.2.3

Multiply $- \frac{1}{2} \cdot \frac{1}{2}$ .

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Step 1.4.2.3.1

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

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

Step 1.4.2.3.2

Multiply $2$ by $2$ .

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

Simplify the numerator.

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Step 1.5.2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 1.5.2.1.1.2

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

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

Step 1.5.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.5.2.1.1.4

One to any power is one.

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

Step 1.5.2.1.1.5

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

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

Step 1.5.2.1.1.6

Multiply $\frac{1}{4}$ by $1$ .

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

Step 1.5.2.1.2

Multiply $4$ by $1$ .

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

Step 1.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5.2.1.4

Multiply $\frac{1}{4} \cdot \frac{1}{4}$ .

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Step 1.5.2.1.4.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 1.5.2.1.4.2

Multiply $4$ by $4$ .

$e = 0 - \frac{1}{16}$

Step 1.5.2.2

Subtract $\frac{1}{16}$ from $0$ .

$e = - \frac{1}{16}$

Step 1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{4})}^{2} - \frac{1}{16}$ .

${(y - \frac{1}{4})}^{2} - \frac{1}{16}$

Step 2

Substitute ${(y - \frac{1}{4})}^{2} - \frac{1}{16}$ for $y^{2} - \frac{1}{2} y$ in the equation $x^{2} + y^{2} - \frac{1}{2} y = \frac{1}{8}$ .

$x^{2} + {(y - \frac{1}{4})}^{2} - \frac{1}{16} = \frac{1}{8}$

Step 3

Move $- \frac{1}{16}$ to the right side of the equation by adding $\frac{1}{16}$ to both sides.

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{1}{8} + \frac{1}{16}$

Step 4

Simplify $\frac{1}{8} + \frac{1}{16}$ .

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Step 4.1

To write $\frac{1}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{1}{8} \cdot \frac{2}{2} + \frac{1}{16}$

Step 4.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{1}{8}$ by $\frac{2}{2}$ .

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{2}{8 \cdot 2} + \frac{1}{16}$

Step 4.2.2

Multiply $8$ by $2$ .

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{2}{16} + \frac{1}{16}$

Step 4.3

Combine the numerators over the common denominator.

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{2 + 1}{16}$

Step 4.4

Add $2$ and $1$ .

$x^{2} + {(y - \frac{1}{4})}^{2} = \frac{3}{16}$

Step 5

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 6

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.

$r = \frac{\sqrt{3}}{4}$

$h = 0$

$k = \frac{1}{4}$

Step 7

The center of the circle is found at $(h, k)$ .

Center: $(0, \frac{1}{4})$

Step 8

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

Center: $(0, \frac{1}{4})$

Radius: $\frac{\sqrt{3}}{4}$

Step 9