Precalculus Examples

$x^{2} - 2 x + 8 y + 9 = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1.1

Subtract $x^{2}$ from both sides of the equation.

$- 2 x + 8 y + 9 = - x^{2}$

Step 1.1.1.2

Add $2 x$ to both sides of the equation.

$8 y + 9 = - x^{2} + 2 x$

Step 1.1.1.3

Subtract $9$ from both sides of the equation.

$8 y = - x^{2} + 2 x - 9$

Step 1.1.2

Divide each term in $8 y = - x^{2} + 2 x - 9$ by $8$ and simplify.

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Step 1.1.2.1

Divide each term in $8 y = - x^{2} + 2 x - 9$ by $8$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $8$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.2.3.1.2

Cancel the common factor of $2$ and $8$ .

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Step 1.1.2.3.1.2.1

Factor $2$ out of $2 x$ .

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

Step 1.1.2.3.1.2.2

Cancel the common factors.

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Step 1.1.2.3.1.2.2.1

Factor $2$ out of $8$ .

$y = - \frac{x^{2}}{8} + \frac{2 x}{2 \cdot 4} + \frac{- 9}{8}$

Step 1.1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{8} + \frac{2 x}{2 \cdot 4} + \frac{- 9}{8}$

Step 1.1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{8} + \frac{x}{4} + \frac{- 9}{8}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{8} + \frac{x}{4} - \frac{9}{8}$

Step 1.2

Complete the square for $- \frac{x^{2}}{8} + \frac{x}{4} - \frac{9}{8}$ .

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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 $c$ .

$a = - \frac{1}{8}$

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

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

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

$d = \frac{\frac{1}{4}}{2 (- \frac{1}{8})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{4} \cdot \frac{1}{2 (- \frac{1}{8})}$

Step 1.2.3.2.2

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

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Step 1.2.3.2.2.1

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

$d = \frac{1}{4} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{8})}$

Step 1.2.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{1}{4} (- \frac{1}{2 (\frac{1}{8})})$

Step 1.2.3.2.3

Combine $2$ and $\frac{1}{8}$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2}{8}})$

Step 1.2.3.2.4

Cancel the common factor of $2$ and $8$ .

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Step 1.2.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{1}{4} (- \frac{1}{\frac{2 (1)}{8}})$

Step 1.2.3.2.4.2

Cancel the common factors.

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Step 1.2.3.2.4.2.1

Factor $2$ out of $8$ .

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

Step 1.2.3.2.4.2.2

Cancel the common factor.

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

Step 1.2.3.2.4.2.3

Rewrite the expression.

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

Step 1.2.3.2.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.3.2.6

Cancel the common factor of $4$ .

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Step 1.2.3.2.6.1

Factor $4$ out of $- (1 \cdot 4)$ .

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

Step 1.2.3.2.6.2

Cancel the common factor.

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

Step 1.2.3.2.6.3

Rewrite the expression.

$d = - 1 \cdot (1)$

Step 1.2.3.2.7

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = - \frac{9}{8} - \frac{{(\frac{1}{4})}^{2}}{4 (- \frac{1}{8})}$

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

Simplify the numerator.

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Step 1.2.4.2.1.1.1

Apply the product rule to $\frac{1}{4}$ .

$e = - \frac{9}{8} - \frac{\frac{1^{2}}{4^{2}}}{4 (- \frac{1}{8})}$

Step 1.2.4.2.1.1.2

One to any power is one.

$e = - \frac{9}{8} - \frac{\frac{1}{4^{2}}}{4 (- \frac{1}{8})}$

Step 1.2.4.2.1.1.3

Raise $4$ to the power of $2$ .

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{4 (- \frac{1}{8})}$

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 $4$ .

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{- 4 (\frac{1}{8})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{8}$ .

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{\frac{- 4}{8}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

Cancel the common factor of $- 4$ and $8$ .

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Step 1.2.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{\frac{4 (- 1)}{8}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $4$ out of $8$ .

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{\frac{4 \cdot - 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{\frac{- 1}{2}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{9}{8} - \frac{\frac{1}{16}}{- \frac{1}{2}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{9}{8} - (\frac{1}{16} (- 1 \cdot 2))$

Step 1.2.4.2.1.5

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.5.1

Factor $2$ out of $16$ .

$e = - \frac{9}{8} - (\frac{1}{2 (8)} (- 1 \cdot 2))$

Step 1.2.4.2.1.5.2

Factor $2$ out of $- 1 \cdot 2$ .

$e = - \frac{9}{8} - (\frac{1}{2 (8)} (2 \cdot - 1))$

Step 1.2.4.2.1.5.3

Cancel the common factor.

$e = - \frac{9}{8} - (\frac{1}{2 \cdot 8} (2 \cdot - 1))$

Step 1.2.4.2.1.5.4

Rewrite the expression.

$e = - \frac{9}{8} - (\frac{1}{8} \cdot - 1)$

Step 1.2.4.2.1.6

Combine $\frac{1}{8}$ and $- 1$ .

$e = - \frac{9}{8} - \frac{- 1}{8}$

Step 1.2.4.2.1.7

Move the negative in front of the fraction.

$e = - \frac{9}{8} - - \frac{1}{8}$

Step 1.2.4.2.1.8

Multiply $- - \frac{1}{8}$ .

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Step 1.2.4.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{9}{8} + 1 (\frac{1}{8})$

Step 1.2.4.2.1.8.2

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

$e = - \frac{9}{8} + \frac{1}{8}$

Step 1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 9 + 1}{8}$

Step 1.2.4.2.3

Add $- 9$ and $1$ .

$e = \frac{- 8}{8}$

Step 1.2.4.2.4

Divide $- 8$ by $8$ .

$e = - 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{8} {(x - 1)}^{2} - 1$ .

$- \frac{1}{8} {(x - 1)}^{2} - 1$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - \frac{1}{8}$

$h = 1$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(1, - 1)$

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.

$\frac{1}{4 \cdot (- \frac{1}{8})}$

Step 4.3

Simplify.

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Step 4.3.1

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

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Step 4.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{8})}$

Step 4.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{8})}$

Step 4.3.2

Combine $4$ and $\frac{1}{8}$ .

$- \frac{1}{\frac{4}{8}}$

Step 4.3.3

Cancel the common factor of $4$ and $8$ .

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Step 4.3.3.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{8}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $4$ out of $8$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 4.3.5

Multiply $- (1 \cdot 2)$ .

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Step 4.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 4.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 5.2

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

$(1, - 3)$

Step 6