Precalculus Examples

$x^{2} - 8 x - 8 y - 16 = 0$

Step 1

Solve for $y$ .

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Step 1.1

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

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Step 1.1.1

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

$- 8 x - 8 y - 16 = - x^{2}$

Step 1.1.2

Add $8 x$ to both sides of the equation.

$- 8 y - 16 = - x^{2} + 8 x$

Step 1.1.3

Add $16$ to both sides of the equation.

$- 8 y = - x^{2} + 8 x + 16$

Step 1.2

Divide each term in $- 8 y = - x^{2} + 8 x + 16$ by $- 8$ and simplify.

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Step 1.2.1

Divide each term in $- 8 y = - x^{2} + 8 x + 16$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 8$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{8} + \frac{8 x}{- 8} + \frac{16}{- 8}$

Step 1.2.3.1.2

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

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Step 1.2.3.1.2.1

Factor $8$ out of $8 x$ .

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

Step 1.2.3.1.2.2

Move the negative one from the denominator of $\frac{x}{- 1}$ .

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

Step 1.2.3.1.3

Rewrite $- 1 \cdot x$ as $- x$ .

$y = \frac{x^{2}}{8} - x + \frac{16}{- 8}$

Step 1.2.3.1.4

Divide $16$ by $- 8$ .

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

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

$b = - 1$

$c = - 2$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

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

$d = \frac{- 1}{\frac{2}{8}}$

Step 2.1.1.3.2.2

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

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Step 2.1.1.3.2.2.1

Factor $2$ out of $2$ .

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

Step 2.1.1.3.2.2.2

Cancel the common factors.

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Step 2.1.1.3.2.2.2.1

Factor $2$ out of $8$ .

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

Step 2.1.1.3.2.2.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.2.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Step 2.1.1.3.2.4

Multiply $- 1$ by $4$ .

$d = - 4$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

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

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

Step 2.1.1.4.2.1.2

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

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

Step 2.1.1.4.2.1.3

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

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Step 2.1.1.4.2.1.3.1

Factor $4$ out of $4$ .

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

Step 2.1.1.4.2.1.3.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.2.1

Factor $4$ out of $8$ .

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

Step 2.1.1.4.2.1.3.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.1.3.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - 2 - (1 \cdot 2)$

Step 2.1.1.4.2.1.5

Multiply $- (1 \cdot 2)$ .

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Step 2.1.1.4.2.1.5.1

Multiply $2$ by $1$ .

$e = - 2 - 1 \cdot 2$

Step 2.1.1.4.2.1.5.2

Multiply $- 1$ by $2$ .

$e = - 2 - 2$

Step 2.1.1.4.2.2

Subtract $2$ from $- 2$ .

$e = - 4$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x - 4)}^{2} - 4$ .

$\frac{1}{8} {(x - 4)}^{2} - 4$

Step 2.1.2

Set $y$ equal to the new right side.

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

Step 2.2

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

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

$h = 4$

$k = - 4$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(4, - 4)$

Step 2.5

Find $p$ , the distance from the vertex to the focus.

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Step 2.5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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Step 2.5.3.2.1

Factor $4$ out of $4$ .

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

Step 2.5.3.2.2

Cancel the common factors.

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Step 2.5.3.2.2.1

Factor $4$ out of $8$ .

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

Step 2.5.3.2.2.2

Cancel the common factor.

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

Step 2.5.3.2.2.3

Rewrite the expression.

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

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 2.5.3.4

Multiply $2$ by $1$ .

$2$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(4, - 2)$

Step 2.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 4$

Step 2.8

Find the directrix.

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Step 2.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 2.8.2

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

$y = - 6$

Step 2.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: $(4, - 4)$

Focus: $(4, - 2)$

Axis of Symmetry: $x = 4$

Directrix: $y = - 6$

Direction: Opens Up

Vertex: $(4, - 4)$

Focus: $(4, - 2)$

Axis of Symmetry: $x = 4$

Directrix: $y = - 6$

Step 3

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 3.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{{(3)}^{2}}{8} - (3) - 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Write $- (3)$ as a fraction with denominator $1$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- (3)}{1} - 2$

Step 3.2.1.2

Multiply $\frac{- (3)}{1}$ by $\frac{8}{8}$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- (3)}{1} \cdot \frac{8}{8} - 2$

Step 3.2.1.3

Multiply $\frac{- (3)}{1}$ by $\frac{8}{8}$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} - 2$

Step 3.2.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.2.1.5

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

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.2.1.6

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

$f (3) = \frac{{(3)}^{2}}{8} + \frac{- 3 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.2.2

Combine the numerators over the common denominator.

$f (3) = \frac{{(3)}^{2} - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Raise $3$ to the power of $2$ .

$f (3) = \frac{9 - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3.2

Multiply $- (3) \cdot 8$ .

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Step 3.2.3.2.1

Multiply $- 1$ by $3$ .

$f (3) = \frac{9 - 3 \cdot 8 - 2 \cdot 8}{8}$

Step 3.2.3.2.2

Multiply $- 3$ by $8$ .

$f (3) = \frac{9 - 24 - 2 \cdot 8}{8}$

Step 3.2.3.3

Multiply $- 2$ by $8$ .

$f (3) = \frac{9 - 24 - 16}{8}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Subtract $24$ from $9$ .

$f (3) = \frac{- 15 - 16}{8}$

Step 3.2.4.2

Subtract $16$ from $- 15$ .

$f (3) = \frac{- 31}{8}$

Step 3.2.4.3

Move the negative in front of the fraction.

$f (3) = - \frac{31}{8}$

Step 3.2.5

The final answer is $- \frac{31}{8}$ .

$- \frac{31}{8}$

Step 3.3

The $y$ value at $x = 3$ is $- \frac{31}{8}$ .

$y = - \frac{31}{8}$

Step 3.4

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{{(2)}^{2}}{8} - (2) - 2$

Step 3.5

Simplify the result.

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Step 3.5.1

Find the common denominator.

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Step 3.5.1.1

Write $- (2)$ as a fraction with denominator $1$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- (2)}{1} - 2$

Step 3.5.1.2

Multiply $\frac{- (2)}{1}$ by $\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- (2)}{1} \cdot \frac{8}{8} - 2$

Step 3.5.1.3

Multiply $\frac{- (2)}{1}$ by $\frac{8}{8}$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} - 2$

Step 3.5.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.5.1.5

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

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.5.1.6

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

$f (2) = \frac{{(2)}^{2}}{8} + \frac{- 2 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.5.2

Combine the numerators over the common denominator.

$f (2) = \frac{{(2)}^{2} - 2 \cdot 8 - 2 \cdot 8}{8}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

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

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

Step 3.5.3.2

Multiply $- (2) \cdot 8$ .

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Step 3.5.3.2.1

Multiply $- 1$ by $2$ .

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

Step 3.5.3.2.2

Multiply $- 2$ by $8$ .

$f (2) = \frac{4 - 16 - 2 \cdot 8}{8}$

Step 3.5.3.3

Multiply $- 2$ by $8$ .

$f (2) = \frac{4 - 16 - 16}{8}$

Step 3.5.4

Reduce the expression by cancelling the common factors.

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Step 3.5.4.1

Subtract $16$ from $4$ .

$f (2) = \frac{- 12 - 16}{8}$

Step 3.5.4.2

Subtract $16$ from $- 12$ .

$f (2) = \frac{- 28}{8}$

Step 3.5.4.3

Cancel the common factor of $- 28$ and $8$ .

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Step 3.5.4.3.1

Factor $4$ out of $- 28$ .

$f (2) = \frac{4 (- 7)}{8}$

Step 3.5.4.3.2

Cancel the common factors.

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Step 3.5.4.3.2.1

Factor $4$ out of $8$ .

$f (2) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.5.4.3.2.2

Cancel the common factor.

$f (2) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.5.4.3.2.3

Rewrite the expression.

$f (2) = \frac{- 7}{2}$

Step 3.5.4.4

Move the negative in front of the fraction.

$f (2) = - \frac{7}{2}$

Step 3.5.5

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 3.6

The $y$ value at $x = 2$ is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 3.7

Replace the variable $x$ with $5$ in the expression.

$f (5) = \frac{{(5)}^{2}}{8} - (5) - 2$

Step 3.8

Simplify the result.

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Step 3.8.1

Find the common denominator.

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Step 3.8.1.1

Write $- (5)$ as a fraction with denominator $1$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- (5)}{1} - 2$

Step 3.8.1.2

Multiply $\frac{- (5)}{1}$ by $\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- (5)}{1} \cdot \frac{8}{8} - 2$

Step 3.8.1.3

Multiply $\frac{- (5)}{1}$ by $\frac{8}{8}$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} - 2$

Step 3.8.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.8.1.5

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

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.8.1.6

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

$f (5) = \frac{{(5)}^{2}}{8} + \frac{- 5 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.8.2

Combine the numerators over the common denominator.

$f (5) = \frac{{(5)}^{2} - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3

Simplify each term.

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Step 3.8.3.1

Raise $5$ to the power of $2$ .

$f (5) = \frac{25 - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3.2

Multiply $- (5) \cdot 8$ .

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Step 3.8.3.2.1

Multiply $- 1$ by $5$ .

$f (5) = \frac{25 - 5 \cdot 8 - 2 \cdot 8}{8}$

Step 3.8.3.2.2

Multiply $- 5$ by $8$ .

$f (5) = \frac{25 - 40 - 2 \cdot 8}{8}$

Step 3.8.3.3

Multiply $- 2$ by $8$ .

$f (5) = \frac{25 - 40 - 16}{8}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Subtract $40$ from $25$ .

$f (5) = \frac{- 15 - 16}{8}$

Step 3.8.4.2

Subtract $16$ from $- 15$ .

$f (5) = \frac{- 31}{8}$

Step 3.8.4.3

Move the negative in front of the fraction.

$f (5) = - \frac{31}{8}$

Step 3.8.5

The final answer is $- \frac{31}{8}$ .

$- \frac{31}{8}$

Step 3.9

The $y$ value at $x = 5$ is $- \frac{31}{8}$ .

$y = - \frac{31}{8}$

Step 3.10

Replace the variable $x$ with $6$ in the expression.

$f (6) = \frac{{(6)}^{2}}{8} - (6) - 2$

Step 3.11

Simplify the result.

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Step 3.11.1

Find the common denominator.

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Step 3.11.1.1

Write $- (6)$ as a fraction with denominator $1$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- (6)}{1} - 2$

Step 3.11.1.2

Multiply $\frac{- (6)}{1}$ by $\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- (6)}{1} \cdot \frac{8}{8} - 2$

Step 3.11.1.3

Multiply $\frac{- (6)}{1}$ by $\frac{8}{8}$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} - 2$

Step 3.11.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2}{1}$

Step 3.11.1.5

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

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2}{1} \cdot \frac{8}{8}$

Step 3.11.1.6

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

$f (6) = \frac{{(6)}^{2}}{8} + \frac{- 6 \cdot 8}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.11.2

Combine the numerators over the common denominator.

$f (6) = \frac{{(6)}^{2} - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3

Simplify each term.

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Step 3.11.3.1

Raise $6$ to the power of $2$ .

$f (6) = \frac{36 - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3.2

Multiply $- (6) \cdot 8$ .

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Step 3.11.3.2.1

Multiply $- 1$ by $6$ .

$f (6) = \frac{36 - 6 \cdot 8 - 2 \cdot 8}{8}$

Step 3.11.3.2.2

Multiply $- 6$ by $8$ .

$f (6) = \frac{36 - 48 - 2 \cdot 8}{8}$

Step 3.11.3.3

Multiply $- 2$ by $8$ .

$f (6) = \frac{36 - 48 - 16}{8}$

Step 3.11.4

Reduce the expression by cancelling the common factors.

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Step 3.11.4.1

Subtract $48$ from $36$ .

$f (6) = \frac{- 12 - 16}{8}$

Step 3.11.4.2

Subtract $16$ from $- 12$ .

$f (6) = \frac{- 28}{8}$

Step 3.11.4.3

Cancel the common factor of $- 28$ and $8$ .

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Step 3.11.4.3.1

Factor $4$ out of $- 28$ .

$f (6) = \frac{4 (- 7)}{8}$

Step 3.11.4.3.2

Cancel the common factors.

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Step 3.11.4.3.2.1

Factor $4$ out of $8$ .

$f (6) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.11.4.3.2.2

Cancel the common factor.

$f (6) = \frac{4 \cdot - 7}{4 \cdot 2}$

Step 3.11.4.3.2.3

Rewrite the expression.

$f (6) = \frac{- 7}{2}$

Step 3.11.4.4

Move the negative in front of the fraction.

$f (6) = - \frac{7}{2}$

Step 3.11.5

The final answer is $- \frac{7}{2}$ .

$- \frac{7}{2}$

Step 3.12

The $y$ value at $x = 6$ is $- \frac{7}{2}$ .

$y = - \frac{7}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 2 & - \frac{7}{2} \\ 3 & - \frac{31}{8} \\ 4 & - 4 \\ 5 & - \frac{31}{8} \\ 6 & - \frac{7}{2} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(4, - 4)$

Focus: $(4, - 2)$

Axis of Symmetry: $x = 4$

Directrix: $y = - 6$

$\begin{array}{c} x & y \\ 2 & - \frac{7}{2} \\ 3 & - \frac{31}{8} \\ 4 & - 4 \\ 5 & - \frac{31}{8} \\ 6 & - \frac{7}{2} \end{array}$

Step 5