Precalculus Examples

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

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $4 y - 8 = x^{2} + 8 x$ .

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

Step 1.2

Add $8$ to both sides of the equation.

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

Step 1.3

Divide each term in $4 y = x^{2} + 8 x + 8$ by $4$ and simplify.

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Step 1.3.1

Divide each term in $4 y = x^{2} + 8 x + 8$ by $4$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

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

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Step 1.3.3.1.1.1

Factor $4$ out of $8 x$ .

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

Step 1.3.3.1.1.2

Cancel the common factors.

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Step 1.3.3.1.1.2.1

Factor $4$ out of $4$ .

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

Step 1.3.3.1.1.2.2

Cancel the common factor.

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

Step 1.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{2 x}{1} + \frac{8}{4}$

Step 1.3.3.1.1.2.4

Divide $2 x$ by $1$ .

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

Step 1.3.3.1.2

Divide $8$ by $4$ .

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

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 $\frac{x^{2}}{4} + 2 x + 2$ .

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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 $c$ .

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

$b = 2$

$c = 2$

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.1.1

Cancel the common factor.

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

Step 2.1.1.3.2.1.2

Rewrite the expression.

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

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 4$

Step 2.1.1.3.2.3

Multiply $4$ by $1$ .

$d = 4$

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

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

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 $2$ to the power of $2$ .

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

Step 2.1.1.4.2.1.2

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

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

Step 2.1.1.4.2.1.3

Divide $4$ by $4$ .

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

Step 2.1.1.4.2.1.4

Divide $4$ by $1$ .

$e = 2 - 1 \cdot 4$

Step 2.1.1.4.2.1.5

Multiply $- 1$ by $4$ .

$e = 2 - 4$

Step 2.1.1.4.2.2

Subtract $4$ from $2$ .

$e = - 2$

Step 2.1.1.5

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

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

Step 2.1.2

Set $y$ equal to the new right side.

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

Step 2.2

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

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

$h = - 4$

$k = - 2$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(- 4, - 2)$

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.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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

Step 2.5.3.2

Simplify by dividing numbers.

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Step 2.5.3.2.1

Divide $4$ by $4$ .

$\frac{1}{1}$

Step 2.5.3.2.2

Divide $1$ by $1$ .

$1$

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.

$(- 4, - 1)$

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.

$y = - 3$

Step 2.9

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

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - 1)$

Axis of Symmetry: $x = - 4$

Directrix: $y = - 3$

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - 1)$

Axis of Symmetry: $x = - 4$

Directrix: $y = - 3$

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 $- 6$ in the expression.

$f (- 6) = \frac{{(- 6)}^{2}}{4} + 2 (- 6) + 2$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (- 6) = \frac{36}{4} + 2 (- 6) + 2$

Step 3.2.1.2

Divide $36$ by $4$ .

$f (- 6) = 9 + 2 (- 6) + 2$

Step 3.2.1.3

Multiply $2$ by $- 6$ .

$f (- 6) = 9 - 12 + 2$

Step 3.2.2

Simplify by adding and subtracting.

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Step 3.2.2.1

Subtract $12$ from $9$ .

$f (- 6) = - 3 + 2$

Step 3.2.2.2

Add $- 3$ and $2$ .

$f (- 6) = - 1$

Step 3.2.3

The final answer is $- 1$ .

$- 1$

Step 3.3

The $y$ value at $x = - 6$ is $- 1$ .

$y = - 1$

Step 3.4

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

$f (- 5) = \frac{{(- 5)}^{2}}{4} + 2 (- 5) + 2$

Step 3.5

Simplify the result.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

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

$f (- 5) = \frac{25}{4} + 2 (- 5) + 2$

Step 3.5.1.2

Multiply $2$ by $- 5$ .

$f (- 5) = \frac{25}{4} - 10 + 2$

Step 3.5.2

Find the common denominator.

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Step 3.5.2.1

Write $- 10$ as a fraction with denominator $1$ .

$f (- 5) = \frac{25}{4} + \frac{- 10}{1} + 2$

Step 3.5.2.2

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10}{1} \cdot \frac{4}{4} + 2$

Step 3.5.2.3

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + 2$

Step 3.5.2.4

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

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{2}{1}$

Step 3.5.2.5

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{2}{1} \cdot \frac{4}{4}$

Step 3.5.2.6

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (- 5) = \frac{25}{4} + \frac{- 10 \cdot 4}{4} + \frac{2 \cdot 4}{4}$

Step 3.5.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{25 - 10 \cdot 4 + 2 \cdot 4}{4}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Multiply $- 10$ by $4$ .

$f (- 5) = \frac{25 - 40 + 2 \cdot 4}{4}$

Step 3.5.4.2

Multiply $2$ by $4$ .

$f (- 5) = \frac{25 - 40 + 8}{4}$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Subtract $40$ from $25$ .

$f (- 5) = \frac{- 15 + 8}{4}$

Step 3.5.5.2

Add $- 15$ and $8$ .

$f (- 5) = \frac{- 7}{4}$

Step 3.5.5.3

Move the negative in front of the fraction.

$f (- 5) = - \frac{7}{4}$

Step 3.5.6

The final answer is $- \frac{7}{4}$ .

$- \frac{7}{4}$

Step 3.6

The $y$ value at $x = - 5$ is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 3.7

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

$f (- 2) = \frac{{(- 2)}^{2}}{4} + 2 (- 2) + 2$

Step 3.8

Simplify the result.

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Step 3.8.1

Simplify each term.

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Step 3.8.1.1

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

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

Step 3.8.1.2

Divide $4$ by $4$ .

$f (- 2) = 1 + 2 (- 2) + 2$

Step 3.8.1.3

Multiply $2$ by $- 2$ .

$f (- 2) = 1 - 4 + 2$

Step 3.8.2

Simplify by adding and subtracting.

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Step 3.8.2.1

Subtract $4$ from $1$ .

$f (- 2) = - 3 + 2$

Step 3.8.2.2

Add $- 3$ and $2$ .

$f (- 2) = - 1$

Step 3.8.3

The final answer is $- 1$ .

$- 1$

Step 3.9

The $y$ value at $x = - 2$ is $- 1$ .

$y = - 1$

Step 3.10

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

$f (- 3) = \frac{{(- 3)}^{2}}{4} + 2 (- 3) + 2$

Step 3.11

Simplify the result.

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Step 3.11.1

Simplify each term.

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Step 3.11.1.1

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

$f (- 3) = \frac{9}{4} + 2 (- 3) + 2$

Step 3.11.1.2

Multiply $2$ by $- 3$ .

$f (- 3) = \frac{9}{4} - 6 + 2$

Step 3.11.2

Find the common denominator.

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Step 3.11.2.1

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

$f (- 3) = \frac{9}{4} + \frac{- 6}{1} + 2$

Step 3.11.2.2

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6}{1} \cdot \frac{4}{4} + 2$

Step 3.11.2.3

Multiply $\frac{- 6}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + 2$

Step 3.11.2.4

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

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{2}{1}$

Step 3.11.2.5

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{2}{1} \cdot \frac{4}{4}$

Step 3.11.2.6

Multiply $\frac{2}{1}$ by $\frac{4}{4}$ .

$f (- 3) = \frac{9}{4} + \frac{- 6 \cdot 4}{4} + \frac{2 \cdot 4}{4}$

Step 3.11.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 - 6 \cdot 4 + 2 \cdot 4}{4}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Multiply $- 6$ by $4$ .

$f (- 3) = \frac{9 - 24 + 2 \cdot 4}{4}$

Step 3.11.4.2

Multiply $2$ by $4$ .

$f (- 3) = \frac{9 - 24 + 8}{4}$

Step 3.11.5

Simplify the expression.

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Step 3.11.5.1

Subtract $24$ from $9$ .

$f (- 3) = \frac{- 15 + 8}{4}$

Step 3.11.5.2

Add $- 15$ and $8$ .

$f (- 3) = \frac{- 7}{4}$

Step 3.11.5.3

Move the negative in front of the fraction.

$f (- 3) = - \frac{7}{4}$

Step 3.11.6

The final answer is $- \frac{7}{4}$ .

$- \frac{7}{4}$

Step 3.12

The $y$ value at $x = - 3$ is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & - 1 \\ - 5 & - \frac{7}{4} \\ - 4 & - 2 \\ - 3 & - \frac{7}{4} \\ - 2 & - 1 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - 1)$

Axis of Symmetry: $x = - 4$

Directrix: $y = - 3$

$\begin{array}{c} x & y \\ - 6 & - 1 \\ - 5 & - \frac{7}{4} \\ - 4 & - 2 \\ - 3 & - \frac{7}{4} \\ - 2 & - 1 \end{array}$

Step 5