Pre-Algebra Examples

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

Step 1

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

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

Step 2

Subtract $4$ from both sides of the equation.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Divide $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Cancel the common factor of $- 4$ and $2$ .

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Step 3.3.1.1.1

Factor $2$ out of $- 4 y$ .

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

Step 3.3.1.1.2

Cancel the common factors.

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Step 3.3.1.1.2.1

Factor $2$ out of $2$ .

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

Step 3.3.1.1.2.2

Cancel the common factor.

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

Step 3.3.1.1.2.3

Rewrite the expression.

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

Step 3.3.1.1.2.4

Divide $- 2 y$ by $1$ .

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

Step 3.3.1.2

Divide $- 4$ by $2$ .

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

Step 4

Find the properties of the given parabola.

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Step 4.1

Rewrite the equation in vertex form.

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Step 4.1.1

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

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Step 4.1.1.1

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

$a = \frac{1}{2}$

$b = - 2$

$c = - 2$

Step 4.1.1.2

Consider the vertex form of a parabola.

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

Step 4.1.1.3

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

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Step 4.1.1.3.1

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

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

Step 4.1.1.3.2

Simplify the right side.

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Step 4.1.1.3.2.1

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

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Step 4.1.1.3.2.1.1

Factor $2$ out of $- 2$ .

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

Step 4.1.1.3.2.1.2

Cancel the common factors.

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Step 4.1.1.3.2.1.2.1

Cancel the common factor.

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

Step 4.1.1.3.2.1.2.2

Rewrite the expression.

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

Step 4.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 2$

Step 4.1.1.3.2.3

Multiply $- 1$ by $2$ .

$d = - 2$

Step 4.1.1.4

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

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Step 4.1.1.4.1

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

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

Step 4.1.1.4.2

Simplify the right side.

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Step 4.1.1.4.2.1

Simplify each term.

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Step 4.1.1.4.2.1.1

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

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

Step 4.1.1.4.2.1.2

Combine $4$ and $\frac{1}{2}$ .

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

Step 4.1.1.4.2.1.3

Divide $4$ by $2$ .

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

Step 4.1.1.4.2.1.4

Divide $4$ by $2$ .

$e = - 2 - 1 \cdot 2$

Step 4.1.1.4.2.1.5

Multiply $- 1$ by $2$ .

$e = - 2 - 2$

Step 4.1.1.4.2.2

Subtract $2$ from $- 2$ .

$e = - 4$

Step 4.1.1.5

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

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

Step 4.1.2

Set $x$ equal to the new right side.

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

Step 4.2

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

$a = \frac{1}{2}$

$h = - 4$

$k = 2$

Step 4.3

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

Opens Right

Step 4.4

Find the vertex $(h, k)$ .

$(- 4, 2)$

Step 4.5

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

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Step 4.5.1

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

$\frac{1}{4 a}$

Step 4.5.2

Substitute the value of $a$ into the formula.

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

Step 4.5.3

Simplify.

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Step 4.5.3.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 4.5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 4.6

Find the focus.

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Step 4.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 4.6.2

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

$(- \frac{7}{2}, 2)$

Step 4.7

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

$y = 2$

Step 4.8

Find the directrix.

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Step 4.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 4.8.2

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

$x = - \frac{9}{2}$

Step 4.9

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

Direction: Opens Right

Vertex: $(- 4, 2)$

Focus: $(- \frac{7}{2}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{9}{2}$

Direction: Opens Right

Vertex: $(- 4, 2)$

Focus: $(- \frac{7}{2}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{9}{2}$

Step 5

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 5.1

Substitute the $x$ value $- 3$ into $f (x) = 2 + \sqrt{- 2 (- x - 4)}$ . In this case, the point is $(- 3, 3.41421356)$ .

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Step 5.1.1

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

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

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Simplify each term.

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Step 5.1.2.1.1

Multiply $- 1$ by $- 3$ .

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

Step 5.1.2.1.2

Subtract $4$ from $3$ .

$f (- 3) = 2 + \sqrt{- 2 \cdot - 1}$

Step 5.1.2.1.3

Multiply $- 2$ by $- 1$ .

$f (- 3) = 2 + \sqrt{2}$

Step 5.1.2.2

The final answer is $2 + \sqrt{2}$ .

$y = 2 + \sqrt{2}$

Step 5.1.3

Convert $2 + \sqrt{2}$ to decimal.

$= 3.41421356$

Step 5.2

Substitute the $x$ value $- 3$ into $f (x) = 2 - \sqrt{- 2 (- x - 4)}$ . In this case, the point is $(- 3, 0.58578643)$ .

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Step 5.2.1

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

$f (- 3) = 2 - \sqrt{- 2 (- (- 3) - 4)}$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Multiply $- 1$ by $- 3$ .

$f (- 3) = 2 - \sqrt{- 2 (3 - 4)}$

Step 5.2.2.1.2

Subtract $4$ from $3$ .

$f (- 3) = 2 - \sqrt{- 2 \cdot - 1}$

Step 5.2.2.1.3

Multiply $- 2$ by $- 1$ .

$f (- 3) = 2 - \sqrt{2}$

Step 5.2.2.2

The final answer is $2 - \sqrt{2}$ .

$y = 2 - \sqrt{2}$

Step 5.2.3

Convert $2 - \sqrt{2}$ to decimal.

$= 0.58578643$

Step 5.3

Substitute the $x$ value $- 2$ into $f (x) = 2 + \sqrt{- 2 (- x - 4)}$ . In this case, the point is $(- 2, 4)$ .

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Step 5.3.1

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

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

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

Multiply $- 1$ by $- 2$ .

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

Step 5.3.2.1.2

Subtract $4$ from $2$ .

$f (- 2) = 2 + \sqrt{- 2 \cdot - 2}$

Step 5.3.2.1.3

Multiply $- 2$ by $- 2$ .

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

Step 5.3.2.1.4

Rewrite $4$ as $2^{2}$ .

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

Step 5.3.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 2) = 2 + 2$

Step 5.3.2.2

Add $2$ and $2$ .

$f (- 2) = 4$

Step 5.3.2.3

The final answer is $4$ .

$y = 4$

Step 5.3.3

Convert $4$ to decimal.

$= 4$

Step 5.4

Substitute the $x$ value $- 2$ into $f (x) = 2 - \sqrt{- 2 (- x - 4)}$ . In this case, the point is $(- 2, 0)$ .

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Step 5.4.1

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

$f (- 2) = 2 - \sqrt{- 2 (- (- 2) - 4)}$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Multiply $- 1$ by $- 2$ .

$f (- 2) = 2 - \sqrt{- 2 (2 - 4)}$

Step 5.4.2.1.2

Subtract $4$ from $2$ .

$f (- 2) = 2 - \sqrt{- 2 \cdot - 2}$

Step 5.4.2.1.3

Multiply $- 2$ by $- 2$ .

$f (- 2) = 2 - \sqrt{4}$

Step 5.4.2.1.4

Rewrite $4$ as $2^{2}$ .

$f (- 2) = 2 - \sqrt{2^{2}}$

Step 5.4.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.4.2.1.6

Multiply $- 1$ by $2$ .

$f (- 2) = 2 - 2$

Step 5.4.2.2

Subtract $2$ from $2$ .

$f (- 2) = 0$

Step 5.4.2.3

The final answer is $0$ .

$y = 0$

Step 5.4.3

Convert $0$ to decimal.

$= 0$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 2 \\ - 3 & 3.41 \\ - 3 & 0.59 \\ - 2 & 4 \\ - 2 & 0 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- 4, 2)$

Focus: $(- \frac{7}{2}, 2)$

Axis of Symmetry: $y = 2$

Directrix: $x = - \frac{9}{2}$

$\begin{array}{c} x & y \\ - 4 & 2 \\ - 3 & 3.41 \\ - 3 & 0.59 \\ - 2 & 4 \\ - 2 & 0 \end{array}$

Step 7