Pre-Algebra Examples

$8 y^{2} + 6 y - 6$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $8 y^{2} + 6 y - 6$ .

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Step 1.1.1.1

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

$a = 8$

$b = 6$

$c = - 6$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{6}{2 \cdot 8}$

Step 1.1.1.3.2

Cancel the common factor of $6$ and $2$ .

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Step 1.1.1.3.2.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 \cdot 8}$

Step 1.1.1.3.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 8$ .

$d = \frac{2 \cdot 3}{2 (8)}$

Step 1.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 \cdot 8}$

Step 1.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{3}{8}$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = - 6 - \frac{6^{2}}{4 \cdot 8}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

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

$e = - 6 - \frac{36}{4 \cdot 8}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $8$ .

$e = - 6 - \frac{36}{32}$

Step 1.1.1.4.2.1.3

Cancel the common factor of $36$ and $32$ .

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Step 1.1.1.4.2.1.3.1

Factor $4$ out of $36$ .

$e = - 6 - \frac{4 (9)}{32}$

Step 1.1.1.4.2.1.3.2

Cancel the common factors.

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Step 1.1.1.4.2.1.3.2.1

Factor $4$ out of $32$ .

$e = - 6 - \frac{4 \cdot 9}{4 \cdot 8}$

Step 1.1.1.4.2.1.3.2.2

Cancel the common factor.

$e = - 6 - \frac{4 \cdot 9}{4 \cdot 8}$

Step 1.1.1.4.2.1.3.2.3

Rewrite the expression.

$e = - 6 - \frac{9}{8}$

Step 1.1.1.4.2.2

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = - 6 \cdot \frac{8}{8} - \frac{9}{8}$

Step 1.1.1.4.2.3

Combine $- 6$ and $\frac{8}{8}$ .

$e = \frac{- 6 \cdot 8}{8} - \frac{9}{8}$

Step 1.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 6 \cdot 8 - 9}{8}$

Step 1.1.1.4.2.5

Simplify the numerator.

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Step 1.1.1.4.2.5.1

Multiply $- 6$ by $8$ .

$e = \frac{- 48 - 9}{8}$

Step 1.1.1.4.2.5.2

Subtract $9$ from $- 48$ .

$e = \frac{- 57}{8}$

Step 1.1.1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{57}{8}$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $8 {(y + \frac{3}{8})}^{2} - \frac{57}{8}$ .

$8 {(y + \frac{3}{8})}^{2} - \frac{57}{8}$

Step 1.1.2

Set $x$ equal to the new right side.

$x = 8 {(y + \frac{3}{8})}^{2} - \frac{57}{8}$

Step 1.2

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

$a = 8$

$h = - \frac{57}{8}$

$k = - \frac{3}{8}$

Step 1.3

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

Opens Right

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{57}{8}, - \frac{3}{8})$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

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

Step 1.5.3

Multiply $4$ by $8$ .

$\frac{1}{32}$

Step 1.6

Find the focus.

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Step 1.6.1

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 1.6.2

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

$(- \frac{227}{32}, - \frac{3}{8})$

Step 1.7

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

$y = - \frac{3}{8}$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$x = - \frac{229}{32}$

Step 1.9

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

Direction: Opens Right

Vertex: $(- \frac{57}{8}, - \frac{3}{8})$

Focus: $(- \frac{227}{32}, - \frac{3}{8})$

Axis of Symmetry: $y = - \frac{3}{8}$

Directrix: $x = - \frac{229}{32}$

Direction: Opens Right

Vertex: $(- \frac{57}{8}, - \frac{3}{8})$

Focus: $(- \frac{227}{32}, - \frac{3}{8})$

Axis of Symmetry: $y = - \frac{3}{8}$

Directrix: $x = - \frac{229}{32}$

Step 2

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 2.1

Substitute the $x$ value $- 6.125$ into $f (x) = - \frac{3 - \sqrt{57 + 8 x}}{8}$ . In this case, the point is $(- 6.125, - \frac{3 - \sqrt{8}}{8})$ .

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Step 2.1.1

Replace the variable $x$ with $- 6.125$ in the expression.

$f (- 6.125) = - \frac{3 - \sqrt{57 + 8 (- 6.125)}}{8}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Multiply $8$ by $- 6.125$ .

$f (- 6.125) = - \frac{3 - \sqrt{57 - 49}}{8}$

Step 2.1.2.1.2

Subtract $49$ from $57$ .

$f (- 6.125) = - \frac{3 - \sqrt{8}}{8}$

Step 2.1.2.2

The final answer is $- \frac{3 - \sqrt{8}}{8}$ .

$y = - \frac{3 - \sqrt{8}}{8}$

Step 2.2

Substitute the $x$ value $- 6.125$ into $f (x) = - \frac{3 + \sqrt{8 x + 57}}{8}$ . In this case, the point is $(- 6.125, - \frac{3 + \sqrt{8}}{8})$ .

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Step 2.2.1

Replace the variable $x$ with $- 6.125$ in the expression.

$f (- 6.125) = - \frac{3 + \sqrt{8 (- 6.125) + 57}}{8}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

Multiply $8$ by $- 6.125$ .

$f (- 6.125) = - \frac{3 + \sqrt{- 49 + 57}}{8}$

Step 2.2.2.1.2

Add $- 49$ and $57$ .

$f (- 6.125) = - \frac{3 + \sqrt{8}}{8}$

Step 2.2.2.2

The final answer is $- \frac{3 + \sqrt{8}}{8}$ .

$y = - \frac{3 + \sqrt{8}}{8}$

Step 2.3

Substitute the $x$ value $- 5.125$ into $f (x) = - \frac{3 - \sqrt{57 + 8 x}}{8}$ . In this case, the point is $(- 5.125, 0.125)$ .

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Step 2.3.1

Replace the variable $x$ with $- 5.125$ in the expression.

$f (- 5.125) = - \frac{3 - \sqrt{57 + 8 (- 5.125)}}{8}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify the numerator.

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Step 2.3.2.1.1

Multiply $8$ by $- 5.125$ .

$f (- 5.125) = - \frac{3 - \sqrt{57 - 41}}{8}$

Step 2.3.2.1.2

Subtract $41$ from $57$ .

$f (- 5.125) = - \frac{3 - \sqrt{16}}{8}$

Step 2.3.2.1.3

Rewrite $16$ as $4^{2}$ .

$f (- 5.125) = - \frac{3 - \sqrt{4^{2}}}{8}$

Step 2.3.2.1.4

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

$f (- 5.125) = - \frac{3 - 1 \cdot 4}{8}$

Step 2.3.2.1.5

Multiply $- 1$ by $4$ .

$f (- 5.125) = - \frac{3 - 4}{8}$

Step 2.3.2.1.6

Subtract $4$ from $3$ .

$f (- 5.125) = - \frac{- 1}{8}$

Step 2.3.2.2

Reduce the expression by cancelling the common factors.

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Step 2.3.2.2.1

Cancel the common factor of $- 1$ and $8$ .

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Step 2.3.2.2.1.1

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

$f (- 5.125) = - \frac{1 (- 1)}{8}$

Step 2.3.2.2.1.2

Cancel the common factors.

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Step 2.3.2.2.1.2.1

Rewrite $8$ as $1 (8)$ .

$f (- 5.125) = - \frac{1 \cdot - 1}{1 \cdot 8}$

Step 2.3.2.2.1.2.2

Cancel the common factor.

$f (- 5.125) = - \frac{1 \cdot - 1}{1 \cdot 8}$

Step 2.3.2.2.1.2.3

Rewrite the expression.

$f (- 5.125) = - \frac{- 1}{8}$

Step 2.3.2.2.2

Move the negative in front of the fraction.

$f (- 5.125) = \frac{1}{8}$

Step 2.3.2.3

The final answer is $\frac{1}{8}$ .

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

Step 2.3.3

Convert $\frac{1}{8}$ to decimal.

$= 0.125$

Step 2.4

Substitute the $x$ value $- 5.125$ into $f (x) = - \frac{3 + \sqrt{8 x + 57}}{8}$ . In this case, the point is $(- 5.125, - 0.875)$ .

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Step 2.4.1

Replace the variable $x$ with $- 5.125$ in the expression.

$f (- 5.125) = - \frac{3 + \sqrt{8 (- 5.125) + 57}}{8}$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

Simplify the numerator.

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Step 2.4.2.1.1

Multiply $8$ by $- 5.125$ .

$f (- 5.125) = - \frac{3 + \sqrt{- 41 + 57}}{8}$

Step 2.4.2.1.2

Add $- 41$ and $57$ .

$f (- 5.125) = - \frac{3 + \sqrt{16}}{8}$

Step 2.4.2.1.3

Rewrite $16$ as $4^{2}$ .

$f (- 5.125) = - \frac{3 + \sqrt{4^{2}}}{8}$

Step 2.4.2.1.4

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

$f (- 5.125) = - \frac{3 + 4}{8}$

Step 2.4.2.1.5

Add $3$ and $4$ .

$f (- 5.125) = - \frac{7}{8}$

Step 2.4.2.2

Cancel the common factor of $7$ and $8$ .

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Step 2.4.2.2.1

Rewrite $7$ as $1 (7)$ .

$f (- 5.125) = - \frac{1 (7)}{8}$

Step 2.4.2.2.2

Cancel the common factors.

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Step 2.4.2.2.2.1

Rewrite $8$ as $1 (8)$ .

$f (- 5.125) = - \frac{1 \cdot 7}{1 \cdot 8}$

Step 2.4.2.2.2.2

Cancel the common factor.

$f (- 5.125) = - \frac{1 \cdot 7}{1 \cdot 8}$

Step 2.4.2.2.2.3

Rewrite the expression.

$f (- 5.125) = - \frac{7}{8}$

Step 2.4.2.3

The final answer is $- \frac{7}{8}$ .

$y = - \frac{7}{8}$

Step 2.4.3

Convert $- \frac{7}{8}$ to decimal.

$= - 0.875$

Step 2.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - \frac{57}{8} & - \frac{3}{8} \\ - 6.125 & - 0.02 \\ - 6.125 & - 0.73 \\ - 5.125 & 0.12 \\ - 5.125 & - 0.88 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(- \frac{57}{8}, - \frac{3}{8})$

Focus: $(- \frac{227}{32}, - \frac{3}{8})$

Axis of Symmetry: $y = - \frac{3}{8}$

Directrix: $x = - \frac{229}{32}$

$\begin{array}{c} x & y \\ - \frac{57}{8} & - \frac{3}{8} \\ - 6.125 & - 0.02 \\ - 6.125 & - 0.73 \\ - 5.125 & 0.12 \\ - 5.125 & - 0.88 \end{array}$

Step 4