Pre-Algebra Examples

$y = - \frac{1}{3} x^{2} + \frac{10}{3} x - \frac{25}{3}$

Step 1

Simplify each term.

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Step 1.1

Combine $x^{2}$ and $\frac{1}{3}$ .

$y = - \frac{x^{2}}{3} + \frac{10}{3} x - \frac{25}{3}$

Step 1.2

Combine $\frac{10}{3}$ and $x$ .

$y = - \frac{x^{2}}{3} + \frac{10 x}{3} - \frac{25}{3}$

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 $- \frac{x^{2}}{3} + \frac{10 x}{3} - \frac{25}{3}$ .

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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 $c$ .

$a = - \frac{1}{3}$

$b = \frac{10}{3}$

$c = - \frac{25}{3}$

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

$d = \frac{\frac{10}{3}}{2 (- \frac{1}{3})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{10}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 2.1.1.3.2.2

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.2.1

Factor $2$ out of $10$ .

$d = \frac{2 (5)}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 2.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 5}{3} \cdot \frac{1}{2 (- \frac{1}{3})}$

Step 2.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{5}{3} \cdot \frac{1}{- \frac{1}{3}}$

Step 2.1.1.3.2.3

Multiply $\frac{5}{3}$ by $\frac{1}{- \frac{1}{3}}$ .

$d = \frac{5}{3 (- \frac{1}{3})}$

Step 2.1.1.3.2.4

Multiply $- 1$ by $3$ .

$d = \frac{5}{- 3 (\frac{1}{3})}$

Step 2.1.1.3.2.5

Combine $- 3$ and $\frac{1}{3}$ .

$d = \frac{5}{\frac{- 3}{3}}$

Step 2.1.1.3.2.6

Cancel the common factor of $- 3$ and $3$ .

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Step 2.1.1.3.2.6.1

Factor $3$ out of $- 3$ .

$d = \frac{5}{\frac{3 \cdot - 1}{3}}$

Step 2.1.1.3.2.6.2

Cancel the common factors.

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Step 2.1.1.3.2.6.2.1

Factor $3$ out of $3$ .

$d = \frac{5}{\frac{3 \cdot - 1}{3 (1)}}$

Step 2.1.1.3.2.6.2.2

Cancel the common factor.

$d = \frac{5}{\frac{3 \cdot - 1}{3 \cdot 1}}$

Step 2.1.1.3.2.6.2.3

Rewrite the expression.

$d = \frac{5}{\frac{- 1}{1}}$

Step 2.1.1.3.2.6.2.4

Divide $- 1$ by $1$ .

$d = \frac{5}{- 1}$

Step 2.1.1.3.2.7

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 2.1.1.3.2.8

Multiply $- 1$ by $5$ .

$d = - 5$

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

$e = - \frac{25}{3} - \frac{{(\frac{10}{3})}^{2}}{4 (- \frac{1}{3})}$

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

Simplify the numerator.

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Step 2.1.1.4.2.1.1.1

Apply the product rule to $\frac{10}{3}$ .

$e = - \frac{25}{3} - \frac{\frac{10^{2}}{3^{2}}}{4 (- \frac{1}{3})}$

Step 2.1.1.4.2.1.1.2

Raise $10$ to the power of $2$ .

$e = - \frac{25}{3} - \frac{\frac{100}{3^{2}}}{4 (- \frac{1}{3})}$

Step 2.1.1.4.2.1.1.3

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

$e = - \frac{25}{3} - \frac{\frac{100}{9}}{4 (- \frac{1}{3})}$

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{25}{3} - \frac{\frac{100}{9}}{- 4 (\frac{1}{3})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{3}$ .

$e = - \frac{25}{3} - \frac{\frac{100}{9}}{\frac{- 4}{3}}$

Step 2.1.1.4.2.1.3

Move the negative in front of the fraction.

$e = - \frac{25}{3} - \frac{\frac{100}{9}}{- \frac{4}{3}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{25}{3} - (\frac{100}{9} (- \frac{3}{4}))$

Step 2.1.1.4.2.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.4.2.1.5.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$e = - \frac{25}{3} - (\frac{100}{9} \cdot \frac{- 3}{4})$

Step 2.1.1.4.2.1.5.2

Factor $4$ out of $100$ .

$e = - \frac{25}{3} - (\frac{4 (25)}{9} \cdot \frac{- 3}{4})$

Step 2.1.1.4.2.1.5.3

Cancel the common factor.

$e = - \frac{25}{3} - (\frac{4 \cdot 25}{9} \cdot \frac{- 3}{4})$

Step 2.1.1.4.2.1.5.4

Rewrite the expression.

$e = - \frac{25}{3} - (\frac{25}{9} \cdot - 3)$

Step 2.1.1.4.2.1.6

Cancel the common factor of $3$ .

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Step 2.1.1.4.2.1.6.1

Factor $3$ out of $9$ .

$e = - \frac{25}{3} - (\frac{25}{3 (3)} \cdot - 3)$

Step 2.1.1.4.2.1.6.2

Factor $3$ out of $- 3$ .

$e = - \frac{25}{3} - (\frac{25}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 2.1.1.4.2.1.6.3

Cancel the common factor.

$e = - \frac{25}{3} - (\frac{25}{3 \cdot 3} \cdot (3 \cdot - 1))$

Step 2.1.1.4.2.1.6.4

Rewrite the expression.

$e = - \frac{25}{3} - (\frac{25}{3} \cdot - 1)$

Step 2.1.1.4.2.1.7

Combine $\frac{25}{3}$ and $- 1$ .

$e = - \frac{25}{3} - \frac{25 \cdot - 1}{3}$

Step 2.1.1.4.2.1.8

Multiply $25$ by $- 1$ .

$e = - \frac{25}{3} - \frac{- 25}{3}$

Step 2.1.1.4.2.1.9

Move the negative in front of the fraction.

$e = - \frac{25}{3} - - \frac{25}{3}$

Step 2.1.1.4.2.1.10

Multiply $- - \frac{25}{3}$ .

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Step 2.1.1.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{25}{3} + 1 (\frac{25}{3})$

Step 2.1.1.4.2.1.10.2

Multiply $\frac{25}{3}$ by $1$ .

$e = - \frac{25}{3} + \frac{25}{3}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 25 + 25}{3}$

Step 2.1.1.4.2.3

Add $- 25$ and $25$ .

$e = \frac{0}{3}$

Step 2.1.1.4.2.4

Divide $0$ by $3$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{3} {(x - 5)}^{2} + 0$ .

$- \frac{1}{3} {(x - 5)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{3} \cdot {(x - 5)}^{2} + 0$

Step 2.2

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

$a = - \frac{1}{3}$

$h = 5$

$k = 0$

Step 2.3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(5, 0)$

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.

$\frac{1}{4 \cdot (- \frac{1}{3})}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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Step 2.5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{3})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{3})}$

Step 2.5.3.2

Combine $4$ and $\frac{1}{3}$ .

$- \frac{1}{\frac{4}{3}}$

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{4}))$

Step 2.5.3.4

Multiply $\frac{3}{4}$ by $1$ .

$- \frac{3}{4}$

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.

$(5, - \frac{3}{4})$

Step 2.7

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

$x = 5$

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.

$y = \frac{3}{4}$

Step 2.9

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

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{3}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{3}{4}$

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{3}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{3}{4}$

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 $4$ in the expression.

$f (4) = - \frac{{(4)}^{2}}{3} + \frac{10 (4)}{3} - \frac{25}{3}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (4) = \frac{- {(4)}^{2} + 10 (4) - 25}{3}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (4) = \frac{- 1 \cdot 16 + 10 (4) - 25}{3}$

Step 3.2.2.2

Multiply $- 1$ by $16$ .

$f (4) = \frac{- 16 + 10 (4) - 25}{3}$

Step 3.2.2.3

Multiply $10$ by $4$ .

$f (4) = \frac{- 16 + 40 - 25}{3}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Add $- 16$ and $40$ .

$f (4) = \frac{24 - 25}{3}$

Step 3.2.3.2

Subtract $25$ from $24$ .

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

Step 3.2.3.3

Move the negative in front of the fraction.

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

Step 3.2.4

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 3.3

The $y$ value at $x = 4$ is $- \frac{1}{3}$ .

$y = - \frac{1}{3}$

Step 3.4

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

$f (3) = - \frac{{(3)}^{2}}{3} + \frac{10 (3)}{3} - \frac{25}{3}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (3) = \frac{- {(3)}^{2} + 10 (3) - 25}{3}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (3) = \frac{- 1 \cdot 9 + 10 (3) - 25}{3}$

Step 3.5.2.2

Multiply $- 1$ by $9$ .

$f (3) = \frac{- 9 + 10 (3) - 25}{3}$

Step 3.5.2.3

Multiply $10$ by $3$ .

$f (3) = \frac{- 9 + 30 - 25}{3}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Add $- 9$ and $30$ .

$f (3) = \frac{21 - 25}{3}$

Step 3.5.3.2

Subtract $25$ from $21$ .

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

Step 3.5.3.3

Move the negative in front of the fraction.

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

Step 3.5.4

The final answer is $- \frac{4}{3}$ .

$- \frac{4}{3}$

Step 3.6

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

$y = - \frac{4}{3}$

Step 3.7

Replace the variable $x$ with $8$ in the expression.

$f (8) = - \frac{{(8)}^{2}}{3} + \frac{10 (8)}{3} - \frac{25}{3}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

$f (8) = \frac{- {(8)}^{2} + 10 (8) - 25}{3}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Raise $8$ to the power of $2$ .

$f (8) = \frac{- 1 \cdot 64 + 10 (8) - 25}{3}$

Step 3.8.2.2

Multiply $- 1$ by $64$ .

$f (8) = \frac{- 64 + 10 (8) - 25}{3}$

Step 3.8.2.3

Multiply $10$ by $8$ .

$f (8) = \frac{- 64 + 80 - 25}{3}$

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Add $- 64$ and $80$ .

$f (8) = \frac{16 - 25}{3}$

Step 3.8.3.2

Subtract $25$ from $16$ .

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

Step 3.8.3.3

Divide $- 9$ by $3$ .

$f (8) = - 3$

Step 3.8.4

The final answer is $- 3$ .

$- 3$

Step 3.9

The $y$ value at $x = 8$ is $- 3$ .

$y = - 3$

Step 3.10

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

$f (6) = - \frac{{(6)}^{2}}{3} + \frac{10 (6)}{3} - \frac{25}{3}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (6) = \frac{- {(6)}^{2} + 10 (6) - 25}{3}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

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

$f (6) = \frac{- 1 \cdot 36 + 10 (6) - 25}{3}$

Step 3.11.2.2

Multiply $- 1$ by $36$ .

$f (6) = \frac{- 36 + 10 (6) - 25}{3}$

Step 3.11.2.3

Multiply $10$ by $6$ .

$f (6) = \frac{- 36 + 60 - 25}{3}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Add $- 36$ and $60$ .

$f (6) = \frac{24 - 25}{3}$

Step 3.11.3.2

Subtract $25$ from $24$ .

$f (6) = \frac{- 1}{3}$

Step 3.11.3.3

Move the negative in front of the fraction.

$f (6) = - \frac{1}{3}$

Step 3.11.4

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 3.12

The $y$ value at $x = 6$ is $- \frac{1}{3}$ .

$y = - \frac{1}{3}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 3 & - \frac{4}{3} \\ 4 & - \frac{1}{3} \\ 5 & 0 \\ 6 & - \frac{1}{3} \\ 8 & - 3 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{3}{4})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{3}{4}$

$\begin{array}{c} x & y \\ 3 & - \frac{4}{3} \\ 4 & - \frac{1}{3} \\ 5 & 0 \\ 6 & - \frac{1}{3} \\ 8 & - 3 \end{array}$

Step 5