Pre-Algebra Examples

$- \frac{2}{3} x^{2} + 8 x - 24$

Step 1

Simplify each term.

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Step 1.1

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

$y = - \frac{x^{2} \cdot 2}{3} + 8 x - 24$

Step 1.2

Move $2$ to the left of $x^{2}$ .

$y = - \frac{2 x^{2}}{3} + 8 x - 24$

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 $- \frac{2 x^{2}}{3} + 8 x - 24$ .

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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 $c$ .

$a = - \frac{2}{3}$

$b = 8$

$c = - 24$

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

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

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 $8$ and $2$ .

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Step 2.1.1.3.2.1.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 (- \frac{2}{3})}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 (- \frac{2}{3})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{4}{- \frac{2}{3}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 4 (- \frac{3}{2})$

Step 2.1.1.3.2.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$d = 4 (\frac{- 3}{2})$

Step 2.1.1.3.2.3.2

Factor $2$ out of $4$ .

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

Step 2.1.1.3.2.3.3

Cancel the common factor.

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

Step 2.1.1.3.2.3.4

Rewrite the expression.

$d = 2 \cdot - 3$

Step 2.1.1.3.2.4

Multiply $2$ by $- 3$ .

$d = - 6$

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

$e = - 24 - \frac{8^{2}}{4 (- \frac{2}{3})}$

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

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

$e = - 24 - \frac{64}{4 (- \frac{2}{3})}$

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 $4$ .

$e = - 24 - \frac{64}{- 4 (\frac{2}{3})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{2}{3}$ .

$e = - 24 - \frac{64}{\frac{- 4 \cdot 2}{3}}$

Step 2.1.1.4.2.1.3

Multiply $- 4$ by $2$ .

$e = - 24 - \frac{64}{\frac{- 8}{3}}$

Step 2.1.1.4.2.1.4

Move the negative in front of the fraction.

$e = - 24 - \frac{64}{- \frac{8}{3}}$

Step 2.1.1.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = - 24 - (64 (- \frac{3}{8}))$

Step 2.1.1.4.2.1.6

Cancel the common factor of $8$ .

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Step 2.1.1.4.2.1.6.1

Move the leading negative in $- \frac{3}{8}$ into the numerator.

$e = - 24 - (64 (\frac{- 3}{8}))$

Step 2.1.1.4.2.1.6.2

Factor $8$ out of $64$ .

$e = - 24 - (8 (8) \frac{- 3}{8})$

Step 2.1.1.4.2.1.6.3

Cancel the common factor.

$e = - 24 - (8 \cdot 8 \frac{- 3}{8})$

Step 2.1.1.4.2.1.6.4

Rewrite the expression.

$e = - 24 - (8 \cdot - 3)$

Step 2.1.1.4.2.1.7

Multiply $8$ by $- 3$ .

$e = - 24 - - 24$

Step 2.1.1.4.2.1.8

Multiply $- 1$ by $- 24$ .

$e = - 24 + 24$

Step 2.1.1.4.2.2

Add $- 24$ and $24$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{2}{3} {(x - 6)}^{2} + 0$ .

$- \frac{2}{3} {(x - 6)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{2}{3} \cdot {(x - 6)}^{2} + 0$

Step 2.2

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

$a = - \frac{2}{3}$

$h = 6$

$k = 0$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(6, 0)$

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.

$\frac{1}{4 \cdot (- \frac{2}{3})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{2}{3})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{2}{3})}$

Step 2.5.3.2

Combine $4$ and $\frac{2}{3}$ .

$- \frac{1}{\frac{4 \cdot 2}{3}}$

Step 2.5.3.3

Multiply $4$ by $2$ .

$- \frac{1}{\frac{8}{3}}$

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{3}{8}))$

Step 2.5.3.5

Multiply $\frac{3}{8}$ by $1$ .

$- \frac{3}{8}$

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.

$(6, - \frac{3}{8})$

Step 2.7

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

$x = 6$

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.

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

Step 2.9

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

Direction: Opens Down

Vertex: $(6, 0)$

Focus: $(6, - \frac{3}{8})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{3}{8}$

Direction: Opens Down

Vertex: $(6, 0)$

Focus: $(6, - \frac{3}{8})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{3}{8}$

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 $5$ in the expression.

$f (5) = - \frac{2 {(5)}^{2}}{3} + 8 (5) - 24$

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 $5$ to the power of $2$ .

$f (5) = - \frac{2 \cdot 25}{3} + 8 (5) - 24$

Step 3.2.1.2

Multiply $2$ by $25$ .

$f (5) = - \frac{50}{3} + 8 (5) - 24$

Step 3.2.1.3

Multiply $8$ by $5$ .

$f (5) = - \frac{50}{3} + 40 - 24$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $40$ as a fraction with denominator $1$ .

$f (5) = - \frac{50}{3} + \frac{40}{1} - 24$

Step 3.2.2.2

Multiply $\frac{40}{1}$ by $\frac{3}{3}$ .

$f (5) = - \frac{50}{3} + \frac{40}{1} \cdot \frac{3}{3} - 24$

Step 3.2.2.3

Multiply $\frac{40}{1}$ by $\frac{3}{3}$ .

$f (5) = - \frac{50}{3} + \frac{40 \cdot 3}{3} - 24$

Step 3.2.2.4

Write $- 24$ as a fraction with denominator $1$ .

$f (5) = - \frac{50}{3} + \frac{40 \cdot 3}{3} + \frac{- 24}{1}$

Step 3.2.2.5

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (5) = - \frac{50}{3} + \frac{40 \cdot 3}{3} + \frac{- 24}{1} \cdot \frac{3}{3}$

Step 3.2.2.6

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (5) = - \frac{50}{3} + \frac{40 \cdot 3}{3} + \frac{- 24 \cdot 3}{3}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (5) = \frac{- 50 + 40 \cdot 3 - 24 \cdot 3}{3}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $40$ by $3$ .

$f (5) = \frac{- 50 + 120 - 24 \cdot 3}{3}$

Step 3.2.4.2

Multiply $- 24$ by $3$ .

$f (5) = \frac{- 50 + 120 - 72}{3}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Add $- 50$ and $120$ .

$f (5) = \frac{70 - 72}{3}$

Step 3.2.5.2

Subtract $72$ from $70$ .

$f (5) = \frac{- 2}{3}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (5) = - \frac{2}{3}$

Step 3.2.6

The final answer is $- \frac{2}{3}$ .

$- \frac{2}{3}$

Step 3.3

The $y$ value at $x = 5$ is $- \frac{2}{3}$ .

$y = - \frac{2}{3}$

Step 3.4

Replace the variable $x$ with $4$ in the expression.

$f (4) = - \frac{2 {(4)}^{2}}{3} + 8 (4) - 24$

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

Simplify the numerator.

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Step 3.5.1.1.1

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

$f (4) = - \frac{2 \cdot {(2^{2})}^{2}}{3} + 8 (4) - 24$

Step 3.5.1.1.2

Multiply the exponents in ${(2^{2})}^{2}$ .

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Step 3.5.1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (4) = - \frac{2 \cdot 2^{2 \cdot 2}}{3} + 8 (4) - 24$

Step 3.5.1.1.2.2

Multiply $2$ by $2$ .

$f (4) = - \frac{2 \cdot 2^{4}}{3} + 8 (4) - 24$

Step 3.5.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (4) = - \frac{2^{1 + 4}}{3} + 8 (4) - 24$

Step 3.5.1.1.4

Add $1$ and $4$ .

$f (4) = - \frac{2^{5}}{3} + 8 (4) - 24$

Step 3.5.1.2

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

$f (4) = - \frac{32}{3} + 8 (4) - 24$

Step 3.5.1.3

Multiply $8$ by $4$ .

$f (4) = - \frac{32}{3} + 32 - 24$

Step 3.5.2

Find the common denominator.

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Step 3.5.2.1

Write $32$ as a fraction with denominator $1$ .

$f (4) = - \frac{32}{3} + \frac{32}{1} - 24$

Step 3.5.2.2

Multiply $\frac{32}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{32}{3} + \frac{32}{1} \cdot \frac{3}{3} - 24$

Step 3.5.2.3

Multiply $\frac{32}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{32}{3} + \frac{32 \cdot 3}{3} - 24$

Step 3.5.2.4

Write $- 24$ as a fraction with denominator $1$ .

$f (4) = - \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 24}{1}$

Step 3.5.2.5

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 24}{1} \cdot \frac{3}{3}$

Step 3.5.2.6

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (4) = - \frac{32}{3} + \frac{32 \cdot 3}{3} + \frac{- 24 \cdot 3}{3}$

Step 3.5.3

Combine the numerators over the common denominator.

$f (4) = \frac{- 32 + 32 \cdot 3 - 24 \cdot 3}{3}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Multiply $32$ by $3$ .

$f (4) = \frac{- 32 + 96 - 24 \cdot 3}{3}$

Step 3.5.4.2

Multiply $- 24$ by $3$ .

$f (4) = \frac{- 32 + 96 - 72}{3}$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Add $- 32$ and $96$ .

$f (4) = \frac{64 - 72}{3}$

Step 3.5.5.2

Subtract $72$ from $64$ .

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

Step 3.5.5.3

Move the negative in front of the fraction.

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

Step 3.5.6

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

$- \frac{8}{3}$

Step 3.6

The $y$ value at $x = 4$ is $- \frac{8}{3}$ .

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

Step 3.7

Replace the variable $x$ with $9$ in the expression.

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

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 $9$ to the power of $2$ .

$f (9) = - \frac{2 \cdot 81}{3} + 8 (9) - 24$

Step 3.8.1.2

Multiply $2$ by $81$ .

$f (9) = - \frac{162}{3} + 8 (9) - 24$

Step 3.8.1.3

Divide $162$ by $3$ .

$f (9) = - 1 \cdot 54 + 8 (9) - 24$

Step 3.8.1.4

Multiply $- 1$ by $54$ .

$f (9) = - 54 + 8 (9) - 24$

Step 3.8.1.5

Multiply $8$ by $9$ .

$f (9) = - 54 + 72 - 24$

Step 3.8.2

Simplify by adding and subtracting.

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Step 3.8.2.1

Add $- 54$ and $72$ .

$f (9) = 18 - 24$

Step 3.8.2.2

Subtract $24$ from $18$ .

$f (9) = - 6$

Step 3.8.3

The final answer is $- 6$ .

$- 6$

Step 3.9

The $y$ value at $x = 9$ is $- 6$ .

$y = - 6$

Step 3.10

Replace the variable $x$ with $7$ in the expression.

$f (7) = - \frac{2 {(7)}^{2}}{3} + 8 (7) - 24$

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 $7$ to the power of $2$ .

$f (7) = - \frac{2 \cdot 49}{3} + 8 (7) - 24$

Step 3.11.1.2

Multiply $2$ by $49$ .

$f (7) = - \frac{98}{3} + 8 (7) - 24$

Step 3.11.1.3

Multiply $8$ by $7$ .

$f (7) = - \frac{98}{3} + 56 - 24$

Step 3.11.2

Find the common denominator.

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Step 3.11.2.1

Write $56$ as a fraction with denominator $1$ .

$f (7) = - \frac{98}{3} + \frac{56}{1} - 24$

Step 3.11.2.2

Multiply $\frac{56}{1}$ by $\frac{3}{3}$ .

$f (7) = - \frac{98}{3} + \frac{56}{1} \cdot \frac{3}{3} - 24$

Step 3.11.2.3

Multiply $\frac{56}{1}$ by $\frac{3}{3}$ .

$f (7) = - \frac{98}{3} + \frac{56 \cdot 3}{3} - 24$

Step 3.11.2.4

Write $- 24$ as a fraction with denominator $1$ .

$f (7) = - \frac{98}{3} + \frac{56 \cdot 3}{3} + \frac{- 24}{1}$

Step 3.11.2.5

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (7) = - \frac{98}{3} + \frac{56 \cdot 3}{3} + \frac{- 24}{1} \cdot \frac{3}{3}$

Step 3.11.2.6

Multiply $\frac{- 24}{1}$ by $\frac{3}{3}$ .

$f (7) = - \frac{98}{3} + \frac{56 \cdot 3}{3} + \frac{- 24 \cdot 3}{3}$

Step 3.11.3

Combine the numerators over the common denominator.

$f (7) = \frac{- 98 + 56 \cdot 3 - 24 \cdot 3}{3}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Multiply $56$ by $3$ .

$f (7) = \frac{- 98 + 168 - 24 \cdot 3}{3}$

Step 3.11.4.2

Multiply $- 24$ by $3$ .

$f (7) = \frac{- 98 + 168 - 72}{3}$

Step 3.11.5

Simplify the expression.

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Step 3.11.5.1

Add $- 98$ and $168$ .

$f (7) = \frac{70 - 72}{3}$

Step 3.11.5.2

Subtract $72$ from $70$ .

$f (7) = \frac{- 2}{3}$

Step 3.11.5.3

Move the negative in front of the fraction.

$f (7) = - \frac{2}{3}$

Step 3.11.6

The final answer is $- \frac{2}{3}$ .

$- \frac{2}{3}$

Step 3.12

The $y$ value at $x = 7$ is $- \frac{2}{3}$ .

$y = - \frac{2}{3}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & - \frac{8}{3} \\ 5 & - \frac{2}{3} \\ 6 & 0 \\ 7 & - \frac{2}{3} \\ 9 & - 6 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(6, 0)$

Focus: $(6, - \frac{3}{8})$

Axis of Symmetry: $x = 6$

Directrix: $y = \frac{3}{8}$

$\begin{array}{c} x & y \\ 4 & - \frac{8}{3} \\ 5 & - \frac{2}{3} \\ 6 & 0 \\ 7 & - \frac{2}{3} \\ 9 & - 6 \end{array}$

Step 5