Pre-Algebra Examples

$y = - \frac{3}{8} \cdot {(x - 5)}^{2}$

Step 1

Simplify $- \frac{3}{8} \cdot {(x - 5)}^{2}$ .

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Step 1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$y = - \frac{3}{8} \cdot ((x - 5) (x - 5))$

Step 1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x (x - 5) - 5 (x - 5))$

Step 1.2.2

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x \cdot x + x \cdot - 5 - 5 (x - 5))$

Step 1.2.3

Apply the distributive property.

$y = - \frac{3}{8} \cdot (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$y = - \frac{3}{8} \cdot (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5)$

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

$y = - \frac{3}{8} \cdot (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5)$

Step 1.3.1.3

Multiply $- 5$ by $- 5$ .

$y = - \frac{3}{8} \cdot (x^{2} - 5 x - 5 x + 25)$

Step 1.3.2

Subtract $5 x$ from $- 5 x$ .

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

Step 1.4

Apply the distributive property.

$y = - \frac{3}{8} x^{2} - \frac{3}{8} (- 10 x) - \frac{3}{8} \cdot 25$

Step 1.5

Simplify.

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Step 1.5.1

Combine $x^{2}$ and $\frac{3}{8}$ .

$y = - \frac{x^{2} \cdot 3}{8} - \frac{3}{8} (- 10 x) - \frac{3}{8} \cdot 25$

Step 1.5.2

Cancel the common factor of $2$ .

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Step 1.5.2.1

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

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

Step 1.5.2.2

Factor $2$ out of $8$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 (4)} (- 10 x) - \frac{3}{8} \cdot 25$

Step 1.5.2.3

Factor $2$ out of $- 10 x$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 (4)} (2 (- 5 x)) - \frac{3}{8} \cdot 25$

Step 1.5.2.4

Cancel the common factor.

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{2 \cdot 4} (2 (- 5 x)) - \frac{3}{8} \cdot 25$

Step 1.5.2.5

Rewrite the expression.

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 3}{4} (- 5 x) - \frac{3}{8} \cdot 25$

Step 1.5.3

Combine $- 5$ and $\frac{- 3}{4}$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{- 5 \cdot - 3}{4} x - \frac{3}{8} \cdot 25$

Step 1.5.4

Multiply $- 5$ by $- 3$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{15}{4} x - \frac{3}{8} \cdot 25$

Step 1.5.5

Combine $\frac{15}{4}$ and $x$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{15 x}{4} - \frac{3}{8} \cdot 25$

Step 1.5.6

Multiply $- \frac{3}{8} \cdot 25$ .

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Step 1.5.6.1

Multiply $25$ by $- 1$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{15 x}{4} - 25 (\frac{3}{8})$

Step 1.5.6.2

Combine $- 25$ and $\frac{3}{8}$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{15 x}{4} + \frac{- 25 \cdot 3}{8}$

Step 1.5.6.3

Multiply $- 25$ by $3$ .

$y = - \frac{x^{2} \cdot 3}{8} + \frac{15 x}{4} + \frac{- 75}{8}$

Step 1.6

Simplify each term.

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Step 1.6.1

Move $3$ to the left of $x^{2}$ .

$y = - \frac{3 \cdot x^{2}}{8} + \frac{15 x}{4} + \frac{- 75}{8}$

Step 1.6.2

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{8} + \frac{15 x}{4} - \frac{75}{8}$

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 $- \frac{3 x^{2}}{8} + \frac{15 x}{4} - \frac{75}{8}$ .

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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 $c$ .

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

$b = \frac{15}{4}$

$c = - \frac{75}{8}$

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

$d = \frac{\frac{15}{4}}{2 (- \frac{3}{8})}$

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.

$d = \frac{15}{4} \cdot \frac{1}{2 (- \frac{3}{8})}$

Step 2.1.1.3.2.2

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

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Step 2.1.1.3.2.2.1

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

$d = \frac{15}{4} \cdot \frac{- 1 (- 1)}{2 (- \frac{3}{8})}$

Step 2.1.1.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{15}{4} (- \frac{1}{2 (\frac{3}{8})})$

Step 2.1.1.3.2.3

Combine $2$ and $\frac{3}{8}$ .

$d = \frac{15}{4} (- \frac{1}{\frac{2 \cdot 3}{8}})$

Step 2.1.1.3.2.4

Multiply $2$ by $3$ .

$d = \frac{15}{4} (- \frac{1}{\frac{6}{8}})$

Step 2.1.1.3.2.5

Cancel the common factor of $6$ and $8$ .

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Step 2.1.1.3.2.5.1

Factor $2$ out of $6$ .

$d = \frac{15}{4} (- \frac{1}{\frac{2 (3)}{8}})$

Step 2.1.1.3.2.5.2

Cancel the common factors.

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Step 2.1.1.3.2.5.2.1

Factor $2$ out of $8$ .

$d = \frac{15}{4} (- \frac{1}{\frac{2 \cdot 3}{2 \cdot 4}})$

Step 2.1.1.3.2.5.2.2

Cancel the common factor.

$d = \frac{15}{4} (- \frac{1}{\frac{2 \cdot 3}{2 \cdot 4}})$

Step 2.1.1.3.2.5.2.3

Rewrite the expression.

$d = \frac{15}{4} (- \frac{1}{\frac{3}{4}})$

Step 2.1.1.3.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{15}{4} (- (1 (\frac{4}{3})))$

Step 2.1.1.3.2.7

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

$d = \frac{15}{4} (- \frac{4}{3})$

Step 2.1.1.3.2.8

Cancel the common factor of $3$ .

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Step 2.1.1.3.2.8.1

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

$d = \frac{15}{4} \cdot \frac{- 4}{3}$

Step 2.1.1.3.2.8.2

Factor $3$ out of $15$ .

$d = \frac{3 (5)}{4} \cdot \frac{- 4}{3}$

Step 2.1.1.3.2.8.3

Cancel the common factor.

$d = \frac{3 \cdot 5}{4} \cdot \frac{- 4}{3}$

Step 2.1.1.3.2.8.4

Rewrite the expression.

$d = \frac{5}{4} \cdot - 4$

Step 2.1.1.3.2.9

Cancel the common factor of $4$ .

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Step 2.1.1.3.2.9.1

Factor $4$ out of $- 4$ .

$d = \frac{5}{4} \cdot (4 (- 1))$

Step 2.1.1.3.2.9.2

Cancel the common factor.

$d = \frac{5}{4} \cdot (4 \cdot - 1)$

Step 2.1.1.3.2.9.3

Rewrite the expression.

$d = 5 \cdot - 1$

Step 2.1.1.3.2.10

Multiply $5$ by $- 1$ .

$d = - 5$

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

$e = - \frac{75}{8} - \frac{{(\frac{15}{4})}^{2}}{4 (- \frac{3}{8})}$

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 $\frac{15}{4}$ .

$e = - \frac{75}{8} - \frac{\frac{15^{2}}{4^{2}}}{4 (- \frac{3}{8})}$

Step 2.1.1.4.2.1.1.2

Raise $15$ to the power of $2$ .

$e = - \frac{75}{8} - \frac{\frac{225}{4^{2}}}{4 (- \frac{3}{8})}$

Step 2.1.1.4.2.1.1.3

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

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{4 (- \frac{3}{8})}$

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 $4$ .

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{- 4 (\frac{3}{8})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{3}{8}$ .

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{- 4 \cdot 3}{8}}$

Step 2.1.1.4.2.1.3

Multiply $- 4$ by $3$ .

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{- 12}{8}}$

Step 2.1.1.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 2.1.1.4.2.1.4.1

Cancel the common factor of $- 12$ and $8$ .

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Step 2.1.1.4.2.1.4.1.1

Factor $4$ out of $- 12$ .

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{4 (- 3)}{8}}$

Step 2.1.1.4.2.1.4.1.2

Cancel the common factors.

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Step 2.1.1.4.2.1.4.1.2.1

Factor $4$ out of $8$ .

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Step 2.1.1.4.2.1.4.1.2.2

Cancel the common factor.

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{4 \cdot - 3}{4 \cdot 2}}$

Step 2.1.1.4.2.1.4.1.2.3

Rewrite the expression.

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{\frac{- 3}{2}}$

Step 2.1.1.4.2.1.4.2

Move the negative in front of the fraction.

$e = - \frac{75}{8} - \frac{\frac{225}{16}}{- \frac{3}{2}}$

Step 2.1.1.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{75}{8} - (\frac{225}{16} (- \frac{2}{3}))$

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

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

$e = - \frac{75}{8} - (\frac{225}{16} \cdot \frac{- 2}{3})$

Step 2.1.1.4.2.1.6.2

Factor $3$ out of $225$ .

$e = - \frac{75}{8} - (\frac{3 (75)}{16} \cdot \frac{- 2}{3})$

Step 2.1.1.4.2.1.6.3

Cancel the common factor.

$e = - \frac{75}{8} - (\frac{3 \cdot 75}{16} \cdot \frac{- 2}{3})$

Step 2.1.1.4.2.1.6.4

Rewrite the expression.

$e = - \frac{75}{8} - (\frac{75}{16} \cdot - 2)$

Step 2.1.1.4.2.1.7

Cancel the common factor of $2$ .

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Step 2.1.1.4.2.1.7.1

Factor $2$ out of $16$ .

$e = - \frac{75}{8} - (\frac{75}{2 (8)} \cdot - 2)$

Step 2.1.1.4.2.1.7.2

Factor $2$ out of $- 2$ .

$e = - \frac{75}{8} - (\frac{75}{2 \cdot 8} \cdot (2 \cdot - 1))$

Step 2.1.1.4.2.1.7.3

Cancel the common factor.

$e = - \frac{75}{8} - (\frac{75}{2 \cdot 8} \cdot (2 \cdot - 1))$

Step 2.1.1.4.2.1.7.4

Rewrite the expression.

$e = - \frac{75}{8} - (\frac{75}{8} \cdot - 1)$

Step 2.1.1.4.2.1.8

Combine $\frac{75}{8}$ and $- 1$ .

$e = - \frac{75}{8} - \frac{75 \cdot - 1}{8}$

Step 2.1.1.4.2.1.9

Multiply $75$ by $- 1$ .

$e = - \frac{75}{8} - \frac{- 75}{8}$

Step 2.1.1.4.2.1.10

Move the negative in front of the fraction.

$e = - \frac{75}{8} - - \frac{75}{8}$

Step 2.1.1.4.2.1.11

Multiply $- - \frac{75}{8}$ .

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Step 2.1.1.4.2.1.11.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{75}{8} + 1 (\frac{75}{8})$

Step 2.1.1.4.2.1.11.2

Multiply $\frac{75}{8}$ by $1$ .

$e = - \frac{75}{8} + \frac{75}{8}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 75 + 75}{8}$

Step 2.1.1.4.2.3

Add $- 75$ and $75$ .

$e = \frac{0}{8}$

Step 2.1.1.4.2.4

Divide $0$ by $8$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{8} {(x - 5)}^{2} + 0$ .

$- \frac{3}{8} {(x - 5)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{3}{8} \cdot {(x - 5)}^{2} + 0$

Step 2.2

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

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

$h = 5$

$k = 0$

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.

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

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 $- 1 (- 1)$ .

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{3}{8})}$

Step 2.5.3.2

Combine $4$ and $\frac{3}{8}$ .

$- \frac{1}{\frac{4 \cdot 3}{8}}$

Step 2.5.3.3

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{8}}$

Step 2.5.3.4

Cancel the common factor of $12$ and $8$ .

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Step 2.5.3.4.1

Factor $4$ out of $12$ .

$- \frac{1}{\frac{4 (3)}{8}}$

Step 2.5.3.4.2

Cancel the common factors.

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Step 2.5.3.4.2.1

Factor $4$ out of $8$ .

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

Step 2.5.3.4.2.2

Cancel the common factor.

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

Step 2.5.3.4.2.3

Rewrite the expression.

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

Step 2.5.3.5

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.3.6

Multiply $\frac{2}{3}$ by $1$ .

$- \frac{2}{3}$

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.

$(5, - \frac{2}{3})$

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.

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

Step 2.9

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

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{2}{3})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{2}{3}$

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{2}{3})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{2}{3}$

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 $4$ in the expression.

$f (4) = - \frac{3 {(4)}^{2}}{8} + \frac{15 (4)}{4} - \frac{75}{8}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (4) = \frac{- 3 {(4)}^{2} - 75}{8} + \frac{15 (4)}{4}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (4) = \frac{- 3 \cdot 16 - 75}{8} + \frac{15 (4)}{4}$

Step 3.2.2.2

Multiply $- 3$ by $16$ .

$f (4) = \frac{- 48 - 75}{8} + \frac{15 (4)}{4}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $75$ from $- 48$ .

$f (4) = \frac{- 123}{8} + \frac{15 (4)}{4}$

Step 3.2.3.2

Multiply $15$ by $4$ .

$f (4) = \frac{- 123}{8} + \frac{60}{4}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Move the negative in front of the fraction.

$f (4) = - \frac{123}{8} + \frac{60}{4}$

Step 3.2.4.2

Divide $60$ by $4$ .

$f (4) = - \frac{123}{8} + 15$

Step 3.2.5

To write $15$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (4) = - \frac{123}{8} + 15 \cdot \frac{8}{8}$

Step 3.2.6

Combine $15$ and $\frac{8}{8}$ .

$f (4) = - \frac{123}{8} + \frac{15 \cdot 8}{8}$

Step 3.2.7

Combine the numerators over the common denominator.

$f (4) = \frac{- 123 + 15 \cdot 8}{8}$

Step 3.2.8

Simplify the numerator.

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Step 3.2.8.1

Multiply $15$ by $8$ .

$f (4) = \frac{- 123 + 120}{8}$

Step 3.2.8.2

Add $- 123$ and $120$ .

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

Step 3.2.9

Move the negative in front of the fraction.

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

Step 3.2.10

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

$- \frac{3}{8}$

Step 3.3

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

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

Step 3.4

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

$f (3) = - \frac{3 {(3)}^{2}}{8} + \frac{15 (3)}{4} - \frac{75}{8}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (3) = \frac{- 3 {(3)}^{2} - 75}{8} + \frac{15 (3)}{4}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (3) = \frac{- 3 \cdot 9 - 75}{8} + \frac{15 (3)}{4}$

Step 3.5.2.2

Multiply $- 3$ by $9$ .

$f (3) = \frac{- 27 - 75}{8} + \frac{15 (3)}{4}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Subtract $75$ from $- 27$ .

$f (3) = \frac{- 102}{8} + \frac{15 (3)}{4}$

Step 3.5.3.2

Multiply $15$ by $3$ .

$f (3) = \frac{- 102}{8} + \frac{45}{4}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Cancel the common factor of $- 102$ and $8$ .

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Step 3.5.4.1.1

Factor $2$ out of $- 102$ .

$f (3) = \frac{2 (- 51)}{8} + \frac{45}{4}$

Step 3.5.4.1.2

Cancel the common factors.

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Step 3.5.4.1.2.1

Factor $2$ out of $8$ .

$f (3) = \frac{2 \cdot - 51}{2 \cdot 4} + \frac{45}{4}$

Step 3.5.4.1.2.2

Cancel the common factor.

$f (3) = \frac{2 \cdot - 51}{2 \cdot 4} + \frac{45}{4}$

Step 3.5.4.1.2.3

Rewrite the expression.

$f (3) = \frac{- 51}{4} + \frac{45}{4}$

Step 3.5.4.2

Move the negative in front of the fraction.

$f (3) = - \frac{51}{4} + \frac{45}{4}$

Step 3.5.5

Simplify terms.

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Step 3.5.5.1

Combine the numerators over the common denominator.

$f (3) = \frac{- 51 + 45}{4}$

Step 3.5.5.2

Add $- 51$ and $45$ .

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

Step 3.5.5.3

Cancel the common factor of $- 6$ and $4$ .

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Step 3.5.5.3.1

Factor $2$ out of $- 6$ .

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

Step 3.5.5.3.2

Cancel the common factors.

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Step 3.5.5.3.2.1

Factor $2$ out of $4$ .

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

Step 3.5.5.3.2.2

Cancel the common factor.

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

Step 3.5.5.3.2.3

Rewrite the expression.

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

Step 3.5.5.4

Move the negative in front of the fraction.

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

Step 3.5.6

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

$- \frac{3}{2}$

Step 3.6

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

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

Step 3.7

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

$f (6) = - \frac{3 {(6)}^{2}}{8} + \frac{15 (6)}{4} - \frac{75}{8}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

$f (6) = \frac{- 3 {(6)}^{2} - 75}{8} + \frac{15 (6)}{4}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

$f (6) = \frac{- 3 \cdot 36 - 75}{8} + \frac{15 (6)}{4}$

Step 3.8.2.2

Multiply $- 3$ by $36$ .

$f (6) = \frac{- 108 - 75}{8} + \frac{15 (6)}{4}$

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Subtract $75$ from $- 108$ .

$f (6) = \frac{- 183}{8} + \frac{15 (6)}{4}$

Step 3.8.3.2

Multiply $15$ by $6$ .

$f (6) = \frac{- 183}{8} + \frac{90}{4}$

Step 3.8.4

Simplify each term.

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Step 3.8.4.1

Move the negative in front of the fraction.

$f (6) = - \frac{183}{8} + \frac{90}{4}$

Step 3.8.4.2

Cancel the common factor of $90$ and $4$ .

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Step 3.8.4.2.1

Factor $2$ out of $90$ .

$f (6) = - \frac{183}{8} + \frac{2 (45)}{4}$

Step 3.8.4.2.2

Cancel the common factors.

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Step 3.8.4.2.2.1

Factor $2$ out of $4$ .

$f (6) = - \frac{183}{8} + \frac{2 \cdot 45}{2 \cdot 2}$

Step 3.8.4.2.2.2

Cancel the common factor.

$f (6) = - \frac{183}{8} + \frac{2 \cdot 45}{2 \cdot 2}$

Step 3.8.4.2.2.3

Rewrite the expression.

$f (6) = - \frac{183}{8} + \frac{45}{2}$

Step 3.8.5

To write $\frac{45}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (6) = - \frac{183}{8} + \frac{45}{2} \cdot \frac{4}{4}$

Step 3.8.6

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.6.1

Multiply $\frac{45}{2}$ by $\frac{4}{4}$ .

$f (6) = - \frac{183}{8} + \frac{45 \cdot 4}{2 \cdot 4}$

Step 3.8.6.2

Multiply $2$ by $4$ .

$f (6) = - \frac{183}{8} + \frac{45 \cdot 4}{8}$

Step 3.8.7

Combine the numerators over the common denominator.

$f (6) = \frac{- 183 + 45 \cdot 4}{8}$

Step 3.8.8

Simplify the numerator.

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Step 3.8.8.1

Multiply $45$ by $4$ .

$f (6) = \frac{- 183 + 180}{8}$

Step 3.8.8.2

Add $- 183$ and $180$ .

$f (6) = \frac{- 3}{8}$

Step 3.8.9

Move the negative in front of the fraction.

$f (6) = - \frac{3}{8}$

Step 3.8.10

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

$- \frac{3}{8}$

Step 3.9

The $y$ value at $x = 6$ is $- \frac{3}{8}$ .

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

Step 3.10

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

$f (7) = - \frac{3 {(7)}^{2}}{8} + \frac{15 (7)}{4} - \frac{75}{8}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (7) = \frac{- 3 {(7)}^{2} - 75}{8} + \frac{15 (7)}{4}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

Raise $7$ to the power of $2$ .

$f (7) = \frac{- 3 \cdot 49 - 75}{8} + \frac{15 (7)}{4}$

Step 3.11.2.2

Multiply $- 3$ by $49$ .

$f (7) = \frac{- 147 - 75}{8} + \frac{15 (7)}{4}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Subtract $75$ from $- 147$ .

$f (7) = \frac{- 222}{8} + \frac{15 (7)}{4}$

Step 3.11.3.2

Multiply $15$ by $7$ .

$f (7) = \frac{- 222}{8} + \frac{105}{4}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Cancel the common factor of $- 222$ and $8$ .

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Step 3.11.4.1.1

Factor $2$ out of $- 222$ .

$f (7) = \frac{2 (- 111)}{8} + \frac{105}{4}$

Step 3.11.4.1.2

Cancel the common factors.

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Step 3.11.4.1.2.1

Factor $2$ out of $8$ .

$f (7) = \frac{2 \cdot - 111}{2 \cdot 4} + \frac{105}{4}$

Step 3.11.4.1.2.2

Cancel the common factor.

$f (7) = \frac{2 \cdot - 111}{2 \cdot 4} + \frac{105}{4}$

Step 3.11.4.1.2.3

Rewrite the expression.

$f (7) = \frac{- 111}{4} + \frac{105}{4}$

Step 3.11.4.2

Move the negative in front of the fraction.

$f (7) = - \frac{111}{4} + \frac{105}{4}$

Step 3.11.5

Simplify terms.

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Step 3.11.5.1

Combine the numerators over the common denominator.

$f (7) = \frac{- 111 + 105}{4}$

Step 3.11.5.2

Add $- 111$ and $105$ .

$f (7) = \frac{- 6}{4}$

Step 3.11.5.3

Cancel the common factor of $- 6$ and $4$ .

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Step 3.11.5.3.1

Factor $2$ out of $- 6$ .

$f (7) = \frac{2 (- 3)}{4}$

Step 3.11.5.3.2

Cancel the common factors.

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Step 3.11.5.3.2.1

Factor $2$ out of $4$ .

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

Step 3.11.5.3.2.2

Cancel the common factor.

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

Step 3.11.5.3.2.3

Rewrite the expression.

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

Step 3.11.5.4

Move the negative in front of the fraction.

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

Step 3.11.6

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

$- \frac{3}{2}$

Step 3.12

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

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

Step 3.13

Graph the parabola using its properties and the selected points.

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

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(5, 0)$

Focus: $(5, - \frac{2}{3})$

Axis of Symmetry: $x = 5$

Directrix: $y = \frac{2}{3}$

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

Step 5