Pre-Algebra Examples

$y + 1 = - \frac{1}{12} \cdot {(x - 3)}^{2}$

Step 1

Solve for $y$ .

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Step 1.1

Simplify $- \frac{1}{12} \cdot {(x - 3)}^{2}$ .

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Step 1.1.1

Rewrite.

$y + 1 = 0 + 0 - \frac{1}{12} \cdot {(x - 3)}^{2}$

Step 1.1.2

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$y + 1 = - \frac{1}{12} \cdot ((x - 3) (x - 3))$

Step 1.1.3

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

$y + 1 = - \frac{1}{12} \cdot (x (x - 3) - 3 (x - 3))$

Step 1.1.3.2

Apply the distributive property.

$y + 1 = - \frac{1}{12} \cdot (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Step 1.1.3.3

Apply the distributive property.

$y + 1 = - \frac{1}{12} \cdot (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

Multiply $x$ by $x$ .

$y + 1 = - \frac{1}{12} \cdot (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.4.1.2

Move $- 3$ to the left of $x$ .

$y + 1 = - \frac{1}{12} \cdot (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Step 1.1.4.1.3

Multiply $- 3$ by $- 3$ .

$y + 1 = - \frac{1}{12} \cdot (x^{2} - 3 x - 3 x + 9)$

Step 1.1.4.2

Subtract $3 x$ from $- 3 x$ .

$y + 1 = - \frac{1}{12} \cdot (x^{2} - 6 x + 9)$

Step 1.1.5

Apply the distributive property.

$y + 1 = - \frac{1}{12} x^{2} - \frac{1}{12} (- 6 x) - \frac{1}{12} \cdot 9$

Step 1.1.6

Simplify.

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Step 1.1.6.1

Combine $x^{2}$ and $\frac{1}{12}$ .

$y + 1 = - \frac{x^{2}}{12} - \frac{1}{12} (- 6 x) - \frac{1}{12} \cdot 9$

Step 1.1.6.2

Cancel the common factor of $6$ .

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Step 1.1.6.2.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$y + 1 = - \frac{x^{2}}{12} + \frac{- 1}{12} (- 6 x) - \frac{1}{12} \cdot 9$

Step 1.1.6.2.2

Factor $6$ out of $12$ .

$y + 1 = - \frac{x^{2}}{12} + \frac{- 1}{6 (2)} (- 6 x) - \frac{1}{12} \cdot 9$

Step 1.1.6.2.3

Factor $6$ out of $- 6 x$ .

$y + 1 = - \frac{x^{2}}{12} + \frac{- 1}{6 (2)} (6 (- x)) - \frac{1}{12} \cdot 9$

Step 1.1.6.2.4

Cancel the common factor.

$y + 1 = - \frac{x^{2}}{12} + \frac{- 1}{6 \cdot 2} (6 (- x)) - \frac{1}{12} \cdot 9$

Step 1.1.6.2.5

Rewrite the expression.

$y + 1 = - \frac{x^{2}}{12} + \frac{- 1}{2} (- x) - \frac{1}{12} \cdot 9$

Step 1.1.6.3

Combine $\frac{- 1}{2}$ and $x$ .

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} - \frac{1}{12} \cdot 9$

Step 1.1.6.4

Cancel the common factor of $3$ .

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Step 1.1.6.4.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1}{12} \cdot 9$

Step 1.1.6.4.2

Factor $3$ out of $12$ .

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1}{3 (4)} \cdot 9$

Step 1.1.6.4.3

Factor $3$ out of $9$ .

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot 3)$

Step 1.1.6.4.4

Cancel the common factor.

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot 3)$

Step 1.1.6.4.5

Rewrite the expression.

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1}{4} \cdot 3$

Step 1.1.6.5

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

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 1 \cdot 3}{4}$

Step 1.1.6.6

Multiply $- 1$ by $3$ .

$y + 1 = - \frac{x^{2}}{12} - \frac{- x}{2} + \frac{- 3}{4}$

Step 1.1.7

Simplify each term.

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Step 1.1.7.1

Move the negative in front of the fraction.

$y + 1 = - \frac{x^{2}}{12} - - \frac{x}{2} + \frac{- 3}{4}$

Step 1.1.7.2

Multiply $- - \frac{x}{2}$ .

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Step 1.1.7.2.1

Multiply $- 1$ by $- 1$ .

$y + 1 = - \frac{x^{2}}{12} + 1 \frac{x}{2} + \frac{- 3}{4}$

Step 1.1.7.2.2

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

$y + 1 = - \frac{x^{2}}{12} + \frac{x}{2} + \frac{- 3}{4}$

Step 1.1.7.3

Move the negative in front of the fraction.

$y + 1 = - \frac{x^{2}}{12} + \frac{x}{2} - \frac{3}{4}$

Step 1.2

Move all terms not containing $y$ to the right side of the equation.

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Step 1.2.1

Subtract $1$ from both sides of the equation.

$y = - \frac{x^{2}}{12} + \frac{x}{2} - \frac{3}{4} - 1$

Step 1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = - \frac{x^{2}}{12} + \frac{x}{2} - \frac{3}{4} - 1 \cdot \frac{4}{4}$

Step 1.2.3

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

$y = - \frac{x^{2}}{12} + \frac{x}{2} - \frac{3}{4} + \frac{- 1 \cdot 4}{4}$

Step 1.2.4

Combine the numerators over the common denominator.

$y = - \frac{x^{2}}{12} + \frac{x}{2} + \frac{- 3 - 1 \cdot 4}{4}$

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $4$ .

$y = - \frac{x^{2}}{12} + \frac{x}{2} + \frac{- 3 - 4}{4}$

Step 1.2.5.2

Subtract $4$ from $- 3$ .

$y = - \frac{x^{2}}{12} + \frac{x}{2} + \frac{- 7}{4}$

Step 1.2.6

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{12} + \frac{x}{2} - \frac{7}{4}$

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 $- \frac{x^{2}}{12} + \frac{x}{2} - \frac{7}{4}$ .

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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 $c$ .

$a = - \frac{1}{12}$

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

$c = - \frac{7}{4}$

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

$d = \frac{\frac{1}{2}}{2 (- \frac{1}{12})}$

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.

$d = \frac{1}{2} \cdot \frac{1}{2 (- \frac{1}{12})}$

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 $- 1 (- 1)$ .

$d = \frac{1}{2} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{12})}$

Step 2.1.1.3.2.2.2

Move the negative in front of the fraction.

$d = \frac{1}{2} (- \frac{1}{2 (\frac{1}{12})})$

Step 2.1.1.3.2.3

Combine $2$ and $\frac{1}{12}$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2}{12}})$

Step 2.1.1.3.2.4

Cancel the common factor of $2$ and $12$ .

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Step 2.1.1.3.2.4.1

Factor $2$ out of $2$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2 (1)}{12}})$

Step 2.1.1.3.2.4.2

Cancel the common factors.

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Step 2.1.1.3.2.4.2.1

Factor $2$ out of $12$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Step 2.1.1.3.2.4.2.2

Cancel the common factor.

$d = \frac{1}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Step 2.1.1.3.2.4.2.3

Rewrite the expression.

$d = \frac{1}{2} (- \frac{1}{\frac{1}{6}})$

Step 2.1.1.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} (- (1 \cdot 6))$

Step 2.1.1.3.2.6

Multiply $- (1 \cdot 6)$ .

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Step 2.1.1.3.2.6.1

Multiply $6$ by $1$ .

$d = \frac{1}{2} (- 1 \cdot 6)$

Step 2.1.1.3.2.6.2

Multiply $- 1$ by $6$ .

$d = \frac{1}{2} \cdot - 6$

Step 2.1.1.3.2.7

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.7.1

Factor $2$ out of $- 6$ .

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

Step 2.1.1.3.2.7.2

Cancel the common factor.

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

Step 2.1.1.3.2.7.3

Rewrite the expression.

$d = - 3$

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

$e = - \frac{7}{4} - \frac{{(\frac{1}{2})}^{2}}{4 (- \frac{1}{12})}$

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 $\frac{1}{2}$ .

$e = - \frac{7}{4} - \frac{\frac{1^{2}}{2^{2}}}{4 (- \frac{1}{12})}$

Step 2.1.1.4.2.1.1.2

One to any power is one.

$e = - \frac{7}{4} - \frac{\frac{1}{2^{2}}}{4 (- \frac{1}{12})}$

Step 2.1.1.4.2.1.1.3

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

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{4 (- \frac{1}{12})}$

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 $4$ .

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{- 4 (\frac{1}{12})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{1}{12}$ .

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{\frac{- 4}{12}}$

Step 2.1.1.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 2.1.1.4.2.1.3.1

Cancel the common factor of $- 4$ and $12$ .

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Step 2.1.1.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{\frac{4 (- 1)}{12}}$

Step 2.1.1.4.2.1.3.1.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 2.1.1.4.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 2.1.1.4.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{\frac{- 1}{3}}$

Step 2.1.1.4.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{7}{4} - \frac{\frac{1}{4}}{- \frac{1}{3}}$

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{7}{4} - (\frac{1}{4} (- 1 \cdot 3))$

Step 2.1.1.4.2.1.5

Multiply $- 1$ by $3$ .

$e = - \frac{7}{4} - (\frac{1}{4} \cdot - 3)$

Step 2.1.1.4.2.1.6

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

$e = - \frac{7}{4} - \frac{- 3}{4}$

Step 2.1.1.4.2.1.7

Move the negative in front of the fraction.

$e = - \frac{7}{4} - - \frac{3}{4}$

Step 2.1.1.4.2.1.8

Multiply $- - \frac{3}{4}$ .

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Step 2.1.1.4.2.1.8.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{7}{4} + 1 (\frac{3}{4})$

Step 2.1.1.4.2.1.8.2

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

$e = - \frac{7}{4} + \frac{3}{4}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 7 + 3}{4}$

Step 2.1.1.4.2.3

Add $- 7$ and $3$ .

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

Step 2.1.1.4.2.4

Divide $- 4$ by $4$ .

$e = - 1$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(x - 3)}^{2} - 1$ .

$- \frac{1}{12} {(x - 3)}^{2} - 1$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{12} \cdot {(x - 3)}^{2} - 1$

Step 2.2

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

$a = - \frac{1}{12}$

$h = 3$

$k = - 1$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(3, - 1)$

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.

$\frac{1}{4 \cdot (- \frac{1}{12})}$

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 $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{12})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{12})}$

Step 2.5.3.2

Combine $4$ and $\frac{1}{12}$ .

$- \frac{1}{\frac{4}{12}}$

Step 2.5.3.3

Cancel the common factor of $4$ and $12$ .

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Step 2.5.3.3.1

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{12}}$

Step 2.5.3.3.2

Cancel the common factors.

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Step 2.5.3.3.2.1

Factor $4$ out of $12$ .

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

Step 2.5.3.3.2.2

Cancel the common factor.

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

Step 2.5.3.3.2.3

Rewrite the expression.

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

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 2.5.3.5

Multiply $- (1 \cdot 3)$ .

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Step 2.5.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 2.5.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

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.

$(3, - 4)$

Step 2.7

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

$x = 3$

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.

$y = 2$

Step 2.9

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

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - 4)$

Axis of Symmetry: $x = 3$

Directrix: $y = 2$

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - 4)$

Axis of Symmetry: $x = 3$

Directrix: $y = 2$

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 $2$ in the expression.

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2}{2} - \frac{7}{4}$

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Multiply $\frac{2}{2}$ by $\frac{6}{6}$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2}{2} \cdot \frac{6}{6} - \frac{7}{4}$

Step 3.2.1.2

Multiply $\frac{2}{2}$ by $\frac{6}{6}$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{2 \cdot 6} - \frac{7}{4}$

Step 3.2.1.3

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{2 \cdot 6} - (\frac{7}{4} \cdot \frac{3}{3})$

Step 3.2.1.4

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{2 \cdot 6} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.2.1.5

Multiply $2$ by $6$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{12} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.2.1.6

Reorder the factors of $4 \cdot 3$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{12} - \frac{7 \cdot 3}{3 \cdot 4}$

Step 3.2.1.7

Multiply $3$ by $4$ .

$f (2) = - \frac{{(2)}^{2}}{12} + \frac{2 \cdot 6}{12} - \frac{7 \cdot 3}{12}$

Step 3.2.2

Combine the numerators over the common denominator.

$f (2) = \frac{- {(2)}^{2} + 2 \cdot 6 - 7 \cdot 3}{12}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

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

$f (2) = \frac{- 1 \cdot 4 + 2 \cdot 6 - 7 \cdot 3}{12}$

Step 3.2.3.2

Multiply $- 1$ by $4$ .

$f (2) = \frac{- 4 + 2 \cdot 6 - 7 \cdot 3}{12}$

Step 3.2.3.3

Multiply $2$ by $6$ .

$f (2) = \frac{- 4 + 12 - 7 \cdot 3}{12}$

Step 3.2.3.4

Multiply $- 7$ by $3$ .

$f (2) = \frac{- 4 + 12 - 21}{12}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $- 4$ and $12$ .

$f (2) = \frac{8 - 21}{12}$

Step 3.2.4.2

Subtract $21$ from $8$ .

$f (2) = \frac{- 13}{12}$

Step 3.2.4.3

Move the negative in front of the fraction.

$f (2) = - \frac{13}{12}$

Step 3.2.5

The final answer is $- \frac{13}{12}$ .

$- \frac{13}{12}$

Step 3.3

The $y$ value at $x = 2$ is $- \frac{13}{12}$ .

$y = - \frac{13}{12}$

Step 3.4

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{1}{2} - \frac{7}{4}$

Step 3.5

Simplify the result.

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Step 3.5.1

Find the common denominator.

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Step 3.5.1.1

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{1}{2} \cdot \frac{6}{6} - \frac{7}{4}$

Step 3.5.1.2

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{2 \cdot 6} - \frac{7}{4}$

Step 3.5.1.3

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{2 \cdot 6} - (\frac{7}{4} \cdot \frac{3}{3})$

Step 3.5.1.4

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{2 \cdot 6} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.5.1.5

Multiply $2$ by $6$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{12} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.5.1.6

Reorder the factors of $4 \cdot 3$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{12} - \frac{7 \cdot 3}{3 \cdot 4}$

Step 3.5.1.7

Multiply $3$ by $4$ .

$f (1) = - \frac{{(1)}^{2}}{12} + \frac{6}{12} - \frac{7 \cdot 3}{12}$

Step 3.5.2

Combine the numerators over the common denominator.

$f (1) = \frac{- {(1)}^{2} + 6 - 7 \cdot 3}{12}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

One to any power is one.

$f (1) = \frac{- 1 \cdot 1 + 6 - 7 \cdot 3}{12}$

Step 3.5.3.2

Multiply $- 1$ by $1$ .

$f (1) = \frac{- 1 + 6 - 7 \cdot 3}{12}$

Step 3.5.3.3

Multiply $- 7$ by $3$ .

$f (1) = \frac{- 1 + 6 - 21}{12}$

Step 3.5.4

Reduce the expression by cancelling the common factors.

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Step 3.5.4.1

Add $- 1$ and $6$ .

$f (1) = \frac{5 - 21}{12}$

Step 3.5.4.2

Subtract $21$ from $5$ .

$f (1) = \frac{- 16}{12}$

Step 3.5.4.3

Cancel the common factor of $- 16$ and $12$ .

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Step 3.5.4.3.1

Factor $4$ out of $- 16$ .

$f (1) = \frac{4 (- 4)}{12}$

Step 3.5.4.3.2

Cancel the common factors.

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Step 3.5.4.3.2.1

Factor $4$ out of $12$ .

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

Step 3.5.4.3.2.2

Cancel the common factor.

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

Step 3.5.4.3.2.3

Rewrite the expression.

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

Step 3.5.4.4

Move the negative in front of the fraction.

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

Step 3.5.5

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

$- \frac{4}{3}$

Step 3.6

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

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

Step 3.7

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

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4}{2} - \frac{7}{4}$

Step 3.8

Simplify the result.

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Step 3.8.1

Find the common denominator.

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Step 3.8.1.1

Multiply $\frac{4}{2}$ by $\frac{6}{6}$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4}{2} \cdot \frac{6}{6} - \frac{7}{4}$

Step 3.8.1.2

Multiply $\frac{4}{2}$ by $\frac{6}{6}$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{2 \cdot 6} - \frac{7}{4}$

Step 3.8.1.3

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{2 \cdot 6} - (\frac{7}{4} \cdot \frac{3}{3})$

Step 3.8.1.4

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{2 \cdot 6} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.8.1.5

Multiply $2$ by $6$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{12} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.8.1.6

Reorder the factors of $4 \cdot 3$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{12} - \frac{7 \cdot 3}{3 \cdot 4}$

Step 3.8.1.7

Multiply $3$ by $4$ .

$f (4) = - \frac{{(4)}^{2}}{12} + \frac{4 \cdot 6}{12} - \frac{7 \cdot 3}{12}$

Step 3.8.2

Combine the numerators over the common denominator.

$f (4) = \frac{- {(4)}^{2} + 4 \cdot 6 - 7 \cdot 3}{12}$

Step 3.8.3

Simplify each term.

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Step 3.8.3.1

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

$f (4) = \frac{- 1 \cdot 16 + 4 \cdot 6 - 7 \cdot 3}{12}$

Step 3.8.3.2

Multiply $- 1$ by $16$ .

$f (4) = \frac{- 16 + 4 \cdot 6 - 7 \cdot 3}{12}$

Step 3.8.3.3

Multiply $4$ by $6$ .

$f (4) = \frac{- 16 + 24 - 7 \cdot 3}{12}$

Step 3.8.3.4

Multiply $- 7$ by $3$ .

$f (4) = \frac{- 16 + 24 - 21}{12}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Add $- 16$ and $24$ .

$f (4) = \frac{8 - 21}{12}$

Step 3.8.4.2

Subtract $21$ from $8$ .

$f (4) = \frac{- 13}{12}$

Step 3.8.4.3

Move the negative in front of the fraction.

$f (4) = - \frac{13}{12}$

Step 3.8.5

The final answer is $- \frac{13}{12}$ .

$- \frac{13}{12}$

Step 3.9

The $y$ value at $x = 4$ is $- \frac{13}{12}$ .

$y = - \frac{13}{12}$

Step 3.10

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5}{2} - \frac{7}{4}$

Step 3.11

Simplify the result.

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Step 3.11.1

Find the common denominator.

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Step 3.11.1.1

Multiply $\frac{5}{2}$ by $\frac{6}{6}$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5}{2} \cdot \frac{6}{6} - \frac{7}{4}$

Step 3.11.1.2

Multiply $\frac{5}{2}$ by $\frac{6}{6}$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{2 \cdot 6} - \frac{7}{4}$

Step 3.11.1.3

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{2 \cdot 6} - (\frac{7}{4} \cdot \frac{3}{3})$

Step 3.11.1.4

Multiply $\frac{7}{4}$ by $\frac{3}{3}$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{2 \cdot 6} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.11.1.5

Multiply $2$ by $6$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{12} - \frac{7 \cdot 3}{4 \cdot 3}$

Step 3.11.1.6

Reorder the factors of $4 \cdot 3$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{12} - \frac{7 \cdot 3}{3 \cdot 4}$

Step 3.11.1.7

Multiply $3$ by $4$ .

$f (5) = - \frac{{(5)}^{2}}{12} + \frac{5 \cdot 6}{12} - \frac{7 \cdot 3}{12}$

Step 3.11.2

Combine the numerators over the common denominator.

$f (5) = \frac{- {(5)}^{2} + 5 \cdot 6 - 7 \cdot 3}{12}$

Step 3.11.3

Simplify each term.

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Step 3.11.3.1

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

$f (5) = \frac{- 1 \cdot 25 + 5 \cdot 6 - 7 \cdot 3}{12}$

Step 3.11.3.2

Multiply $- 1$ by $25$ .

$f (5) = \frac{- 25 + 5 \cdot 6 - 7 \cdot 3}{12}$

Step 3.11.3.3

Multiply $5$ by $6$ .

$f (5) = \frac{- 25 + 30 - 7 \cdot 3}{12}$

Step 3.11.3.4

Multiply $- 7$ by $3$ .

$f (5) = \frac{- 25 + 30 - 21}{12}$

Step 3.11.4

Reduce the expression by cancelling the common factors.

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Step 3.11.4.1

Add $- 25$ and $30$ .

$f (5) = \frac{5 - 21}{12}$

Step 3.11.4.2

Subtract $21$ from $5$ .

$f (5) = \frac{- 16}{12}$

Step 3.11.4.3

Cancel the common factor of $- 16$ and $12$ .

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Step 3.11.4.3.1

Factor $4$ out of $- 16$ .

$f (5) = \frac{4 (- 4)}{12}$

Step 3.11.4.3.2

Cancel the common factors.

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Step 3.11.4.3.2.1

Factor $4$ out of $12$ .

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

Step 3.11.4.3.2.2

Cancel the common factor.

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

Step 3.11.4.3.2.3

Rewrite the expression.

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

Step 3.11.4.4

Move the negative in front of the fraction.

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

Step 3.11.5

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

$- \frac{4}{3}$

Step 3.12

The $y$ value at $x = 5$ is $- \frac{4}{3}$ .

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

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 1 & - \frac{4}{3} \\ 2 & - \frac{13}{12} \\ 3 & - 1 \\ 4 & - \frac{13}{12} \\ 5 & - \frac{4}{3} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(3, - 1)$

Focus: $(3, - 4)$

Axis of Symmetry: $x = 3$

Directrix: $y = 2$

$\begin{array}{c} x & y \\ 1 & - \frac{4}{3} \\ 2 & - \frac{13}{12} \\ 3 & - 1 \\ 4 & - \frac{13}{12} \\ 5 & - \frac{4}{3} \end{array}$

Step 5