Pre-Algebra Examples

$3 - 15 y - 5 y = 6 x^{2} + 7 x$

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

Isolate $y$ to the left side of the equation.

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Step 1.1.1.1

Subtract $5 y$ from $- 15 y$ .

$3 - 20 y = 6 x^{2} + 7 x$

Step 1.1.1.2

Subtract $3$ from both sides of the equation.

$- 20 y = 6 x^{2} + 7 x - 3$

Step 1.1.1.3

Divide each term in $- 20 y = 6 x^{2} + 7 x - 3$ by $- 20$ and simplify.

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Step 1.1.1.3.1

Divide each term in $- 20 y = 6 x^{2} + 7 x - 3$ by $- 20$ .

$\frac{- 20 y}{- 20} = \frac{6 x^{2}}{- 20} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.2

Simplify the left side.

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Step 1.1.1.3.2.1

Cancel the common factor of $- 20$ .

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Step 1.1.1.3.2.1.1

Cancel the common factor.

$\frac{- 20 y}{- 20} = \frac{6 x^{2}}{- 20} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6 x^{2}}{- 20} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3

Simplify the right side.

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Step 1.1.1.3.3.1

Simplify each term.

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Step 1.1.1.3.3.1.1

Cancel the common factor of $6$ and $- 20$ .

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Step 1.1.1.3.3.1.1.1

Factor $2$ out of $6 x^{2}$ .

$y = \frac{2 (3 x^{2})}{- 20} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.1.2

Cancel the common factors.

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Step 1.1.1.3.3.1.1.2.1

Factor $2$ out of $- 20$ .

$y = \frac{2 (3 x^{2})}{2 (- 10)} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 (3 x^{2})}{2 \cdot - 10} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.1.2.3

Rewrite the expression.

$y = \frac{3 x^{2}}{- 10} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.2

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{10} + \frac{7 x}{- 20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.3

Move the negative in front of the fraction.

$y = - \frac{3 x^{2}}{10} - \frac{7 x}{20} + \frac{- 3}{- 20}$

Step 1.1.1.3.3.1.4

Dividing two negative values results in a positive value.

$y = - \frac{3 x^{2}}{10} - \frac{7 x}{20} + \frac{3}{20}$

Step 1.1.2

Complete the square for $- \frac{3 x^{2}}{10} - \frac{7 x}{20} + \frac{3}{20}$ .

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Step 1.1.2.1

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

$a = - \frac{3}{10}$

$b = - \frac{7}{20}$

$c = \frac{3}{20}$

Step 1.1.2.2

Consider the vertex form of a parabola.

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

$d = \frac{- \frac{7}{20}}{2 (- \frac{3}{10})}$

Step 1.1.2.3.2

Simplify the right side.

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Step 1.1.2.3.2.1

Dividing two negative values results in a positive value.

$d = \frac{\frac{7}{20}}{2 (\frac{3}{10})}$

Step 1.1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{7}{20} \cdot \frac{1}{2 (\frac{3}{10})}$

Step 1.1.2.3.2.3

Combine $2$ and $\frac{3}{10}$ .

$d = \frac{7}{20} \cdot \frac{1}{\frac{2 \cdot 3}{10}}$

Step 1.1.2.3.2.4

Multiply $2$ by $3$ .

$d = \frac{7}{20} \cdot \frac{1}{\frac{6}{10}}$

Step 1.1.2.3.2.5

Cancel the common factor of $6$ and $10$ .

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Step 1.1.2.3.2.5.1

Factor $2$ out of $6$ .

$d = \frac{7}{20} \cdot \frac{1}{\frac{2 (3)}{10}}$

Step 1.1.2.3.2.5.2

Cancel the common factors.

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Step 1.1.2.3.2.5.2.1

Factor $2$ out of $10$ .

$d = \frac{7}{20} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 5}}$

Step 1.1.2.3.2.5.2.2

Cancel the common factor.

$d = \frac{7}{20} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 5}}$

Step 1.1.2.3.2.5.2.3

Rewrite the expression.

$d = \frac{7}{20} \cdot \frac{1}{\frac{3}{5}}$

Step 1.1.2.3.2.6

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{7}{20} (1 (\frac{5}{3}))$

Step 1.1.2.3.2.7

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

$d = \frac{7}{20} \cdot \frac{5}{3}$

Step 1.1.2.3.2.8

Cancel the common factor of $5$ .

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Step 1.1.2.3.2.8.1

Factor $5$ out of $20$ .

$d = \frac{7}{5 (4)} \cdot \frac{5}{3}$

Step 1.1.2.3.2.8.2

Cancel the common factor.

$d = \frac{7}{5 \cdot 4} \cdot \frac{5}{3}$

Step 1.1.2.3.2.8.3

Rewrite the expression.

$d = \frac{7}{4} \cdot \frac{1}{3}$

Step 1.1.2.3.2.9

Multiply $\frac{7}{4}$ by $\frac{1}{3}$ .

$d = \frac{7}{4 \cdot 3}$

Step 1.1.2.3.2.10

Multiply $4$ by $3$ .

$d = \frac{7}{12}$

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

$e = \frac{3}{20} - \frac{{(- \frac{7}{20})}^{2}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2

Simplify the right side.

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Step 1.1.2.4.2.1

Simplify each term.

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Step 1.1.2.4.2.1.1

Simplify the numerator.

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Step 1.1.2.4.2.1.1.1

Apply the product rule to $- \frac{7}{20}$ .

$e = \frac{3}{20} - \frac{{(- 1)}^{2} {(\frac{7}{20})}^{2}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = \frac{3}{20} - \frac{1 {(\frac{7}{20})}^{2}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.1.3

Apply the product rule to $\frac{7}{20}$ .

$e = \frac{3}{20} - \frac{1 \frac{7^{2}}{20^{2}}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.1.4

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

$e = \frac{3}{20} - \frac{1 \frac{49}{20^{2}}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.1.5

Raise $20$ to the power of $2$ .

$e = \frac{3}{20} - \frac{1 (\frac{49}{400})}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.1.6

Multiply $\frac{49}{400}$ by $1$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{4 (- \frac{3}{10})}$

Step 1.1.2.4.2.1.2

Simplify the denominator.

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Step 1.1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{- 4 (\frac{3}{10})}$

Step 1.1.2.4.2.1.2.2

Combine $- 4$ and $\frac{3}{10}$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{- 4 \cdot 3}{10}}$

Step 1.1.2.4.2.1.3

Multiply $- 4$ by $3$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{- 12}{10}}$

Step 1.1.2.4.2.1.4

Reduce the expression by cancelling the common factors.

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Step 1.1.2.4.2.1.4.1

Cancel the common factor of $- 12$ and $10$ .

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Step 1.1.2.4.2.1.4.1.1

Factor $2$ out of $- 12$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{2 (- 6)}{10}}$

Step 1.1.2.4.2.1.4.1.2

Cancel the common factors.

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Step 1.1.2.4.2.1.4.1.2.1

Factor $2$ out of $10$ .

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{2 \cdot - 6}{2 \cdot 5}}$

Step 1.1.2.4.2.1.4.1.2.2

Cancel the common factor.

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{2 \cdot - 6}{2 \cdot 5}}$

Step 1.1.2.4.2.1.4.1.2.3

Rewrite the expression.

$e = \frac{3}{20} - \frac{\frac{49}{400}}{\frac{- 6}{5}}$

Step 1.1.2.4.2.1.4.2

Move the negative in front of the fraction.

$e = \frac{3}{20} - \frac{\frac{49}{400}}{- \frac{6}{5}}$

Step 1.1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{3}{20} - (\frac{49}{400} (- \frac{5}{6}))$

Step 1.1.2.4.2.1.6

Cancel the common factor of $5$ .

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Step 1.1.2.4.2.1.6.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$e = \frac{3}{20} - (\frac{49}{400} \cdot \frac{- 5}{6})$

Step 1.1.2.4.2.1.6.2

Factor $5$ out of $400$ .

$e = \frac{3}{20} - (\frac{49}{5 (80)} \cdot \frac{- 5}{6})$

Step 1.1.2.4.2.1.6.3

Factor $5$ out of $- 5$ .

$e = \frac{3}{20} - (\frac{49}{5 \cdot 80} \cdot \frac{5 \cdot - 1}{6})$

Step 1.1.2.4.2.1.6.4

Cancel the common factor.

$e = \frac{3}{20} - (\frac{49}{5 \cdot 80} \cdot \frac{5 \cdot - 1}{6})$

Step 1.1.2.4.2.1.6.5

Rewrite the expression.

$e = \frac{3}{20} - (\frac{49}{80} \cdot \frac{- 1}{6})$

Step 1.1.2.4.2.1.7

Multiply $\frac{49}{80}$ by $\frac{- 1}{6}$ .

$e = \frac{3}{20} - \frac{49 \cdot - 1}{80 \cdot 6}$

Step 1.1.2.4.2.1.8

Multiply $49$ by $- 1$ .

$e = \frac{3}{20} - \frac{- 49}{80 \cdot 6}$

Step 1.1.2.4.2.1.9

Multiply $80$ by $6$ .

$e = \frac{3}{20} - \frac{- 49}{480}$

Step 1.1.2.4.2.1.10

Move the negative in front of the fraction.

$e = \frac{3}{20} - - \frac{49}{480}$

Step 1.1.2.4.2.1.11

Multiply $- - \frac{49}{480}$ .

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Step 1.1.2.4.2.1.11.1

Multiply $- 1$ by $- 1$ .

$e = \frac{3}{20} + 1 (\frac{49}{480})$

Step 1.1.2.4.2.1.11.2

Multiply $\frac{49}{480}$ by $1$ .

$e = \frac{3}{20} + \frac{49}{480}$

Step 1.1.2.4.2.2

To write $\frac{3}{20}$ as a fraction with a common denominator, multiply by $\frac{24}{24}$ .

$e = \frac{3}{20} \cdot \frac{24}{24} + \frac{49}{480}$

Step 1.1.2.4.2.3

Write each expression with a common denominator of $480$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.1.2.4.2.3.1

Multiply $\frac{3}{20}$ by $\frac{24}{24}$ .

$e = \frac{3 \cdot 24}{20 \cdot 24} + \frac{49}{480}$

Step 1.1.2.4.2.3.2

Multiply $20$ by $24$ .

$e = \frac{3 \cdot 24}{480} + \frac{49}{480}$

Step 1.1.2.4.2.4

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 24 + 49}{480}$

Step 1.1.2.4.2.5

Simplify the numerator.

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Step 1.1.2.4.2.5.1

Multiply $3$ by $24$ .

$e = \frac{72 + 49}{480}$

Step 1.1.2.4.2.5.2

Add $72$ and $49$ .

$e = \frac{121}{480}$

Step 1.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{3}{10} {(x + \frac{7}{12})}^{2} + \frac{121}{480}$ .

$- \frac{3}{10} {(x + \frac{7}{12})}^{2} + \frac{121}{480}$

Step 1.1.3

Set $y$ equal to the new right side.

$y = - \frac{3}{10} \cdot {(x + \frac{7}{12})}^{2} + \frac{121}{480}$

Step 1.2

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

$a = - \frac{3}{10}$

$h = - \frac{7}{12}$

$k = \frac{121}{480}$

Step 1.3

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

Opens Down

Step 1.4

Find the vertex $(h, k)$ .

$(- \frac{7}{12}, \frac{121}{480})$

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.

$\frac{1}{4 \cdot (- \frac{3}{10})}$

Step 1.5.3

Simplify.

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Step 1.5.3.1

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

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Step 1.5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{3}{10})}$

Step 1.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{3}{10})}$

Step 1.5.3.2

Combine $4$ and $\frac{3}{10}$ .

$- \frac{1}{\frac{4 \cdot 3}{10}}$

Step 1.5.3.3

Multiply $4$ by $3$ .

$- \frac{1}{\frac{12}{10}}$

Step 1.5.3.4

Cancel the common factor of $12$ and $10$ .

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Step 1.5.3.4.1

Factor $2$ out of $12$ .

$- \frac{1}{\frac{2 (6)}{10}}$

Step 1.5.3.4.2

Cancel the common factors.

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Step 1.5.3.4.2.1

Factor $2$ out of $10$ .

$- \frac{1}{\frac{2 \cdot 6}{2 \cdot 5}}$

Step 1.5.3.4.2.2

Cancel the common factor.

$- \frac{1}{\frac{2 \cdot 6}{2 \cdot 5}}$

Step 1.5.3.4.2.3

Rewrite the expression.

$- \frac{1}{\frac{6}{5}}$

Step 1.5.3.5

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{6}))$

Step 1.5.3.6

Multiply $\frac{5}{6}$ by $1$ .

$- \frac{5}{6}$

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 1.6.2

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

$(- \frac{7}{12}, - \frac{93}{160})$

Step 1.7

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

$x = - \frac{7}{12}$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$y = \frac{521}{480}$

Step 1.9

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

Direction: Opens Down

Vertex: $(- \frac{7}{12}, \frac{121}{480})$

Focus: $(- \frac{7}{12}, - \frac{93}{160})$

Axis of Symmetry: $x = - \frac{7}{12}$

Directrix: $y = \frac{521}{480}$

Direction: Opens Down

Vertex: $(- \frac{7}{12}, \frac{121}{480})$

Focus: $(- \frac{7}{12}, - \frac{93}{160})$

Axis of Symmetry: $x = - \frac{7}{12}$

Directrix: $y = \frac{521}{480}$

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

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = - \frac{3 {(- 2)}^{2}}{10} - \frac{7 (- 2)}{20} + \frac{3}{20}$

Step 2.2

Simplify the result.

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Step 2.2.1

Combine fractions.

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Step 2.2.1.1

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 3 {(- 2)}^{2}}{10} + \frac{- 7 \cdot - 2 + 3}{20}$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Multiply $- 7$ by $- 2$ .

$f (- 2) = \frac{- 3 {(- 2)}^{2}}{10} + \frac{14 + 3}{20}$

Step 2.2.1.2.2

Add $14$ and $3$ .

$f (- 2) = \frac{- 3 {(- 2)}^{2}}{10} + \frac{17}{20}$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

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

$f (- 2) = \frac{- 3 \cdot 4}{10} + \frac{17}{20}$

Step 2.2.2.2

Multiply $- 3$ by $4$ .

$f (- 2) = \frac{- 12}{10} + \frac{17}{20}$

Step 2.2.2.3

Cancel the common factor of $- 12$ and $10$ .

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Step 2.2.2.3.1

Factor $2$ out of $- 12$ .

$f (- 2) = \frac{2 (- 6)}{10} + \frac{17}{20}$

Step 2.2.2.3.2

Cancel the common factors.

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Step 2.2.2.3.2.1

Factor $2$ out of $10$ .

$f (- 2) = \frac{2 \cdot - 6}{2 \cdot 5} + \frac{17}{20}$

Step 2.2.2.3.2.2

Cancel the common factor.

$f (- 2) = \frac{2 \cdot - 6}{2 \cdot 5} + \frac{17}{20}$

Step 2.2.2.3.2.3

Rewrite the expression.

$f (- 2) = \frac{- 6}{5} + \frac{17}{20}$

Step 2.2.2.4

Move the negative in front of the fraction.

$f (- 2) = - \frac{6}{5} + \frac{17}{20}$

Step 2.2.3

To write $- \frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (- 2) = - \frac{6}{5} \cdot \frac{4}{4} + \frac{17}{20}$

Step 2.2.4

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.4.1

Multiply $\frac{6}{5}$ by $\frac{4}{4}$ .

$f (- 2) = - \frac{6 \cdot 4}{5 \cdot 4} + \frac{17}{20}$

Step 2.2.4.2

Multiply $5$ by $4$ .

$f (- 2) = - \frac{6 \cdot 4}{20} + \frac{17}{20}$

Step 2.2.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 6 \cdot 4 + 17}{20}$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 6$ by $4$ .

$f (- 2) = \frac{- 24 + 17}{20}$

Step 2.2.6.2

Add $- 24$ and $17$ .

$f (- 2) = \frac{- 7}{20}$

Step 2.2.7

Move the negative in front of the fraction.

$f (- 2) = - \frac{7}{20}$

Step 2.2.8

The final answer is $- \frac{7}{20}$ .

$- \frac{7}{20}$

Step 2.3

The $y$ value at $x = - 2$ is $- \frac{7}{20}$ .

$y = - \frac{7}{20}$

Step 2.4

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

$f (- 3) = - \frac{3 {(- 3)}^{2}}{10} - \frac{7 (- 3)}{20} + \frac{3}{20}$

Step 2.5

Simplify the result.

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Step 2.5.1

Combine fractions.

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Step 2.5.1.1

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 3 {(- 3)}^{2}}{10} + \frac{- 7 \cdot - 3 + 3}{20}$

Step 2.5.1.2

Simplify the expression.

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Step 2.5.1.2.1

Multiply $- 7$ by $- 3$ .

$f (- 3) = \frac{- 3 {(- 3)}^{2}}{10} + \frac{21 + 3}{20}$

Step 2.5.1.2.2

Add $21$ and $3$ .

$f (- 3) = \frac{- 3 {(- 3)}^{2}}{10} + \frac{24}{20}$

Step 2.5.2

Simplify each term.

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Step 2.5.2.1

Multiply $- 3$ by ${(- 3)}^{2}$ by adding the exponents.

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Step 2.5.2.1.1

Multiply $- 3$ by ${(- 3)}^{2}$ .

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Step 2.5.2.1.1.1

Raise $- 3$ to the power of $1$ .

$f (- 3) = \frac{(- 3) {(- 3)}^{2}}{10} + \frac{24}{20}$

Step 2.5.2.1.1.2

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

$f (- 3) = \frac{{(- 3)}^{1 + 2}}{10} + \frac{24}{20}$

Step 2.5.2.1.2

Add $1$ and $2$ .

$f (- 3) = \frac{{(- 3)}^{3}}{10} + \frac{24}{20}$

Step 2.5.2.2

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

$f (- 3) = \frac{- 27}{10} + \frac{24}{20}$

Step 2.5.2.3

Move the negative in front of the fraction.

$f (- 3) = - \frac{27}{10} + \frac{24}{20}$

Step 2.5.2.4

Cancel the common factor of $24$ and $20$ .

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Step 2.5.2.4.1

Factor $4$ out of $24$ .

$f (- 3) = - \frac{27}{10} + \frac{4 (6)}{20}$

Step 2.5.2.4.2

Cancel the common factors.

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Step 2.5.2.4.2.1

Factor $4$ out of $20$ .

$f (- 3) = - \frac{27}{10} + \frac{4 \cdot 6}{4 \cdot 5}$

Step 2.5.2.4.2.2

Cancel the common factor.

$f (- 3) = - \frac{27}{10} + \frac{4 \cdot 6}{4 \cdot 5}$

Step 2.5.2.4.2.3

Rewrite the expression.

$f (- 3) = - \frac{27}{10} + \frac{6}{5}$

Step 2.5.3

To write $\frac{6}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 3) = - \frac{27}{10} + \frac{6}{5} \cdot \frac{2}{2}$

Step 2.5.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.4.1

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

$f (- 3) = - \frac{27}{10} + \frac{6 \cdot 2}{5 \cdot 2}$

Step 2.5.4.2

Multiply $5$ by $2$ .

$f (- 3) = - \frac{27}{10} + \frac{6 \cdot 2}{10}$

Step 2.5.5

Combine the numerators over the common denominator.

$f (- 3) = \frac{- 27 + 6 \cdot 2}{10}$

Step 2.5.6

Simplify the numerator.

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Step 2.5.6.1

Multiply $6$ by $2$ .

$f (- 3) = \frac{- 27 + 12}{10}$

Step 2.5.6.2

Add $- 27$ and $12$ .

$f (- 3) = \frac{- 15}{10}$

Step 2.5.7

Cancel the common factor of $- 15$ and $10$ .

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Step 2.5.7.1

Factor $5$ out of $- 15$ .

$f (- 3) = \frac{5 (- 3)}{10}$

Step 2.5.7.2

Cancel the common factors.

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Step 2.5.7.2.1

Factor $5$ out of $10$ .

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

Step 2.5.7.2.2

Cancel the common factor.

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

Step 2.5.7.2.3

Rewrite the expression.

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

Step 2.5.8

Move the negative in front of the fraction.

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

Step 2.5.9

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

$- \frac{3}{2}$

Step 2.6

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

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

Step 2.7

Replace the variable $x$ with $0$ in the expression.

$f (0) = - \frac{3 {(0)}^{2}}{10} - \frac{7 (0)}{20} + \frac{3}{20}$

Step 2.8

Simplify the result.

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Step 2.8.1

Combine fractions.

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Step 2.8.1.1

Combine the numerators over the common denominator.

$f (0) = \frac{- 3 {(0)}^{2}}{10} + \frac{- 7 \cdot 0 + 3}{20}$

Step 2.8.1.2

Simplify the expression.

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Step 2.8.1.2.1

Multiply $- 7$ by $0$ .

$f (0) = \frac{- 3 {(0)}^{2}}{10} + \frac{0 + 3}{20}$

Step 2.8.1.2.2

Add $0$ and $3$ .

$f (0) = \frac{- 3 {(0)}^{2}}{10} + \frac{3}{20}$

Step 2.8.2

Simplify each term.

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Step 2.8.2.1

Raising $0$ to any positive power yields $0$ .

$f (0) = \frac{- 3 \cdot 0}{10} + \frac{3}{20}$

Step 2.8.2.2

Multiply $- 3$ by $0$ .

$f (0) = \frac{0}{10} + \frac{3}{20}$

Step 2.8.2.3

Divide $0$ by $10$ .

$f (0) = 0 + \frac{3}{20}$

Step 2.8.3

Add $0$ and $\frac{3}{20}$ .

$f (0) = \frac{3}{20}$

Step 2.8.4

The final answer is $\frac{3}{20}$ .

$\frac{3}{20}$

Step 2.9

The $y$ value at $x = 0$ is $\frac{3}{20}$ .

$y = \frac{3}{20}$

Step 2.10

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

$f (1) = - \frac{3 {(1)}^{2}}{10} - \frac{7 (1)}{20} + \frac{3}{20}$

Step 2.11

Simplify the result.

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Step 2.11.1

Combine fractions.

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Step 2.11.1.1

Combine the numerators over the common denominator.

$f (1) = \frac{- 3 {(1)}^{2}}{10} + \frac{- 7 \cdot 1 + 3}{20}$

Step 2.11.1.2

Simplify the expression.

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Step 2.11.1.2.1

Multiply $- 7$ by $1$ .

$f (1) = \frac{- 3 {(1)}^{2}}{10} + \frac{- 7 + 3}{20}$

Step 2.11.1.2.2

Add $- 7$ and $3$ .

$f (1) = \frac{- 3 {(1)}^{2}}{10} + \frac{- 4}{20}$

Step 2.11.2

Simplify each term.

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Step 2.11.2.1

One to any power is one.

$f (1) = \frac{- 3 \cdot 1}{10} + \frac{- 4}{20}$

Step 2.11.2.2

Multiply $- 3$ by $1$ .

$f (1) = \frac{- 3}{10} + \frac{- 4}{20}$

Step 2.11.2.3

Move the negative in front of the fraction.

$f (1) = - \frac{3}{10} + \frac{- 4}{20}$

Step 2.11.2.4

Cancel the common factor of $- 4$ and $20$ .

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Step 2.11.2.4.1

Factor $4$ out of $- 4$ .

$f (1) = - \frac{3}{10} + \frac{4 (- 1)}{20}$

Step 2.11.2.4.2

Cancel the common factors.

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Step 2.11.2.4.2.1

Factor $4$ out of $20$ .

$f (1) = - \frac{3}{10} + \frac{4 \cdot - 1}{4 \cdot 5}$

Step 2.11.2.4.2.2

Cancel the common factor.

$f (1) = - \frac{3}{10} + \frac{4 \cdot - 1}{4 \cdot 5}$

Step 2.11.2.4.2.3

Rewrite the expression.

$f (1) = - \frac{3}{10} + \frac{- 1}{5}$

Step 2.11.2.5

Move the negative in front of the fraction.

$f (1) = - \frac{3}{10} - \frac{1}{5}$

Step 2.11.3

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (1) = - \frac{3}{10} - \frac{1}{5} \cdot \frac{2}{2}$

Step 2.11.4

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.11.4.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$f (1) = - \frac{3}{10} - \frac{2}{5 \cdot 2}$

Step 2.11.4.2

Multiply $5$ by $2$ .

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

Step 2.11.5

Simplify the expression.

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Step 2.11.5.1

Combine the numerators over the common denominator.

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

Step 2.11.5.2

Subtract $2$ from $- 3$ .

$f (1) = \frac{- 5}{10}$

Step 2.11.6

Cancel the common factor of $- 5$ and $10$ .

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Step 2.11.6.1

Factor $5$ out of $- 5$ .

$f (1) = \frac{5 (- 1)}{10}$

Step 2.11.6.2

Cancel the common factors.

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Step 2.11.6.2.1

Factor $5$ out of $10$ .

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

Step 2.11.6.2.2

Cancel the common factor.

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

Step 2.11.6.2.3

Rewrite the expression.

$f (1) = \frac{- 1}{2}$

Step 2.11.7

Move the negative in front of the fraction.

$f (1) = - \frac{1}{2}$

Step 2.11.8

The final answer is $- \frac{1}{2}$ .

$- \frac{1}{2}$

Step 2.12

The $y$ value at $x = 1$ is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 3 & - \frac{3}{2} \\ - 2 & - \frac{7}{20} \\ - \frac{7}{12} & \frac{121}{480} \\ 0 & \frac{3}{20} \\ 1 & - \frac{1}{2} \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{7}{12}, \frac{121}{480})$

Focus: $(- \frac{7}{12}, - \frac{93}{160})$

Axis of Symmetry: $x = - \frac{7}{12}$

Directrix: $y = \frac{521}{480}$

$\begin{array}{c} x & y \\ - 3 & - \frac{3}{2} \\ - 2 & - \frac{7}{20} \\ - \frac{7}{12} & \frac{121}{480} \\ 0 & \frac{3}{20} \\ 1 & - \frac{1}{2} \end{array}$

Step 4