Trigonometry Examples

$25 x^{2} - 50 x - 4 y - 8 y - 79 = 0$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $8 y$ from $- 4 y$ .

$25 x^{2} - 50 x - 12 y - 79 = 0$

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 $25 x^{2}$ from both sides of the equation.

$- 50 x - 12 y - 79 = - 25 x^{2}$

Step 1.2.2

Add $50 x$ to both sides of the equation.

$- 12 y - 79 = - 25 x^{2} + 50 x$

Step 1.2.3

Add $79$ to both sides of the equation.

$- 12 y = - 25 x^{2} + 50 x + 79$

Step 1.3

Divide each term in $- 12 y = - 25 x^{2} + 50 x + 79$ by $- 12$ and simplify.

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Step 1.3.1

Divide each term in $- 12 y = - 25 x^{2} + 50 x + 79$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- 25 x^{2}}{- 12} + \frac{50 x}{- 12} + \frac{79}{- 12}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $- 12$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- 25 x^{2}}{- 12} + \frac{50 x}{- 12} + \frac{79}{- 12}$

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 25 x^{2}}{- 12} + \frac{50 x}{- 12} + \frac{79}{- 12}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Dividing two negative values results in a positive value.

$y = \frac{25 x^{2}}{12} + \frac{50 x}{- 12} + \frac{79}{- 12}$

Step 1.3.3.1.2

Cancel the common factor of $50$ and $- 12$ .

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Step 1.3.3.1.2.1

Factor $2$ out of $50 x$ .

$y = \frac{25 x^{2}}{12} + \frac{2 (25 x)}{- 12} + \frac{79}{- 12}$

Step 1.3.3.1.2.2

Cancel the common factors.

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Step 1.3.3.1.2.2.1

Factor $2$ out of $- 12$ .

$y = \frac{25 x^{2}}{12} + \frac{2 (25 x)}{2 (- 6)} + \frac{79}{- 12}$

Step 1.3.3.1.2.2.2

Cancel the common factor.

$y = \frac{25 x^{2}}{12} + \frac{2 (25 x)}{2 \cdot - 6} + \frac{79}{- 12}$

Step 1.3.3.1.2.2.3

Rewrite the expression.

$y = \frac{25 x^{2}}{12} + \frac{25 x}{- 6} + \frac{79}{- 12}$

Step 1.3.3.1.3

Move the negative in front of the fraction.

$y = \frac{25 x^{2}}{12} - \frac{25 x}{6} + \frac{79}{- 12}$

Step 1.3.3.1.4

Move the negative in front of the fraction.

$y = \frac{25 x^{2}}{12} - \frac{25 x}{6} - \frac{79}{12}$

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 $\frac{25 x^{2}}{12} - \frac{25 x}{6} - \frac{79}{12}$ .

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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 $c$ .

$a = \frac{25}{12}$

$b = - \frac{25}{6}$

$c = - \frac{79}{12}$

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

$d = \frac{- \frac{25}{6}}{2 (\frac{25}{12})}$

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.

$d = - \frac{25}{6} \cdot \frac{1}{2 (\frac{25}{12})}$

Step 2.1.1.3.2.2

Combine $2$ and $\frac{25}{12}$ .

$d = - \frac{25}{6} \cdot \frac{1}{\frac{2 \cdot 25}{12}}$

Step 2.1.1.3.2.3

Multiply $2$ by $25$ .

$d = - \frac{25}{6} \cdot \frac{1}{\frac{50}{12}}$

Step 2.1.1.3.2.4

Cancel the common factor of $50$ and $12$ .

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Step 2.1.1.3.2.4.1

Factor $2$ out of $50$ .

$d = - \frac{25}{6} \cdot \frac{1}{\frac{2 (25)}{12}}$

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 $12$ .

$d = - \frac{25}{6} \cdot \frac{1}{\frac{2 \cdot 25}{2 \cdot 6}}$

Step 2.1.1.3.2.4.2.2

Cancel the common factor.

$d = - \frac{25}{6} \cdot \frac{1}{\frac{2 \cdot 25}{2 \cdot 6}}$

Step 2.1.1.3.2.4.2.3

Rewrite the expression.

$d = - \frac{25}{6} \cdot \frac{1}{\frac{25}{6}}$

Step 2.1.1.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{25}{6} (1 (\frac{6}{25}))$

Step 2.1.1.3.2.6

Multiply $\frac{6}{25}$ by $1$ .

$d = - \frac{25}{6} \cdot \frac{6}{25}$

Step 2.1.1.3.2.7

Cancel the common factor of $25$ .

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Step 2.1.1.3.2.7.1

Move the leading negative in $- \frac{25}{6}$ into the numerator.

$d = \frac{- 25}{6} \cdot \frac{6}{25}$

Step 2.1.1.3.2.7.2

Factor $25$ out of $- 25$ .

$d = \frac{25 (- 1)}{6} \cdot \frac{6}{25}$

Step 2.1.1.3.2.7.3

Cancel the common factor.

$d = \frac{25 \cdot - 1}{6} \cdot \frac{6}{25}$

Step 2.1.1.3.2.7.4

Rewrite the expression.

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

Step 2.1.1.3.2.8

Cancel the common factor of $6$ .

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Step 2.1.1.3.2.8.1

Cancel the common factor.

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

Step 2.1.1.3.2.8.2

Rewrite the expression.

$d = - 1$

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

$e = - \frac{79}{12} - \frac{{(- \frac{25}{6})}^{2}}{4 (\frac{25}{12})}$

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 $- \frac{25}{6}$ .

$e = - \frac{79}{12} - \frac{{(- 1)}^{2} {(\frac{25}{6})}^{2}}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.1.2

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

$e = - \frac{79}{12} - \frac{1 {(\frac{25}{6})}^{2}}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.1.3

Apply the product rule to $\frac{25}{6}$ .

$e = - \frac{79}{12} - \frac{1 \frac{25^{2}}{6^{2}}}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.1.4

Raise $25$ to the power of $2$ .

$e = - \frac{79}{12} - \frac{1 \frac{625}{6^{2}}}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.1.5

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

$e = - \frac{79}{12} - \frac{1 (\frac{625}{36})}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.1.6

Multiply $\frac{625}{36}$ by $1$ .

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{4 (\frac{25}{12})}$

Step 2.1.1.4.2.1.2

Combine $4$ and $\frac{25}{12}$ .

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{4 \cdot 25}{12}}$

Step 2.1.1.4.2.1.3

Multiply $4$ by $25$ .

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{100}{12}}$

Step 2.1.1.4.2.1.4

Cancel the common factor of $100$ and $12$ .

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Step 2.1.1.4.2.1.4.1

Factor $4$ out of $100$ .

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{4 (25)}{12}}$

Step 2.1.1.4.2.1.4.2

Cancel the common factors.

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Step 2.1.1.4.2.1.4.2.1

Factor $4$ out of $12$ .

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{4 \cdot 25}{4 \cdot 3}}$

Step 2.1.1.4.2.1.4.2.2

Cancel the common factor.

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{4 \cdot 25}{4 \cdot 3}}$

Step 2.1.1.4.2.1.4.2.3

Rewrite the expression.

$e = - \frac{79}{12} - \frac{\frac{625}{36}}{\frac{25}{3}}$

Step 2.1.1.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{79}{12} - (\frac{625}{36} \cdot \frac{3}{25})$

Step 2.1.1.4.2.1.6

Cancel the common factor of $25$ .

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Step 2.1.1.4.2.1.6.1

Factor $25$ out of $625$ .

$e = - \frac{79}{12} - (\frac{25 (25)}{36} \cdot \frac{3}{25})$

Step 2.1.1.4.2.1.6.2

Cancel the common factor.

$e = - \frac{79}{12} - (\frac{25 \cdot 25}{36} \cdot \frac{3}{25})$

Step 2.1.1.4.2.1.6.3

Rewrite the expression.

$e = - \frac{79}{12} - (\frac{25}{36} \cdot 3)$

Step 2.1.1.4.2.1.7

Cancel the common factor of $3$ .

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Step 2.1.1.4.2.1.7.1

Factor $3$ out of $36$ .

$e = - \frac{79}{12} - (\frac{25}{3 (12)} \cdot 3)$

Step 2.1.1.4.2.1.7.2

Cancel the common factor.

$e = - \frac{79}{12} - (\frac{25}{3 \cdot 12} \cdot 3)$

Step 2.1.1.4.2.1.7.3

Rewrite the expression.

$e = - \frac{79}{12} - \frac{25}{12}$

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 79 - 25}{12}$

Step 2.1.1.4.2.3

Subtract $25$ from $- 79$ .

$e = \frac{- 104}{12}$

Step 2.1.1.4.2.4

Cancel the common factor of $- 104$ and $12$ .

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Step 2.1.1.4.2.4.1

Factor $4$ out of $- 104$ .

$e = \frac{4 (- 26)}{12}$

Step 2.1.1.4.2.4.2

Cancel the common factors.

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Step 2.1.1.4.2.4.2.1

Factor $4$ out of $12$ .

$e = \frac{4 \cdot - 26}{4 \cdot 3}$

Step 2.1.1.4.2.4.2.2

Cancel the common factor.

$e = \frac{4 \cdot - 26}{4 \cdot 3}$

Step 2.1.1.4.2.4.2.3

Rewrite the expression.

$e = \frac{- 26}{3}$

Step 2.1.1.4.2.5

Move the negative in front of the fraction.

$e = - \frac{26}{3}$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{25}{12} {(x - 1)}^{2} - \frac{26}{3}$ .

$\frac{25}{12} {(x - 1)}^{2} - \frac{26}{3}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{25}{12} \cdot {(x - 1)}^{2} - \frac{26}{3}$

Step 2.2

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

$a = \frac{25}{12}$

$h = 1$

$k = - \frac{26}{3}$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(1, - \frac{26}{3})$

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.

$\frac{1}{4 \cdot \frac{25}{12}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{25}{12}$ .

$\frac{1}{\frac{4 \cdot 25}{12}}$

Step 2.5.3.2

Reduce the expression by cancelling the common factors.

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Step 2.5.3.2.1

Multiply $4$ by $25$ .

$\frac{1}{\frac{100}{12}}$

Step 2.5.3.2.2

Cancel the common factor of $100$ and $12$ .

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Step 2.5.3.2.2.1

Factor $4$ out of $100$ .

$\frac{1}{\frac{4 (25)}{12}}$

Step 2.5.3.2.2.2

Cancel the common factors.

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Step 2.5.3.2.2.2.1

Factor $4$ out of $12$ .

$\frac{1}{\frac{4 \cdot 25}{4 \cdot 3}}$

Step 2.5.3.2.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 25}{4 \cdot 3}}$

Step 2.5.3.2.2.2.3

Rewrite the expression.

$\frac{1}{\frac{25}{3}}$

Step 2.5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{3}{25})$

Step 2.5.3.4

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

$\frac{3}{25}$

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.

$(1, - \frac{641}{75})$

Step 2.7

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

$x = 1$

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.

$y = - \frac{659}{75}$

Step 2.9

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

Direction: Opens Up

Vertex: $(1, - \frac{26}{3})$

Focus: $(1, - \frac{641}{75})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{659}{75}$

Direction: Opens Up

Vertex: $(1, - \frac{26}{3})$

Focus: $(1, - \frac{641}{75})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{659}{75}$

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 $0$ in the expression.

$f (0) = \frac{25 {(0)}^{2}}{12} - \frac{25 (0)}{6} - \frac{79}{12}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

$f (0) = \frac{25 {(0)}^{2} - 79}{12} + \frac{- 25 \cdot 0}{6}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

$f (0) = \frac{25 \cdot 0 - 79}{12} + \frac{- 25 \cdot 0}{6}$

Step 3.2.2.2

Multiply $25$ by $0$ .

$f (0) = \frac{0 - 79}{12} + \frac{- 25 \cdot 0}{6}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $79$ from $0$ .

$f (0) = \frac{- 79}{12} + \frac{- 25 \cdot 0}{6}$

Step 3.2.3.2

Multiply $- 25$ by $0$ .

$f (0) = \frac{- 79}{12} + \frac{0}{6}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Move the negative in front of the fraction.

$f (0) = - \frac{79}{12} + \frac{0}{6}$

Step 3.2.4.2

Divide $0$ by $6$ .

$f (0) = - \frac{79}{12} + 0$

Step 3.2.5

Add $- \frac{79}{12}$ and $0$ .

$f (0) = - \frac{79}{12}$

Step 3.2.6

The final answer is $- \frac{79}{12}$ .

$- \frac{79}{12}$

Step 3.3

The $y$ value at $x = 0$ is $- \frac{79}{12}$ .

$y = - \frac{79}{12}$

Step 3.4

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

$f (- 1) = \frac{25 {(- 1)}^{2}}{12} - \frac{25 (- 1)}{6} - \frac{79}{12}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

$f (- 1) = \frac{25 {(- 1)}^{2} - 79}{12} + \frac{- 25 \cdot - 1}{6}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (- 1) = \frac{25 \cdot 1 - 79}{12} + \frac{- 25 \cdot - 1}{6}$

Step 3.5.2.2

Multiply $25$ by $1$ .

$f (- 1) = \frac{25 - 79}{12} + \frac{- 25 \cdot - 1}{6}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Subtract $79$ from $25$ .

$f (- 1) = \frac{- 54}{12} + \frac{- 25 \cdot - 1}{6}$

Step 3.5.3.2

Multiply $- 25$ by $- 1$ .

$f (- 1) = \frac{- 54}{12} + \frac{25}{6}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Cancel the common factor of $- 54$ and $12$ .

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Step 3.5.4.1.1

Factor $6$ out of $- 54$ .

$f (- 1) = \frac{6 (- 9)}{12} + \frac{25}{6}$

Step 3.5.4.1.2

Cancel the common factors.

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Step 3.5.4.1.2.1

Factor $6$ out of $12$ .

$f (- 1) = \frac{6 \cdot - 9}{6 \cdot 2} + \frac{25}{6}$

Step 3.5.4.1.2.2

Cancel the common factor.

$f (- 1) = \frac{6 \cdot - 9}{6 \cdot 2} + \frac{25}{6}$

Step 3.5.4.1.2.3

Rewrite the expression.

$f (- 1) = \frac{- 9}{2} + \frac{25}{6}$

Step 3.5.4.2

Move the negative in front of the fraction.

$f (- 1) = - \frac{9}{2} + \frac{25}{6}$

Step 3.5.5

To write $- \frac{9}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 1) = - \frac{9}{2} \cdot \frac{3}{3} + \frac{25}{6}$

Step 3.5.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.6.1

Multiply $\frac{9}{2}$ by $\frac{3}{3}$ .

$f (- 1) = - \frac{9 \cdot 3}{2 \cdot 3} + \frac{25}{6}$

Step 3.5.6.2

Multiply $2$ by $3$ .

$f (- 1) = - \frac{9 \cdot 3}{6} + \frac{25}{6}$

Step 3.5.7

Combine the numerators over the common denominator.

$f (- 1) = \frac{- 9 \cdot 3 + 25}{6}$

Step 3.5.8

Simplify the numerator.

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Step 3.5.8.1

Multiply $- 9$ by $3$ .

$f (- 1) = \frac{- 27 + 25}{6}$

Step 3.5.8.2

Add $- 27$ and $25$ .

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

Step 3.5.9

Cancel the common factor of $- 2$ and $6$ .

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Step 3.5.9.1

Factor $2$ out of $- 2$ .

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

Step 3.5.9.2

Cancel the common factors.

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Step 3.5.9.2.1

Factor $2$ out of $6$ .

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

Step 3.5.9.2.2

Cancel the common factor.

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

Step 3.5.9.2.3

Rewrite the expression.

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

Step 3.5.10

Move the negative in front of the fraction.

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

Step 3.5.11

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

$- \frac{1}{3}$

Step 3.6

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

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

Step 3.7

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

$f (2) = \frac{25 {(2)}^{2}}{12} - \frac{25 (2)}{6} - \frac{79}{12}$

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

$f (2) = \frac{25 {(2)}^{2} - 79}{12} + \frac{- 25 \cdot 2}{6}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

$f (2) = \frac{25 \cdot 4 - 79}{12} + \frac{- 25 \cdot 2}{6}$

Step 3.8.2.2

Multiply $25$ by $4$ .

$f (2) = \frac{100 - 79}{12} + \frac{- 25 \cdot 2}{6}$

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Subtract $79$ from $100$ .

$f (2) = \frac{21}{12} + \frac{- 25 \cdot 2}{6}$

Step 3.8.3.2

Multiply $- 25$ by $2$ .

$f (2) = \frac{21}{12} + \frac{- 50}{6}$

Step 3.8.4

Simplify each term.

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Step 3.8.4.1

Cancel the common factor of $21$ and $12$ .

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Step 3.8.4.1.1

Factor $3$ out of $21$ .

$f (2) = \frac{3 (7)}{12} + \frac{- 50}{6}$

Step 3.8.4.1.2

Cancel the common factors.

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Step 3.8.4.1.2.1

Factor $3$ out of $12$ .

$f (2) = \frac{3 \cdot 7}{3 \cdot 4} + \frac{- 50}{6}$

Step 3.8.4.1.2.2

Cancel the common factor.

$f (2) = \frac{3 \cdot 7}{3 \cdot 4} + \frac{- 50}{6}$

Step 3.8.4.1.2.3

Rewrite the expression.

$f (2) = \frac{7}{4} + \frac{- 50}{6}$

Step 3.8.4.2

Cancel the common factor of $- 50$ and $6$ .

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Step 3.8.4.2.1

Factor $2$ out of $- 50$ .

$f (2) = \frac{7}{4} + \frac{2 (- 25)}{6}$

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 $6$ .

$f (2) = \frac{7}{4} + \frac{2 \cdot - 25}{2 \cdot 3}$

Step 3.8.4.2.2.2

Cancel the common factor.

$f (2) = \frac{7}{4} + \frac{2 \cdot - 25}{2 \cdot 3}$

Step 3.8.4.2.2.3

Rewrite the expression.

$f (2) = \frac{7}{4} + \frac{- 25}{3}$

Step 3.8.4.3

Move the negative in front of the fraction.

$f (2) = \frac{7}{4} - \frac{25}{3}$

Step 3.8.5

To write $\frac{7}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (2) = \frac{7}{4} \cdot \frac{3}{3} - \frac{25}{3}$

Step 3.8.6

To write $- \frac{25}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (2) = \frac{7}{4} \cdot \frac{3}{3} - \frac{25}{3} \cdot \frac{4}{4}$

Step 3.8.7

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.7.1

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

$f (2) = \frac{7 \cdot 3}{4 \cdot 3} - \frac{25}{3} \cdot \frac{4}{4}$

Step 3.8.7.2

Multiply $4$ by $3$ .

$f (2) = \frac{7 \cdot 3}{12} - \frac{25}{3} \cdot \frac{4}{4}$

Step 3.8.7.3

Multiply $\frac{25}{3}$ by $\frac{4}{4}$ .

$f (2) = \frac{7 \cdot 3}{12} - \frac{25 \cdot 4}{3 \cdot 4}$

Step 3.8.7.4

Multiply $3$ by $4$ .

$f (2) = \frac{7 \cdot 3}{12} - \frac{25 \cdot 4}{12}$

Step 3.8.8

Combine the numerators over the common denominator.

$f (2) = \frac{7 \cdot 3 - 25 \cdot 4}{12}$

Step 3.8.9

Simplify the numerator.

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Step 3.8.9.1

Multiply $7$ by $3$ .

$f (2) = \frac{21 - 25 \cdot 4}{12}$

Step 3.8.9.2

Multiply $- 25$ by $4$ .

$f (2) = \frac{21 - 100}{12}$

Step 3.8.9.3

Subtract $100$ from $21$ .

$f (2) = \frac{- 79}{12}$

Step 3.8.10

Move the negative in front of the fraction.

$f (2) = - \frac{79}{12}$

Step 3.8.11

The final answer is $- \frac{79}{12}$ .

$- \frac{79}{12}$

Step 3.9

The $y$ value at $x = 2$ is $- \frac{79}{12}$ .

$y = - \frac{79}{12}$

Step 3.10

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

$f (3) = \frac{25 {(3)}^{2}}{12} - \frac{25 (3)}{6} - \frac{79}{12}$

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

$f (3) = \frac{25 {(3)}^{2} - 79}{12} + \frac{- 25 \cdot 3}{6}$

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

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

$f (3) = \frac{25 \cdot 9 - 79}{12} + \frac{- 25 \cdot 3}{6}$

Step 3.11.2.2

Multiply $25$ by $9$ .

$f (3) = \frac{225 - 79}{12} + \frac{- 25 \cdot 3}{6}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Subtract $79$ from $225$ .

$f (3) = \frac{146}{12} + \frac{- 25 \cdot 3}{6}$

Step 3.11.3.2

Multiply $- 25$ by $3$ .

$f (3) = \frac{146}{12} + \frac{- 75}{6}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Cancel the common factor of $146$ and $12$ .

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Step 3.11.4.1.1

Factor $2$ out of $146$ .

$f (3) = \frac{2 (73)}{12} + \frac{- 75}{6}$

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 $12$ .

$f (3) = \frac{2 \cdot 73}{2 \cdot 6} + \frac{- 75}{6}$

Step 3.11.4.1.2.2

Cancel the common factor.

$f (3) = \frac{2 \cdot 73}{2 \cdot 6} + \frac{- 75}{6}$

Step 3.11.4.1.2.3

Rewrite the expression.

$f (3) = \frac{73}{6} + \frac{- 75}{6}$

Step 3.11.4.2

Cancel the common factor of $- 75$ and $6$ .

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Step 3.11.4.2.1

Factor $3$ out of $- 75$ .

$f (3) = \frac{73}{6} + \frac{3 (- 25)}{6}$

Step 3.11.4.2.2

Cancel the common factors.

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Step 3.11.4.2.2.1

Factor $3$ out of $6$ .

$f (3) = \frac{73}{6} + \frac{3 \cdot - 25}{3 \cdot 2}$

Step 3.11.4.2.2.2

Cancel the common factor.

$f (3) = \frac{73}{6} + \frac{3 \cdot - 25}{3 \cdot 2}$

Step 3.11.4.2.2.3

Rewrite the expression.

$f (3) = \frac{73}{6} + \frac{- 25}{2}$

Step 3.11.4.3

Move the negative in front of the fraction.

$f (3) = \frac{73}{6} - \frac{25}{2}$

Step 3.11.5

To write $- \frac{25}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (3) = \frac{73}{6} - \frac{25}{2} \cdot \frac{3}{3}$

Step 3.11.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.6.1

Multiply $\frac{25}{2}$ by $\frac{3}{3}$ .

$f (3) = \frac{73}{6} - \frac{25 \cdot 3}{2 \cdot 3}$

Step 3.11.6.2

Multiply $2$ by $3$ .

$f (3) = \frac{73}{6} - \frac{25 \cdot 3}{6}$

Step 3.11.7

Combine the numerators over the common denominator.

$f (3) = \frac{73 - 25 \cdot 3}{6}$

Step 3.11.8

Simplify the numerator.

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Step 3.11.8.1

Multiply $- 25$ by $3$ .

$f (3) = \frac{73 - 75}{6}$

Step 3.11.8.2

Subtract $75$ from $73$ .

$f (3) = \frac{- 2}{6}$

Step 3.11.9

Cancel the common factor of $- 2$ and $6$ .

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Step 3.11.9.1

Factor $2$ out of $- 2$ .

$f (3) = \frac{2 (- 1)}{6}$

Step 3.11.9.2

Cancel the common factors.

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Step 3.11.9.2.1

Factor $2$ out of $6$ .

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

Step 3.11.9.2.2

Cancel the common factor.

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

Step 3.11.9.2.3

Rewrite the expression.

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

Step 3.11.10

Move the negative in front of the fraction.

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

Step 3.11.11

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

$- \frac{1}{3}$

Step 3.12

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

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

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - \frac{1}{3} \\ 0 & - \frac{79}{12} \\ 1 & - \frac{26}{3} \\ 2 & - \frac{79}{12} \\ 3 & - \frac{1}{3} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(1, - \frac{26}{3})$

Focus: $(1, - \frac{641}{75})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{659}{75}$

$\begin{array}{c} x & y \\ - 1 & - \frac{1}{3} \\ 0 & - \frac{79}{12} \\ 1 & - \frac{26}{3} \\ 2 & - \frac{79}{12} \\ 3 & - \frac{1}{3} \end{array}$

Step 5