Algebra Examples

$2 x - 2 y = x^{2} + 9$

Step 1

Solve for $y$ .

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Step 1.1

Subtract $2 x$ from both sides of the equation.

$- 2 y = x^{2} + 9 - 2 x$

Step 1.2

Divide each term in $- 2 y = x^{2} + 9 - 2 x$ by $- 2$ and simplify.

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Step 1.2.1

Divide each term in $- 2 y = x^{2} + 9 - 2 x$ by $- 2$ .

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

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.1.2

Divide $y$ by $1$ .

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

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

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

Step 1.2.3.1.2

Move the negative in front of the fraction.

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

Step 1.2.3.1.3

Cancel the common factor of $- 2$ .

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Step 1.2.3.1.3.1

Cancel the common factor.

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

Step 1.2.3.1.3.2

Divide $x$ by $1$ .

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

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

Move $- \frac{9}{2}$ .

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

Step 2.1.2

Complete the square for $- \frac{x^{2}}{2} + x - \frac{9}{2}$ .

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Step 2.1.2.1

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

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

$b = 1$

$c = - \frac{9}{2}$

Step 2.1.2.2

Consider the vertex form of a parabola.

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

Step 2.1.2.3

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

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Step 2.1.2.3.1

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

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

Step 2.1.2.3.2

Simplify the right side.

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Step 2.1.2.3.2.1

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

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Step 2.1.2.3.2.1.1

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

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

Step 2.1.2.3.2.1.2

Move the negative in front of the fraction.

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

Step 2.1.2.3.2.2

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

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

Step 2.1.2.3.2.3

Divide $2$ by $2$ .

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

Step 2.1.2.3.2.4

Cancel the common factor of $1$ .

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Step 2.1.2.3.2.4.1

Cancel the common factor.

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

Step 2.1.2.3.2.4.2

Rewrite the expression.

$d = - 1 \cdot 1$

Step 2.1.2.3.2.5

Multiply $- 1$ by $1$ .

$d = - 1$

Step 2.1.2.4

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

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Step 2.1.2.4.1

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

$e = - \frac{9}{2} - \frac{1^{2}}{4 (- \frac{1}{2})}$

Step 2.1.2.4.2

Simplify the right side.

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Step 2.1.2.4.2.1

Simplify each term.

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Step 2.1.2.4.2.1.1

One to any power is one.

$e = - \frac{9}{2} - \frac{1}{4 (- \frac{1}{2})}$

Step 2.1.2.4.2.1.2

Simplify the denominator.

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Step 2.1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{9}{2} - \frac{1}{- 4 (\frac{1}{2})}$

Step 2.1.2.4.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = - \frac{9}{2} - \frac{1}{\frac{- 4}{2}}$

Step 2.1.2.4.2.1.3

Divide $- 4$ by $2$ .

$e = - \frac{9}{2} - \frac{1}{- 2}$

Step 2.1.2.4.2.1.4

Move the negative in front of the fraction.

$e = - \frac{9}{2} - - \frac{1}{2}$

Step 2.1.2.4.2.1.5

Multiply $- - \frac{1}{2}$ .

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Step 2.1.2.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{9}{2} + 1 (\frac{1}{2})$

Step 2.1.2.4.2.1.5.2

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

$e = - \frac{9}{2} + \frac{1}{2}$

Step 2.1.2.4.2.2

Combine the numerators over the common denominator.

$e = \frac{- 9 + 1}{2}$

Step 2.1.2.4.2.3

Add $- 9$ and $1$ .

$e = \frac{- 8}{2}$

Step 2.1.2.4.2.4

Divide $- 8$ by $2$ .

$e = - 4$

Step 2.1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 1)}^{2} - 4$ .

$- \frac{1}{2} {(x - 1)}^{2} - 4$

Step 2.1.3

Set $y$ equal to the new right side.

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

Step 2.2

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

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

$h = 1$

$k = - 4$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(1, - 4)$

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.

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

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 $- 1 (- 1)$ .

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

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

Step 2.5.3.2

Combine $4$ and $\frac{1}{2}$ .

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

Step 2.5.3.3

Divide $4$ by $2$ .

$- \frac{1}{2}$

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.

$(1, - \frac{9}{2})$

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.

$y = - \frac{7}{2}$

Step 2.9

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

Direction: Opens Down

Vertex: $(1, - 4)$

Focus: $(1, - \frac{9}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{7}{2}$

Direction: Opens Down

Vertex: $(1, - 4)$

Focus: $(1, - \frac{9}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{7}{2}$

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 $- 1$ in the expression.

$f (- 1) = - \frac{{(- 1)}^{2}}{2} - 1 - \frac{9}{2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Combine fractions.

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Step 3.2.1.1

Remove parentheses.

$f (- 1) = - \frac{{(- 1)}^{2}}{2} - 1 - \frac{9}{2}$

Step 3.2.1.2

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.2.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.2.2.1.1.1

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

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

Step 3.2.2.1.1.2

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

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

Step 3.2.2.1.2

Add $1$ and $2$ .

$f (- 1) = - 1 + \frac{{(- 1)}^{3} - 9}{2}$

Step 3.2.2.2

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

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

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Subtract $9$ from $- 1$ .

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

Step 3.2.3.2

Divide $- 10$ by $2$ .

$f (- 1) = - 1 - 5$

Step 3.2.3.3

Subtract $5$ from $- 1$ .

$f (- 1) = - 6$

Step 3.2.4

The final answer is $- 6$ .

$- 6$

Step 3.3

The $y$ value at $x = - 1$ is $- 6$ .

$y = - 6$

Step 3.4

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

$f (0) = - \frac{{(0)}^{2}}{2} + 0 - \frac{9}{2}$

Step 3.5

Simplify the result.

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Step 3.5.1

Combine fractions.

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Step 3.5.1.1

Remove parentheses.

$f (0) = - \frac{{(0)}^{2}}{2} + 0 - \frac{9}{2}$

Step 3.5.1.2

Combine the numerators over the common denominator.

$f (0) = \frac{- {(0)}^{2} - 9}{2}$

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (0) = \frac{- 0 - 9}{2}$

Step 3.5.2.2

Multiply $- 1$ by $0$ .

$f (0) = \frac{0 - 9}{2}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Subtract $9$ from $0$ .

$f (0) = \frac{- 9}{2}$

Step 3.5.3.2

Move the negative in front of the fraction.

$f (0) = - \frac{9}{2}$

Step 3.5.4

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

Step 3.6

The $y$ value at $x = 0$ is $- \frac{9}{2}$ .

$y = - \frac{9}{2}$

Step 3.7

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

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

Step 3.8

Simplify the result.

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Step 3.8.1

Combine fractions.

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Step 3.8.1.1

Remove parentheses.

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

Step 3.8.1.2

Combine the numerators over the common denominator.

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

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

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

Step 3.8.2.2

Multiply $- 1$ by $9$ .

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

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Subtract $9$ from $- 9$ .

$f (3) = 3 + \frac{- 18}{2}$

Step 3.8.3.2

Divide $- 18$ by $2$ .

$f (3) = 3 - 9$

Step 3.8.3.3

Subtract $9$ from $3$ .

$f (3) = - 6$

Step 3.8.4

The final answer is $- 6$ .

$- 6$

Step 3.9

The $y$ value at $x = 3$ is $- 6$ .

$y = - 6$

Step 3.10

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

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

Step 3.11

Simplify the result.

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Step 3.11.1

Combine fractions.

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Step 3.11.1.1

Remove parentheses.

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

Step 3.11.1.2

Combine the numerators over the common denominator.

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

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

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

$f (2) = 2 + \frac{- 1 \cdot 4 - 9}{2}$

Step 3.11.2.2

Multiply $- 1$ by $4$ .

$f (2) = 2 + \frac{- 4 - 9}{2}$

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Subtract $9$ from $- 4$ .

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

Step 3.11.3.2

Move the negative in front of the fraction.

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

Step 3.11.4

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

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

Step 3.11.5

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

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

Step 3.11.6

Combine the numerators over the common denominator.

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

Step 3.11.7

Simplify the numerator.

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Step 3.11.7.1

Multiply $2$ by $2$ .

$f (2) = \frac{4 - 13}{2}$

Step 3.11.7.2

Subtract $13$ from $4$ .

$f (2) = \frac{- 9}{2}$

Step 3.11.8

Move the negative in front of the fraction.

$f (2) = - \frac{9}{2}$

Step 3.11.9

The final answer is $- \frac{9}{2}$ .

$- \frac{9}{2}$

Step 3.12

The $y$ value at $x = 2$ is $- \frac{9}{2}$ .

$y = - \frac{9}{2}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 1 & - 6 \\ 0 & - \frac{9}{2} \\ 1 & - 4 \\ 2 & - \frac{9}{2} \\ 3 & - 6 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(1, - 4)$

Focus: $(1, - \frac{9}{2})$

Axis of Symmetry: $x = 1$

Directrix: $y = - \frac{7}{2}$

$\begin{array}{c} x & y \\ - 1 & - 6 \\ 0 & - \frac{9}{2} \\ 1 & - 4 \\ 2 & - \frac{9}{2} \\ 3 & - 6 \end{array}$

Step 5