Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

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 $- \frac{x^{2}}{2} - \frac{3 x}{2} + 2$ .

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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 $c$ .

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

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

$c = 2$

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.3

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

$d = \frac{3}{2} \cdot \frac{1}{\frac{2}{2}}$

Step 2.1.1.3.2.4

Divide $2$ by $2$ .

$d = \frac{3}{2} \cdot \frac{1}{1}$

Step 2.1.1.3.2.5

Cancel the common factor of $1$ .

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Step 2.1.1.3.2.5.1

Cancel the common factor.

$d = \frac{3}{2} \cdot \frac{1}{1}$

Step 2.1.1.3.2.5.2

Rewrite the expression.

$d = \frac{3}{2} \cdot 1$

Step 2.1.1.3.2.6

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

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

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

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

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

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

Step 2.1.1.4.2.1.1.2

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

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

Step 2.1.1.4.2.1.1.3

Apply the product rule to $\frac{3}{2}$ .

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

Step 2.1.1.4.2.1.1.4

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

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

Step 2.1.1.4.2.1.1.5

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

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

Step 2.1.1.4.2.1.1.6

Multiply $\frac{9}{4}$ by $1$ .

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

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 $4$ .

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

Step 2.1.1.4.2.1.2.2

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

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

Step 2.1.1.4.2.1.3

Divide $- 4$ by $2$ .

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

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.4.2.1.5

Move the negative in front of the fraction.

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

Step 2.1.1.4.2.1.6

Multiply $\frac{9}{4} (- \frac{1}{2})$ .

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Step 2.1.1.4.2.1.6.1

Multiply $\frac{9}{4}$ by $\frac{1}{2}$ .

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

Step 2.1.1.4.2.1.6.2

Multiply $4$ by $2$ .

$e = 2 - - \frac{9}{8}$

Step 2.1.1.4.2.1.7

Multiply $- - \frac{9}{8}$ .

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Step 2.1.1.4.2.1.7.1

Multiply $- 1$ by $- 1$ .

$e = 2 + 1 (\frac{9}{8})$

Step 2.1.1.4.2.1.7.2

Multiply $\frac{9}{8}$ by $1$ .

$e = 2 + \frac{9}{8}$

Step 2.1.1.4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e = 2 \cdot \frac{8}{8} + \frac{9}{8}$

Step 2.1.1.4.2.3

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

$e = \frac{2 \cdot 8}{8} + \frac{9}{8}$

Step 2.1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{2 \cdot 8 + 9}{8}$

Step 2.1.1.4.2.5

Simplify the numerator.

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Step 2.1.1.4.2.5.1

Multiply $2$ by $8$ .

$e = \frac{16 + 9}{8}$

Step 2.1.1.4.2.5.2

Add $16$ and $9$ .

$e = \frac{25}{8}$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + \frac{3}{2})}^{2} + \frac{25}{8}$ .

$- \frac{1}{2} {(x + \frac{3}{2})}^{2} + \frac{25}{8}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x + \frac{3}{2})}^{2} + \frac{25}{8}$

Step 2.2

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

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

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

$k = \frac{25}{8}$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, \frac{25}{8})$

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.

$(- \frac{3}{2}, \frac{21}{8})$

Step 2.7

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

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

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.

$y = \frac{29}{8}$

Step 2.9

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

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{29}{8}$

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{29}{8}$

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 $- 3$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Combine the numerators over the common denominator.

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

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $- 1$ by $9$ .

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

Step 3.2.2.3

Multiply $- 3$ by $- 3$ .

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

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Add $- 9$ and $9$ .

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

Step 3.2.3.2

Divide $0$ by $2$ .

$f (- 3) = 2 + 0$

Step 3.2.3.3

Add $2$ and $0$ .

$f (- 3) = 2$

Step 3.2.4

The final answer is $2$ .

$2$

Step 3.3

The $y$ value at $x = - 3$ is $2$ .

$y = 2$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

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

Step 3.5.2

Simplify each term.

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Step 3.5.2.1

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

$f (- 4) = 2 + \frac{- 1 \cdot 16 - 3 \cdot - 4}{2}$

Step 3.5.2.2

Multiply $- 1$ by $16$ .

$f (- 4) = 2 + \frac{- 16 - 3 \cdot - 4}{2}$

Step 3.5.2.3

Multiply $- 3$ by $- 4$ .

$f (- 4) = 2 + \frac{- 16 + 12}{2}$

Step 3.5.3

Simplify the expression.

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Step 3.5.3.1

Add $- 16$ and $12$ .

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

Step 3.5.3.2

Divide $- 4$ by $2$ .

$f (- 4) = 2 - 2$

Step 3.5.3.3

Subtract $2$ from $2$ .

$f (- 4) = 0$

Step 3.5.4

The final answer is $0$ .

$0$

Step 3.6

The $y$ value at $x = - 4$ is $0$ .

$y = 0$

Step 3.7

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

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

Step 3.8

Simplify the result.

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Step 3.8.1

Combine the numerators over the common denominator.

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

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

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

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Step 3.8.2.1.1

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

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Step 3.8.2.1.1.1

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

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

Step 3.8.2.1.1.2

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

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

Step 3.8.2.1.2

Add $1$ and $2$ .

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

Step 3.8.2.2

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

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

Step 3.8.2.3

Multiply $- 3$ by $- 1$ .

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

Step 3.8.3

Simplify the expression.

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Step 3.8.3.1

Add $- 1$ and $3$ .

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

Step 3.8.3.2

Divide $2$ by $2$ .

$f (- 1) = 2 + 1$

Step 3.8.3.3

Add $2$ and $1$ .

$f (- 1) = 3$

Step 3.8.4

The final answer is $3$ .

$3$

Step 3.9

The $y$ value at $x = - 1$ is $3$ .

$y = 3$

Step 3.10

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

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

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

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

Step 3.11.2

Simplify each term.

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Step 3.11.2.1

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

$f (0) = 2 + \frac{- 0 - 3 \cdot 0}{2}$

Step 3.11.2.2

Multiply $- 1$ by $0$ .

$f (0) = 2 + \frac{0 - 3 \cdot 0}{2}$

Step 3.11.2.3

Multiply $- 3$ by $0$ .

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

Step 3.11.3

Simplify the expression.

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Step 3.11.3.1

Add $0$ and $0$ .

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

Step 3.11.3.2

Divide $0$ by $2$ .

$f (0) = 2 + 0$

Step 3.11.3.3

Add $2$ and $0$ .

$f (0) = 2$

Step 3.11.4

The final answer is $2$ .

$2$

Step 3.12

The $y$ value at $x = 0$ is $2$ .

$y = 2$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & 2 \\ - \frac{3}{2} & \frac{25}{8} \\ - 1 & 3 \\ 0 & 2 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- \frac{3}{2}, \frac{25}{8})$

Focus: $(- \frac{3}{2}, \frac{21}{8})$

Axis of Symmetry: $x = - \frac{3}{2}$

Directrix: $y = \frac{29}{8}$

$\begin{array}{c} x & y \\ - 4 & 0 \\ - 3 & 2 \\ - \frac{3}{2} & \frac{25}{8} \\ - 1 & 3 \\ 0 & 2 \end{array}$

Step 5