Algebra Examples

$g (x) = \frac{1}{2} {(x + 1)}^{2} - 2$

Step 1

Find the properties of the given parabola.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Reorder terms.

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

Step 1.2

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

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

$h = - 1$

$k = - 2$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(- 1, - 2)$

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.

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

Step 1.5.3

Simplify.

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Step 1.5.3.1

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

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

Step 1.5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

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.

$(- 1, - \frac{3}{2})$

Step 1.7

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

$x = - 1$

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.

$y = - \frac{5}{2}$

Step 1.9

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

Direction: Opens Up

Vertex: $(- 1, - 2)$

Focus: $(- 1, - \frac{3}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

Direction: Opens Up

Vertex: $(- 1, - 2)$

Focus: $(- 1, - \frac{3}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

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 $- 3$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Combine the numerators over the common denominator.

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

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

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

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

Step 2.2.3.2

Subtract $3$ from $9$ .

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

Step 2.2.3.3

Divide $6$ by $2$ .

$f (- 3) = - 3 + 3$

Step 2.2.3.4

Add $- 3$ and $3$ .

$f (- 3) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

The $y$ value at $x = - 3$ is $0$ .

$y = 0$

Step 2.4

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

Remove parentheses.

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

Step 2.5.2

Combine the numerators over the common denominator.

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

Step 2.5.3

Simplify the expression.

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Step 2.5.3.1

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

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

Step 2.5.3.2

Subtract $3$ from $4$ .

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

Step 2.5.4

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

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

Step 2.5.5

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

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

Step 2.5.6

Combine the numerators over the common denominator.

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

Step 2.5.7

Simplify the numerator.

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Step 2.5.7.1

Multiply $- 2$ by $2$ .

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

Step 2.5.7.2

Add $- 4$ and $1$ .

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

Step 2.5.8

Move the negative in front of the fraction.

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

Step 2.5.9

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

$- \frac{3}{2}$

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

Remove parentheses.

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

Step 2.8.2

Combine the numerators over the common denominator.

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

Step 2.8.3

Simplify the expression.

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Step 2.8.3.1

One to any power is one.

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

Step 2.8.3.2

Subtract $3$ from $1$ .

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

Step 2.8.3.3

Divide $- 2$ by $2$ .

$f (1) = 1 - 1$

Step 2.8.3.4

Subtract $1$ from $1$ .

$f (1) = 0$

Step 2.8.4

The final answer is $0$ .

$0$

Step 2.9

The $y$ value at $x = 1$ is $0$ .

$y = 0$

Step 2.10

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Remove parentheses.

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

Step 2.11.2

Combine the numerators over the common denominator.

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

Step 2.11.3

Simplify the expression.

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Step 2.11.3.1

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

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

Step 2.11.3.2

Subtract $3$ from $0$ .

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

Step 2.11.3.3

Move the negative in front of the fraction.

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

Step 2.11.4

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

$- \frac{3}{2}$

Step 2.12

The $y$ value at $x = 0$ is $- \frac{3}{2}$ .

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

Step 2.13

Graph the parabola using its properties and the selected points.

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

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 1, - 2)$

Focus: $(- 1, - \frac{3}{2})$

Axis of Symmetry: $x = - 1$

Directrix: $y = - \frac{5}{2}$

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

Step 4