Algebra Examples

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

Step 1

Find the properties of the given parabola.

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Step 1.1

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

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

$h = - 4$

$k = - 2$

Step 1.2

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

Opens Up

Step 1.3

Find the vertex $(h, k)$ .

$(- 4, - 2)$

Step 1.4

Find $p$ , the distance from the vertex to the focus.

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Step 1.4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 1.4.2

Substitute the value of $a$ into the formula.

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

Step 1.4.3

Simplify.

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Step 1.4.3.1

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

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

Step 1.4.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 1.5

Find the focus.

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Step 1.5.1

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 1.5.2

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

$(- 4, - \frac{3}{2})$

Step 1.6

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

$x = - 4$

Step 1.7

Find the directrix.

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Step 1.7.1

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 1.7.2

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

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

Step 1.8

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

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - \frac{3}{2})$

Axis of Symmetry: $x = - 4$

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

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - \frac{3}{2})$

Axis of Symmetry: $x = - 4$

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 $- 6$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

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

$f (- 6) = \frac{36}{2} + 4 (- 6) + 6$

Step 2.2.1.2

Divide $36$ by $2$ .

$f (- 6) = 18 + 4 (- 6) + 6$

Step 2.2.1.3

Multiply $4$ by $- 6$ .

$f (- 6) = 18 - 24 + 6$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $24$ from $18$ .

$f (- 6) = - 6 + 6$

Step 2.2.2.2

Add $- 6$ and $6$ .

$f (- 6) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

The $y$ value at $x = - 6$ is $0$ .

$y = 0$

Step 2.4

Replace the variable $x$ with $- 5$ in the expression.

$f (- 5) = \frac{{(- 5)}^{2}}{2} + 4 (- 5) + 6$

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Raise $- 5$ to the power of $2$ .

$f (- 5) = \frac{25}{2} + 4 (- 5) + 6$

Step 2.5.1.2

Multiply $4$ by $- 5$ .

$f (- 5) = \frac{25}{2} - 20 + 6$

Step 2.5.2

Find the common denominator.

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Step 2.5.2.1

Write $- 20$ as a fraction with denominator $1$ .

$f (- 5) = \frac{25}{2} + \frac{- 20}{1} + 6$

Step 2.5.2.2

Multiply $\frac{- 20}{1}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{25}{2} + \frac{- 20}{1} \cdot \frac{2}{2} + 6$

Step 2.5.2.3

Multiply $\frac{- 20}{1}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + 6$

Step 2.5.2.4

Write $6$ as a fraction with denominator $1$ .

$f (- 5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6}{1}$

Step 2.5.2.5

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

Step 2.5.2.6

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (- 5) = \frac{25}{2} + \frac{- 20 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Step 2.5.3

Combine the numerators over the common denominator.

$f (- 5) = \frac{25 - 20 \cdot 2 + 6 \cdot 2}{2}$

Step 2.5.4

Simplify each term.

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Step 2.5.4.1

Multiply $- 20$ by $2$ .

$f (- 5) = \frac{25 - 40 + 6 \cdot 2}{2}$

Step 2.5.4.2

Multiply $6$ by $2$ .

$f (- 5) = \frac{25 - 40 + 12}{2}$

Step 2.5.5

Simplify the expression.

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Step 2.5.5.1

Subtract $40$ from $25$ .

$f (- 5) = \frac{- 15 + 12}{2}$

Step 2.5.5.2

Add $- 15$ and $12$ .

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

Step 2.5.5.3

Move the negative in front of the fraction.

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

Step 2.5.6

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

$- \frac{3}{2}$

Step 2.6

The $y$ value at $x = - 5$ is $- \frac{3}{2}$ .

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

Step 2.7

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Cancel the common factor of ${(- 2)}^{2}$ and $2$ .

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Step 2.8.1.1.1

Rewrite $- 2$ as $- 1 (2)$ .

$f (- 2) = \frac{{(- 1 \cdot 2)}^{2}}{2} + 4 (- 2) + 6$

Step 2.8.1.1.2

Apply the product rule to $- 1 (2)$ .

$f (- 2) = \frac{{(- 1)}^{2} \cdot 2^{2}}{2} + 4 (- 2) + 6$

Step 2.8.1.1.3

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

$f (- 2) = \frac{1 \cdot 2^{2}}{2} + 4 (- 2) + 6$

Step 2.8.1.1.4

Multiply $2^{2}$ by $1$ .

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

Step 2.8.1.1.5

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

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

Step 2.8.1.1.6

Cancel the common factors.

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Step 2.8.1.1.6.1

Factor $2$ out of $2$ .

$f (- 2) = \frac{2 \cdot 2}{2 (1)} + 4 (- 2) + 6$

Step 2.8.1.1.6.2

Cancel the common factor.

$f (- 2) = \frac{2 \cdot 2}{2 \cdot 1} + 4 (- 2) + 6$

Step 2.8.1.1.6.3

Rewrite the expression.

$f (- 2) = \frac{2}{1} + 4 (- 2) + 6$

Step 2.8.1.1.6.4

Divide $2$ by $1$ .

$f (- 2) = 2 + 4 (- 2) + 6$

Step 2.8.1.2

Multiply $4$ by $- 2$ .

$f (- 2) = 2 - 8 + 6$

Step 2.8.2

Simplify by adding and subtracting.

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Step 2.8.2.1

Subtract $8$ from $2$ .

$f (- 2) = - 6 + 6$

Step 2.8.2.2

Add $- 6$ and $6$ .

$f (- 2) = 0$

Step 2.8.3

The final answer is $0$ .

$0$

Step 2.9

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

$y = 0$

Step 2.10

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

$f (- 3) = \frac{{(- 3)}^{2}}{2} + 4 (- 3) + 6$

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify each term.

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Step 2.11.1.1

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

$f (- 3) = \frac{9}{2} + 4 (- 3) + 6$

Step 2.11.1.2

Multiply $4$ by $- 3$ .

$f (- 3) = \frac{9}{2} - 12 + 6$

Step 2.11.2

Find the common denominator.

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Step 2.11.2.1

Write $- 12$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{2} + \frac{- 12}{1} + 6$

Step 2.11.2.2

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 12}{1} \cdot \frac{2}{2} + 6$

Step 2.11.2.3

Multiply $\frac{- 12}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + 6$

Step 2.11.2.4

Write $6$ as a fraction with denominator $1$ .

$f (- 3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6}{1}$

Step 2.11.2.5

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6}{1} \cdot \frac{2}{2}$

Step 2.11.2.6

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$f (- 3) = \frac{9}{2} + \frac{- 12 \cdot 2}{2} + \frac{6 \cdot 2}{2}$

Step 2.11.3

Combine the numerators over the common denominator.

$f (- 3) = \frac{9 - 12 \cdot 2 + 6 \cdot 2}{2}$

Step 2.11.4

Simplify each term.

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Step 2.11.4.1

Multiply $- 12$ by $2$ .

$f (- 3) = \frac{9 - 24 + 6 \cdot 2}{2}$

Step 2.11.4.2

Multiply $6$ by $2$ .

$f (- 3) = \frac{9 - 24 + 12}{2}$

Step 2.11.5

Simplify the expression.

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Step 2.11.5.1

Subtract $24$ from $9$ .

$f (- 3) = \frac{- 15 + 12}{2}$

Step 2.11.5.2

Add $- 15$ and $12$ .

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

Step 2.11.5.3

Move the negative in front of the fraction.

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

Step 2.11.6

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

$- \frac{3}{2}$

Step 2.12

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

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

Step 2.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 6 & 0 \\ - 5 & - \frac{3}{2} \\ - 4 & - 2 \\ - 3 & - \frac{3}{2} \\ - 2 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 4, - 2)$

Focus: $(- 4, - \frac{3}{2})$

Axis of Symmetry: $x = - 4$

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

$\begin{array}{c} x & y \\ - 6 & 0 \\ - 5 & - \frac{3}{2} \\ - 4 & - 2 \\ - 3 & - \frac{3}{2} \\ - 2 & 0 \end{array}$

Step 4