Pre-Algebra Examples

$y = \frac{1}{2} x^{2} + 3 x + \frac{9}{2}$

Step 1

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

$y = \frac{x^{2}}{2} + 3 x + \frac{9}{2}$

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 $\frac{x^{2}}{2} + 3 x + \frac{9}{2}$ .

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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 $c$ .

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

$b = 3$

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

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

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

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

Step 2.1.1.3.2.2

Divide $2$ by $2$ .

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

Step 2.1.1.3.2.3

Divide $3$ by $1$ .

$d = 3$

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

$e = \frac{9}{2} - \frac{3^{2}}{4 (\frac{1}{2})}$

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

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

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

Step 2.1.1.4.2.1.2

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

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

Step 2.1.1.4.2.1.3

Divide $4$ by $2$ .

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

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

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

Step 2.1.1.4.2.3

Subtract $9$ from $9$ .

$e = \frac{0}{2}$

Step 2.1.1.4.2.4

Divide $0$ by $2$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x + 3)}^{2} + 0$ .

$\frac{1}{2} {(x + 3)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 3)}^{2} + 0$

Step 2.2

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

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

$h = - 3$

$k = 0$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(- 3, 0)$

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

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

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

Step 2.5.3.2

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.

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

Step 2.7

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

$x = - 3$

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.

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

Step 2.9

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

Direction: Opens Up

Vertex: $(- 3, 0)$

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

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{1}{2}$

Direction: Opens Up

Vertex: $(- 3, 0)$

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

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{1}{2}$

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 $- 5$ in the expression.

$f (- 5) = \frac{{(- 5)}^{2}}{2} + 3 (- 5) + \frac{9}{2}$

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

Combine the numerators over the common denominator.

$f (- 5) = 3 (- 5) + \frac{{(- 5)}^{2} + 9}{2}$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

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

$f (- 5) = 3 (- 5) + \frac{25 + 9}{2}$

Step 3.2.1.2.2

Add $25$ and $9$ .

$f (- 5) = 3 (- 5) + \frac{34}{2}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply $3$ by $- 5$ .

$f (- 5) = - 15 + \frac{34}{2}$

Step 3.2.2.2

Divide $34$ by $2$ .

$f (- 5) = - 15 + 17$

Step 3.2.3

Add $- 15$ and $17$ .

$f (- 5) = 2$

Step 3.2.4

The final answer is $2$ .

$2$

Step 3.3

The $y$ value at $x = - 5$ is $2$ .

$y = 2$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

Combine the numerators over the common denominator.

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

Step 3.5.2

Simplify the expression.

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Step 3.5.2.1

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

$f (- 4) = 3 (- 4) + \frac{16 + 9}{2}$

Step 3.5.2.2

Add $16$ and $9$ .

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

Step 3.5.2.3

Multiply $3$ by $- 4$ .

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

Step 3.5.3

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.5.4

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

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

Step 3.5.5

Combine the numerators over the common denominator.

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

Step 3.5.6

Simplify the numerator.

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Step 3.5.6.1

Multiply $- 12$ by $2$ .

$f (- 4) = \frac{- 24 + 25}{2}$

Step 3.5.6.2

Add $- 24$ and $25$ .

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

Step 3.5.7

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3.6

The $y$ value at $x = - 4$ is $\frac{1}{2}$ .

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

Step 3.7

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

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

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

Combine the numerators over the common denominator.

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

Step 3.8.1.2

Simplify the expression.

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Step 3.8.1.2.1

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

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

Step 3.8.1.2.2

Add $1$ and $9$ .

$f (- 1) = 3 (- 1) + \frac{10}{2}$

Step 3.8.2

Simplify each term.

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Step 3.8.2.1

Multiply $3$ by $- 1$ .

$f (- 1) = - 3 + \frac{10}{2}$

Step 3.8.2.2

Divide $10$ by $2$ .

$f (- 1) = - 3 + 5$

Step 3.8.3

Add $- 3$ and $5$ .

$f (- 1) = 2$

Step 3.8.4

The final answer is $2$ .

$2$

Step 3.9

The $y$ value at $x = - 1$ is $2$ .

$y = 2$

Step 3.10

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

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

Step 3.11

Simplify the result.

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Step 3.11.1

Combine the numerators over the common denominator.

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

Step 3.11.2

Simplify the expression.

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Step 3.11.2.1

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

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

Step 3.11.2.2

Add $4$ and $9$ .

$f (- 2) = 3 (- 2) + \frac{13}{2}$

Step 3.11.2.3

Multiply $3$ by $- 2$ .

$f (- 2) = - 6 + \frac{13}{2}$

Step 3.11.3

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (- 2) = - 6 \cdot \frac{2}{2} + \frac{13}{2}$

Step 3.11.4

Combine $- 6$ and $\frac{2}{2}$ .

$f (- 2) = \frac{- 6 \cdot 2}{2} + \frac{13}{2}$

Step 3.11.5

Combine the numerators over the common denominator.

$f (- 2) = \frac{- 6 \cdot 2 + 13}{2}$

Step 3.11.6

Simplify the numerator.

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Step 3.11.6.1

Multiply $- 6$ by $2$ .

$f (- 2) = \frac{- 12 + 13}{2}$

Step 3.11.6.2

Add $- 12$ and $13$ .

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

Step 3.11.7

The final answer is $\frac{1}{2}$ .

$\frac{1}{2}$

Step 3.12

The $y$ value at $x = - 2$ is $\frac{1}{2}$ .

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

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & 2 \\ - 4 & \frac{1}{2} \\ - 3 & 0 \\ - 2 & \frac{1}{2} \\ - 1 & 2 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 3, 0)$

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

Axis of Symmetry: $x = - 3$

Directrix: $y = - \frac{1}{2}$

$\begin{array}{c} x & y \\ - 5 & 2 \\ - 4 & \frac{1}{2} \\ - 3 & 0 \\ - 2 & \frac{1}{2} \\ - 1 & 2 \end{array}$

Step 5