Algebra Examples

$x = \frac{1}{2} y^{2} + 2 y + 11$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

$x = \frac{y^{2}}{2} + 2 y + 11$

Step 1.2

Complete the square for $\frac{y^{2}}{2} + 2 y + 11$ .

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Step 1.2.1

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

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

$b = 2$

$c = 11$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

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

Step 1.2.3.2.1.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Step 1.2.3.2.3

Multiply $2$ by $1$ .

$d = 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = 11 - \frac{2^{2}}{4 (\frac{1}{2})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

$e = 11 - \frac{4}{4 (\frac{1}{2})}$

Step 1.2.4.2.1.2

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

$e = 11 - \frac{4}{\frac{4}{2}}$

Step 1.2.4.2.1.3

Divide $4$ by $2$ .

$e = 11 - \frac{4}{2}$

Step 1.2.4.2.1.4

Divide $4$ by $2$ .

$e = 11 - 1 \cdot 2$

Step 1.2.4.2.1.5

Multiply $- 1$ by $2$ .

$e = 11 - 2$

Step 1.2.4.2.2

Subtract $2$ from $11$ .

$e = 9$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(y + 2)}^{2} + 9$ .

$\frac{1}{2} {(y + 2)}^{2} + 9$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{2} \cdot {(y + 2)}^{2} + 9$

Step 2

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

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

$h = 9$

$k = - 2$

Step 3

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

Opens Right

Step 4

Find the vertex $(h, k)$ .

$(9, - 2)$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

Divide $4$ by $2$ .

$\frac{1}{2}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.2

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

$(\frac{19}{2}, - 2)$

Step 7

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

$y = - 2$

Step 8