Pre-Algebra Examples

${(x - 6)}^{2} = 9 y$

Step 1

Solve for $y$ .

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Step 1.1

Rewrite the equation as $9 y = {(x - 6)}^{2}$ .

$9 y = {(x - 6)}^{2}$

Step 1.2

Divide each term in $9 y = {(x - 6)}^{2}$ by $9$ and simplify.

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Step 1.2.1

Divide each term in $9 y = {(x - 6)}^{2}$ by $9$ .

$\frac{9 y}{9} = \frac{{(x - 6)}^{2}}{9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{{(x - 6)}^{2}}{9}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{{(x - 6)}^{2}}{9}$

Step 2

Find the properties of the given parabola.

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Step 2.1

Reorder terms.

$y = \frac{1}{9} \cdot {(x - 6)}^{2}$

Step 2.2

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

$a = \frac{1}{9}$

$h = 6$

$k = 0$

Step 2.3

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

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(6, 0)$

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.

$\frac{1}{4 \cdot \frac{1}{9}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{1}{9}$ .

$\frac{1}{\frac{4}{9}}$

Step 2.5.3.2

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{9}{4})$

Step 2.5.3.3

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

$\frac{9}{4}$

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.

$(6, \frac{9}{4})$

Step 2.7

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

$x = 6$

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.

$y = - \frac{9}{4}$

Step 2.9

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

Direction: Opens Up

Vertex: $(6, 0)$

Focus: $(6, \frac{9}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{9}{4}$

Direction: Opens Up

Vertex: $(6, 0)$

Focus: $(6, \frac{9}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{9}{4}$

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.

$f (5) = \frac{{((5) - 6)}^{2}}{9}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Subtract $6$ from $5$ .

$f (5) = \frac{{(- 1)}^{2}}{9}$

Step 3.2.1.2

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

$f (5) = \frac{1}{9}$

Step 3.2.2

The final answer is $\frac{1}{9}$ .

$\frac{1}{9}$

Step 3.3

The $y$ value at $x = 5$ is $\frac{1}{9}$ .

$y = \frac{1}{9}$

Step 3.4

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

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

Step 3.5

Simplify the result.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Subtract $6$ from $4$ .

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

Step 3.5.1.2

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

$f (4) = \frac{4}{9}$

Step 3.5.2

The final answer is $\frac{4}{9}$ .

$\frac{4}{9}$

Step 3.6

The $y$ value at $x = 4$ is $\frac{4}{9}$ .

$y = \frac{4}{9}$

Step 3.7

Replace the variable $x$ with $9$ in the expression.

$f (9) = \frac{{((9) - 6)}^{2}}{9}$

Step 3.8

Simplify the result.

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

Subtract $6$ from $9$ .

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

Step 3.8.1.2

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

$f (9) = \frac{9}{9}$

Step 3.8.2

Divide $9$ by $9$ .

$f (9) = 1$

Step 3.8.3

The final answer is $1$ .

$1$

Step 3.9

The $y$ value at $x = 9$ is $1$ .

$y = 1$

Step 3.10

Replace the variable $x$ with $7$ in the expression.

$f (7) = \frac{{((7) - 6)}^{2}}{9}$

Step 3.11

Simplify the result.

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Step 3.11.1

Simplify the numerator.

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Step 3.11.1.1

Subtract $6$ from $7$ .

$f (7) = \frac{1^{2}}{9}$

Step 3.11.1.2

One to any power is one.

$f (7) = \frac{1}{9}$

Step 3.11.2

The final answer is $\frac{1}{9}$ .

$\frac{1}{9}$

Step 3.12

The $y$ value at $x = 7$ is $\frac{1}{9}$ .

$y = \frac{1}{9}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 4 & \frac{4}{9} \\ 5 & \frac{1}{9} \\ 6 & 0 \\ 7 & \frac{1}{9} \\ 9 & 1 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(6, 0)$

Focus: $(6, \frac{9}{4})$

Axis of Symmetry: $x = 6$

Directrix: $y = - \frac{9}{4}$

$\begin{array}{c} x & y \\ 4 & \frac{4}{9} \\ 5 & \frac{1}{9} \\ 6 & 0 \\ 7 & \frac{1}{9} \\ 9 & 1 \end{array}$

Step 5