Algebra Examples

$2 y = {(x - 3)}^{2}$

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

Divide each term in $2 y = {(x - 3)}^{2}$ by $2$ and simplify.

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Step 1.1.1.1

Divide each term in $2 y = {(x - 3)}^{2}$ by $2$ .

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

Step 1.1.1.2

Simplify the left side.

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Step 1.1.1.2.1

Cancel the common factor of $2$ .

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Step 1.1.1.2.1.1

Cancel the common factor.

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

Step 1.1.1.2.1.2

Divide $y$ by $1$ .

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

Step 1.1.2

Reorder terms.

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

Step 1.2

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

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

$h = 3$

$k = 0$

Step 1.3

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

Opens Up

Step 1.4

Find the vertex $(h, k)$ .

$(3, 0)$

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.

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

Step 1.7

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

$x = 3$

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.

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

Step 1.9

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 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 $1$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Subtract $3$ from $1$ .

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

Step 2.2.1.2

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

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

Step 2.2.2

Divide $4$ by $2$ .

$f (1) = 2$

Step 2.2.3

The final answer is $2$ .

$2$

Step 2.3

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

$y = 2$

Step 2.4

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

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

Step 2.5

Simplify the result.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Subtract $3$ from $2$ .

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

Step 2.5.1.2

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

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

Step 2.5.2

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

$\frac{1}{2}$

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the result.

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

Subtract $3$ from $5$ .

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

Step 2.8.1.2

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

$f (5) = \frac{4}{2}$

Step 2.8.2

Divide $4$ by $2$ .

$f (5) = 2$

Step 2.8.3

The final answer is $2$ .

$2$

Step 2.9

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

$y = 2$

Step 2.10

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

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

Step 2.11

Simplify the result.

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Step 2.11.1

Simplify the numerator.

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Step 2.11.1.1

Subtract $3$ from $4$ .

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

Step 2.11.1.2

One to any power is one.

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

Step 2.11.2

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

$\frac{1}{2}$

Step 2.12

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

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

Step 2.13

Graph the parabola using its properties and the selected points.

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

Step 3

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

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

Step 4