Precalculus Examples

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

Step 1

Rewrite the equation as $\frac{1}{2} x = y^{2}$ .

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

Step 2

Multiply both sides of the equation by $2$ .

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $2 (\frac{1}{2} x)$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Rewrite the expression.

$x = 2 y^{2}$

Step 4

Find the properties of the given parabola.

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Step 4.1

Rewrite the equation in vertex form.

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Step 4.1.1

Complete the square for $2 y^{2}$ .

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Step 4.1.1.1

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

$a = 2$

$b = 0$

$c = 0$

Step 4.1.1.2

Consider the vertex form of a parabola.

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

Step 4.1.1.3

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

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Step 4.1.1.3.1

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

$d = \frac{0}{2 \cdot 2}$

Step 4.1.1.3.2

Simplify the right side.

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Step 4.1.1.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 4.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 2}$

Step 4.1.1.3.2.1.2

Cancel the common factors.

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Step 4.1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 2$ .

$d = \frac{2 (0)}{2 (2)}$

Step 4.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 2}$

Step 4.1.1.3.2.1.2.3

Rewrite the expression.

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

Step 4.1.1.3.2.2

Cancel the common factor of $0$ and $2$ .

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Step 4.1.1.3.2.2.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2}$

Step 4.1.1.3.2.2.2

Cancel the common factors.

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Step 4.1.1.3.2.2.2.1

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 4.1.1.3.2.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 4.1.1.3.2.2.2.3

Rewrite the expression.

$d = \frac{0}{1}$

Step 4.1.1.3.2.2.2.4

Divide $0$ by $1$ .

$d = 0$

Step 4.1.1.4

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

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Step 4.1.1.4.1

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

$e = 0 - \frac{0^{2}}{4 \cdot 2}$

Step 4.1.1.4.2

Simplify the right side.

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Step 4.1.1.4.2.1

Simplify each term.

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Step 4.1.1.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 \cdot 2}$

Step 4.1.1.4.2.1.2

Multiply $4$ by $2$ .

$e = 0 - \frac{0}{8}$

Step 4.1.1.4.2.1.3

Divide $0$ by $8$ .

$e = 0 - 0$

Step 4.1.1.4.2.1.4

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 4.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 4.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 y^{2}$ .

$2 y^{2}$

Step 4.1.2

Set $x$ equal to the new right side.

$x = 2 y^{2}$

Step 4.2

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

$a = 2$

$h = 0$

$k = 0$

Step 4.3

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

Opens Right

Step 4.4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 4.5

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

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Step 4.5.1

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

$\frac{1}{4 a}$

Step 4.5.2

Substitute the value of $a$ into the formula.

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

Step 4.5.3

Multiply $4$ by $2$ .

$\frac{1}{8}$

Step 4.6

Find the focus.

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Step 4.6.1

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 4.6.2

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

$(\frac{1}{8}, 0)$

Step 4.7

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

$y = 0$

Step 4.8

Find the directrix.

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Step 4.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 4.8.2

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

$x = - \frac{1}{8}$

Step 4.9

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

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{8}$

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{8}$

Step 5

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 5.1

Substitute the $x$ value $1$ into $f (x) = \frac{\sqrt{2 x}}{2}$ . In this case, the point is $(1, 0.70710678)$ .

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Step 5.1.1

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

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

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Multiply $2$ by $1$ .

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

Step 5.1.2.2

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

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

Step 5.1.3

Convert $\frac{\sqrt{2}}{2}$ to decimal.

$= 0.70710678$

Step 5.2

Substitute the $x$ value $1$ into $f (x) = - \frac{\sqrt{2 x}}{2}$ . In this case, the point is $(1, - 0.70710678)$ .

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Step 5.2.1

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

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

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Multiply $2$ by $1$ .

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

Step 5.2.2.2

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

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

Step 5.2.3

Convert $- \frac{\sqrt{2}}{2}$ to decimal.

$= - 0.70710678$

Step 5.3

Substitute the $x$ value $2$ into $f (x) = \frac{\sqrt{2 x}}{2}$ . In this case, the point is $(2, 1)$ .

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Step 5.3.1

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

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

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Simplify the numerator.

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Step 5.3.2.1.1

Multiply $2$ by $2$ .

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

Step 5.3.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.3.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.3.2.2

Divide $2$ by $2$ .

$f (2) = 1$

Step 5.3.2.3

The final answer is $1$ .

$y = 1$

Step 5.3.3

Convert $1$ to decimal.

$= 1$

Step 5.4

Substitute the $x$ value $2$ into $f (x) = - \frac{\sqrt{2 x}}{2}$ . In this case, the point is $(2, - 1)$ .

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Step 5.4.1

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

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

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Simplify the numerator.

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Step 5.4.2.1.1

Multiply $2$ by $2$ .

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

Step 5.4.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 5.4.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

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

Step 5.4.2.2

Reduce the expression by cancelling the common factors.

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Step 5.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

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

Step 5.4.2.2.1.2

Rewrite the expression.

$f (2) = - 1 \cdot 1$

Step 5.4.2.2.2

Multiply $- 1$ by $1$ .

$f (2) = - 1$

Step 5.4.2.3

The final answer is $- 1$ .

$y = - 1$

Step 5.4.3

Convert $- 1$ to decimal.

$= - 1$

Step 5.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0.71 \\ 1 & - 0.71 \\ 2 & 1 \\ 2 & - 1 \end{array}$

Step 6

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{8}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{8}$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0.71 \\ 1 & - 0.71 \\ 2 & 1 \\ 2 & - 1 \end{array}$

Step 7