Precalculus Examples

y^{2} = 8 x

$y^{2} = 8 x$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate

x

$x$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as

8 x = y^{2}

$8 x = y^{2}$ .

8 x = y^{2}

$8 x = y^{2}$

Step 1.1.2

Divide each term in

8 x = y^{2}

$8 x = y^{2}$ by

8

$8$ and simplify.

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Step 1.1.2.1

Divide each term in

8 x = y^{2}

$8 x = y^{2}$ by

8

$8$ .

\frac{8 x}{8} = \frac{y^{2}}{8}

$\frac{8 x}{8} = \frac{y^{2}}{8}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of

8

$8$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{y^{2}}{8}$

Step 1.1.2.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{y^{2}}{8}$

Step 1.2

Complete the square for

$\frac{y^{2}}{8}$ .

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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

$c$ .

$a = \frac{1}{8}$

$b = 0$

$c = 0$

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

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

$d = \frac{0}{2 (\frac{1}{8})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of

$0$ and

$2$ .

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Step 1.2.3.2.1.1

Factor

$2$ out of

$0$ .

$d = \frac{2 (0)}{2 (\frac{1}{8})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{8})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{1}{8}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 8$

Step 1.2.3.2.3

Multiply

$0$ by

$8$ .

$d = 0$

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

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

$e = 0 - \frac{0^{2}}{4 (\frac{1}{8})}$

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

Raising

$0$ to any positive power yields

$0$ .

$e = 0 - \frac{0}{4 (\frac{1}{8})}$

Step 1.2.4.2.1.2

Combine

$4$ and

$\frac{1}{8}$ .

$e = 0 - \frac{0}{\frac{4}{8}}$

Step 1.2.4.2.1.3

Cancel the common factor of

$4$ and

$8$ .

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Step 1.2.4.2.1.3.1

Factor

$4$ out of

$4$ .

$e = 0 - \frac{0}{\frac{4 (1)}{8}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor

$4$ out of

$8$ .

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 2}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{0}{\frac{1}{2}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 \cdot 2)$

Step 1.2.4.2.1.5

Multiply

$- (0 \cdot 2)$ .

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Step 1.2.4.2.1.5.1

Multiply

$0$ by

$2$ .

$e = 0 - 0$

Step 1.2.4.2.1.5.2

Multiply

$- 1$ by

$0$ .

$e = 0 + 0$

Step 1.2.4.2.2

Add

$0$ and

$0$ .

$e = 0$

Step 1.2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$\frac{1}{8} y^{2}$ .

$\frac{1}{8} y^{2}$

Step 1.3

Set

$x$ equal to the new right side.

$x = \frac{1}{8} y^{2}$

Step 2

Use the vertex form,

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

$a$ ,

$h$ , and

$k$ .

$a = \frac{1}{8}$

$h = 0$

$k = 0$

Step 3

Find the vertex

$(h, k)$ .

$(0, 0)$

Step 4

Find

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

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Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of

$a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{8}}$

Step 4.3

Simplify.

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Step 4.3.1

Combine

$4$ and

$\frac{1}{8}$ .

$\frac{1}{\frac{4}{8}}$

Step 4.3.2

Cancel the common factor of

$4$ and

$8$ .

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Step 4.3.2.1

Factor

$4$ out of

$4$ .

$\frac{1}{\frac{4 (1)}{8}}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor

$4$ out of

$8$ .

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

Step 4.3.2.2.2

Cancel the common factor.

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

Step 4.3.2.2.3

Rewrite the expression.

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

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 4.3.4

Multiply

$2$ by

$1$ .

$2$

Step 5

Find the focus.

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Step 5.1

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 5.2

Substitute the known values of

$h$ ,

$p$ , and

$k$ into the formula and simplify.

$(2, 0)$

Step 6