Precalculus Examples

$x^{2} = 8 (y - 1)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Rewrite the equation as $8 (y - 1) = x^{2}$ .

$8 (y - 1) = x^{2}$

Step 1.1.2

Divide each term in $8 (y - 1) = x^{2}$ by $8$ and simplify.

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Step 1.1.2.1

Divide each term in $8 (y - 1) = x^{2}$ by $8$ .

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

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $8$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

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

Step 1.1.2.2.1.2

Divide $y - 1$ by $1$ .

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

Step 1.1.3

Add $1$ to both sides of the equation.

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

Step 1.2

Complete the square for $\frac{x^{2}}{8} + 1$ .

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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 $c$ .

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

$b = 0$

$c = 1$

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

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

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 $0$ .

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

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{8}$ .

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

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 $4$ .

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

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 $8$ .

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

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

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

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

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

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 1 - (0 \cdot 2)$

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 $2$ .

$e = 1 - 0$

Step 1.2.4.2.1.5.2

Multiply $- 1$ by $0$ .

$e = 1 + 0$

Step 1.2.4.2.2

Add $1$ and $0$ .

$e = 1$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{8} {(x + 0)}^{2} + 1$ .

$\frac{1}{8} {(x + 0)}^{2} + 1$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{8} \cdot {(x + 0)}^{2} + 1$

Step 2

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

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

$h = 0$

$k = 1$

Step 3

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

Opens Up

Step 4

Find the vertex $(h, k)$ .

$(0, 1)$

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.

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

Step 5.3

Simplify.

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Step 5.3.1

Combine $4$ and $\frac{1}{8}$ .

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

Step 5.3.2

Cancel the common factor of $4$ and $8$ .

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Step 5.3.2.1

Factor $4$ out of $4$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $4$ out of $8$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 5.3.4

Multiply $2$ by $1$ .

$2$

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 y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 6.2

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

$(0, 3)$

Step 7

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

$x = 0$

Step 8

Find the directrix.

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Step 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 8.2

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

$y = - 1$

Step 9

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

Direction: Opens Up

Vertex: $(0, 1)$

Focus: $(0, 3)$

Axis of Symmetry: $x = 0$

Directrix: $y = - 1$

Step 10