Precalculus Examples

${(x - 4)}^{2} = 8 (y - 2)$

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

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Rewrite the equation as $8 (y - 2) = {(x - 4)}^{2}$ .

$8 (y - 2) = {(x - 4)}^{2}$

Divide each term in $8 (y - 2) = {(x - 4)}^{2}$ by $8$ and simplify.

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Divide each term in $8 (y - 2) = {(x - 4)}^{2}$ by $8$ .

$\frac{8 (y - 2)}{8} = \frac{{(x - 4)}^{2}}{8}$

Simplify the left side.

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Cancel the common factor of $8$ .

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Cancel the common factor.

$\frac{8 (y - 2)}{8} = \frac{{(x - 4)}^{2}}{8}$

Divide $y - 2$ by $1$ .

$y - 2 = \frac{{(x - 4)}^{2}}{8}$

Add $2$ to both sides of the equation.

$y = \frac{{(x - 4)}^{2}}{8} + 2$

Reorder terms.

$y = \frac{1}{8} \cdot {(x - 4)}^{2} + 2$

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

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

$h = 4$

$k = 2$

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

Opens Up

Find the vertex $(h, k)$ .

$(4, 2)$

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Simplify.

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

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

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

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Factor $4$ out of $4$ .

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

Cancel the common factors.

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Factor $4$ out of $8$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Multiply $2$ by $1$ .

$2$

Find the focus.

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The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(4, 4)$

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

$x = 4$