Find the Focus 1/8(y-5)^2=x+3

$\frac{1}{8} {(y - 5)}^{2} = x + 3$

Step 1

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

Rewrite the equation as $x + 3 = \frac{1}{8} {(y - 5)}^{2}$ .

$x + 3 = \frac{1}{8} {(y - 5)}^{2}$

Simplify $\frac{1}{8} {(y - 5)}^{2}$ .

Rewrite.

$x + 3 = 0 + 0 + \frac{1}{8} {(y - 5)}^{2}$

Simplify by adding zeros.

$x + 3 = \frac{1}{8} {(y - 5)}^{2}$

Combine $\frac{1}{8}$ and ${(y - 5)}^{2}$ .

$x + 3 = \frac{{(y - 5)}^{2}}{8}$

Subtract $3$ from both sides of the equation.

$x = \frac{{(y - 5)}^{2}}{8} - 3$

Reorder terms.

$x = \frac{1}{8} \cdot {(y - 5)}^{2} - 3$

Step 2

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

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

$h = - 3$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(- 3, 5)$

Step 4

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

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.

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

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

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

Factor $4$ out of $4$ .

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

Cancel the common factors.

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 $1$ .

$2$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(- 1, 5)$

Step 6

Find the Focus 1/8(y-5)^2=x+3

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