Algebra Examples

$y = \frac{1}{28} {(x - 4)}^{2} - 5$

Step 1

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

$a = \frac{1}{28}$

$h = 4$

$k = - 5$

Step 2

Find the vertex $(h, k)$ .

$(4, - 5)$

Step 3

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

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Step 3.1

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

$\frac{1}{4 a}$

Step 3.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{28}}$

Step 3.3

Simplify.

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Step 3.3.1

Combine $4$ and $\frac{1}{28}$ .

$\frac{1}{\frac{4}{28}}$

Step 3.3.2

Cancel the common factor of $4$ and $28$ .

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Step 3.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{28}}$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $4$ out of $28$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 7}}$

Step 3.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 7}}$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{7}}$

Step 3.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 7$

Step 3.3.4

Multiply $7$ by $1$ .

$7$

Step 4

Find the focus.

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Step 4.1

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 4.2

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

$(4, 2)$

Step 5