Algebra Examples

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

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

Step 1

Use the vertex form,

y = a {(x - h)}^{2} + k

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

a

$a$ ,

h

$h$ , and

k

$k$ .

a = \frac{1}{28}

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

h = 4

$h = 4$

k = - 5

$k = - 5$

Step 2

Find the vertex

(h, k)

$(h, k)$ .

(4, - 5)

$(4, - 5)$

Step 3

Find

p

$p$ , 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}

$\frac{1}{4 a}$

Step 3.2

Substitute the value of

a

$a$ into the formula.

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

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

Step 3.3

Simplify.

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Step 3.3.1

Combine

4

$4$ and

\frac{1}{28}

$\frac{1}{28}$ .

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

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

Step 3.3.2

Cancel the common factor of

4

$4$ and

28

$28$ .

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Step 3.3.2.1

Factor

4

$4$ out of

4

$4$ .

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

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor

4

$4$ out of

28

$28$ .

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

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

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