Algebra Examples

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

Step 1

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

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

$h = 4$

$k = 0$

Step 2

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 3

Find the vertex $(h, k)$ .

$(4, 0)$

Step 4

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

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Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

Simplify.

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Step 4.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

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

Step 4.3.3

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

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Step 4.3.3.1

Factor $4$ out of $4$ .

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $4$ out of $8$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Step 4.3.5

Multiply $- (1 \cdot 2)$ .

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Step 4.3.5.1

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Step 4.3.5.2

Multiply $- 1$ by $2$ .

$- 2$

Step 5

Find the focus.

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Step 5.1

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 5.2

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

$(4, - 2)$

Step 6

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

$x = 4$

Step 7