Algebra Examples

$y = - 3 {(x - 6)}^{2} + 5$

Step 1

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

$a = - 3$

$h = 6$

$k = 5$

Step 2

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

Opens Down

Step 3

Find the vertex $(h, k)$ .

$(6, 5)$

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.

$\frac{1}{4 \cdot - 3}$

Step 4.3

Simplify.

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Step 4.3.1

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{1}{12}$

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.

$(6, \frac{59}{12})$

Step 6

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

$x = 6$

Step 7