Algebra Examples

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

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

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 = - 3

$a = - 3$

h = 6

$h = 6$

k = 5

$k = 5$

Step 2

Since the value of

a

$a$ is negative, the parabola opens down.

Opens Down

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(6, 5)

$(6, 5)$

Step 4

Find

p

$p$ , 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}

$\frac{1}{4 a}$

Step 4.2

Substitute the value of

a

$a$ into the formula.

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

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

Step 4.3

Simplify.

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Step 4.3.1

Multiply

4

$4$ by

- 3

$- 3$ .

\frac{1}{- 12}

$\frac{1}{- 12}$

Step 4.3.2

Move the negative in front of the fraction.

- \frac{1}{12}

$- \frac{1}{12}$

- \frac{1}{12}

$- \frac{1}{12}$

- \frac{1}{12}

$- \frac{1}{12}$

Step 5

Find the focus.

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Step 5.1

The focus of a parabola can be found by adding

p

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 5.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

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

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

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

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

Step 6

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

x = 6

$x = 6$

Step 7