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Precalculus Examples

(0, 0)

$(0, 0)$ ,

(- 6, 6)

$(- 6, 6)$

Step 1

The general equation of a parabola with vertex

(h, k)

$(h, k)$ is

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

$y = a {(x - h)}^{2} + k$ . In this case we have

(0, 0)

$(0, 0)$ as the vertex

(h, k)

$(h, k)$ and

(- 6, 6)

$(- 6, 6)$ is a point

(x, y)

$(x, y)$ on the parabola. To find

a

$a$ , substitute the two points in

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

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

6 = a {(- 6 - (0))}^{2} + 0

$6 = a {(- 6 - (0))}^{2} + 0$

Step 2

Using

6 = a {(- 6 - (0))}^{2} + 0

$6 = a {(- 6 - (0))}^{2} + 0$ to solve for

a

$a$ ,

a = \frac{1}{6}

$a = \frac{1}{6}$ .

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Step 2.1

Rewrite the equation as

a {(- 6 - (0))}^{2} + 0 = 6

$a {(- 6 - (0))}^{2} + 0 = 6$ .

a {(- 6 - (0))}^{2} + 0 = 6

$a {(- 6 - (0))}^{2} + 0 = 6$

Step 2.2

Simplify

a {(- 6 - (0))}^{2} + 0

$a {(- 6 - (0))}^{2} + 0$ .

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Step 2.2.1

Add

a {(- 6 - (0))}^{2}

$a {(- 6 - (0))}^{2}$ and

0

$0$ .

a {(- 6 - (0))}^{2} = 6

$a {(- 6 - (0))}^{2} = 6$

Step 2.2.2

Subtract

0

$0$ from

- 6

$- 6$ .

a {(- 6)}^{2} = 6

$a {(- 6)}^{2} = 6$

Step 2.2.3

Raise

- 6

$- 6$ to the power of

2

$2$ .

a \cdot 36 = 6

$a \cdot 36 = 6$

Step 2.2.4

Move

36

$36$ to the left of

a

$a$ .

36 a = 6

$36 a = 6$

36 a = 6

$36 a = 6$

Step 2.3

Divide each term in

36 a = 6

$36 a = 6$ by

36

$36$ and simplify.

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Step 2.3.1

Divide each term in

36 a = 6

$36 a = 6$ by

36

$36$ .

\frac{36 a}{36} = \frac{6}{36}

$\frac{36 a}{36} = \frac{6}{36}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of

36

$36$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{36 a}{36} = \frac{6}{36}$

Step 2.3.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{6}{36}$

$a = \frac{6}{36}$

$a = \frac{6}{36}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of

$6$ and

$36$ .

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Step 2.3.3.1.1

Factor

$6$ out of

$6$ .

$a = \frac{6 (1)}{36}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor

$6$ out of

$36$ .

$a = \frac{6 \cdot 1}{6 \cdot 6}$

Step 2.3.3.1.2.2

Cancel the common factor.

$a = \frac{6 \cdot 1}{6 \cdot 6}$

Step 2.3.3.1.2.3

Rewrite the expression.

$a = \frac{1}{6}$

$a = \frac{1}{6}$

$a = \frac{1}{6}$

$a = \frac{1}{6}$

$a = \frac{1}{6}$

$a = \frac{1}{6}$

Step 3

Using

$y = a {(x - h)}^{2} + k$ , the general equation of the parabola with the vertex

$(0, 0)$ and

$a = \frac{1}{6}$ is

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$ .

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$

Step 4

Solve

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$ for

$y$ .

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Step 4.1

Remove parentheses.

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$

Step 4.2

Multiply

$\frac{1}{6}$ by

${(x - (0))}^{2}$ .

$y = \frac{1}{6} \cdot {(x - (0))}^{2} + 0$

Step 4.3

Remove parentheses.

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$

Step 4.4

Simplify

$(\frac{1}{6}) {(x - (0))}^{2} + 0$ .

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Step 4.4.1

Simplify by adding zeros.

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Step 4.4.1.1

Add

$(\frac{1}{6}) {(x - (0))}^{2}$ and

$0$ .

$y = (\frac{1}{6}) {(x - (0))}^{2}$

Step 4.4.1.2

Subtract

$0$ from

$x$ .

$y = \frac{1}{6} x^{2}$

$y = \frac{1}{6} x^{2}$

Step 4.4.2

Combine

$\frac{1}{6}$ and

$x^{2}$ .

$y = \frac{x^{2}}{6}$

$y = \frac{x^{2}}{6}$

$y = \frac{x^{2}}{6}$

Step 5

The standard form and vertex form are as follows.

Standard Form:

$y = \frac{1}{6} x^{2}$

Vertex Form:

$y = (\frac{1}{6}) {(x - (0))}^{2} + 0$

Step 6

Simplify the standard form.

Standard Form:

$y = \frac{1}{6} x^{2}$

Vertex Form:

$y = \frac{1}{6} x^{2}$

Step 7

Enter YOUR Problem

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$