Precalculus Examples

$(0, 0)$ , $(- 6, 6)$

Step 1

The general equation of a parabola with vertex $(h, k)$ is $y = a {(x - h)}^{2} + k$ . In this case we have $(0, 0)$ as the vertex $(h, k)$ and $(- 6, 6)$ is a point $(x, y)$ on the parabola. To find $a$ , substitute the two points in $y = a {(x - h)}^{2} + k$ .

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

Step 2

Using $6 = a {(- 6 - (0))}^{2} + 0$ to solve for $a$ , $a = \frac{1}{6}$ .

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Step 2.1

Rewrite the equation as $a {(- 6 - (0))}^{2} + 0 = 6$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Subtract $0$ from $- 6$ .

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

Step 2.2.3

Raise $- 6$ to the power of $2$ .

$a \cdot 36 = 6$

Step 2.2.4

Move $36$ to the left of $a$ .

$36 a = 6$

Step 2.3

Divide each term in $36 a = 6$ by $36$ and simplify.

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Step 2.3.1

Divide each term in $36 a = 6$ by $36$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $36$ .

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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 $1$ .

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

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.

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

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 $x$ .

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

Step 4.4.2

Combine $\frac{1}{6}$ and $x^{2}$ .

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

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

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