Precalculus Examples

$4 x^{2} - 128 y = 0$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $y$ to the left side of the equation.

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Step 1.1.1

Subtract $4 x^{2}$ from both sides of the equation.

$- 128 y = - 4 x^{2}$

Step 1.1.2

Divide each term in $- 128 y = - 4 x^{2}$ by $- 128$ and simplify.

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Step 1.1.2.1

Divide each term in $- 128 y = - 4 x^{2}$ by $- 128$ .

$\frac{- 128 y}{- 128} = \frac{- 4 x^{2}}{- 128}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 128$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 128 y}{- 128} = \frac{- 4 x^{2}}{- 128}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 4 x^{2}}{- 128}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Cancel the common factor of $- 4$ and $- 128$ .

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Step 1.1.2.3.1.1

Factor $- 4$ out of $- 4 x^{2}$ .

$y = \frac{- 4 (x^{2})}{- 128}$

Step 1.1.2.3.1.2

Cancel the common factors.

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Step 1.1.2.3.1.2.1

Factor $- 4$ out of $- 128$ .

$y = \frac{- 4 (x^{2})}{- 4 (32)}$

Step 1.1.2.3.1.2.2

Cancel the common factor.

$y = \frac{- 4 x^{2}}{- 4 \cdot 32}$

Step 1.1.2.3.1.2.3

Rewrite the expression.

$y = \frac{x^{2}}{32}$

Step 1.2

Complete the square for $\frac{x^{2}}{32}$ .

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Step 1.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{1}{32}$

$b = 0$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 (\frac{1}{32})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{1}{32})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{1}{32})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{1}{32}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 \cdot 32$

Step 1.2.3.2.3

Multiply $0$ by $32$ .

$d = 0$

Step 1.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{0^{2}}{4 (\frac{1}{32})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 (\frac{1}{32})}$

Step 1.2.4.2.1.2

Combine $4$ and $\frac{1}{32}$ .

$e = 0 - \frac{0}{\frac{4}{32}}$

Step 1.2.4.2.1.3

Cancel the common factor of $4$ and $32$ .

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Step 1.2.4.2.1.3.1

Factor $4$ out of $4$ .

$e = 0 - \frac{0}{\frac{4 (1)}{32}}$

Step 1.2.4.2.1.3.2

Cancel the common factors.

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Step 1.2.4.2.1.3.2.1

Factor $4$ out of $32$ .

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 8}}$

Step 1.2.4.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot 1}{4 \cdot 8}}$

Step 1.2.4.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{0}{\frac{1}{8}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 \cdot 8)$

Step 1.2.4.2.1.5

Multiply $- (0 \cdot 8)$ .

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Step 1.2.4.2.1.5.1

Multiply $0$ by $8$ .

$e = 0 - 0$

Step 1.2.4.2.1.5.2

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{32} x^{2}$ .

$\frac{1}{32} x^{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{32} x^{2}$

Step 2

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

$a = \frac{1}{32}$

$h = 0$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(0, 0)$

Step 4