Algebra Examples

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

Step 1

Simplify.

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Step 1.1

Add $x^{2}$ to both sides of the equation.

$x^{2} y + 4 y = x^{2}$

Step 1.2

Factor $y$ out of $x^{2} y + 4 y$ .

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Step 1.2.1

Factor $y$ out of $x^{2} y$ .

$y x^{2} + 4 y = x^{2}$

Step 1.2.2

Factor $y$ out of $4 y$ .

$y x^{2} + y \cdot 4 = x^{2}$

Step 1.2.3

Factor $y$ out of $y x^{2} + y \cdot 4$ .

$y (x^{2} + 4) = x^{2}$

Step 1.3

Divide each term in $y (x^{2} + 4) = x^{2}$ by $x^{2} + 4$ and simplify.

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Step 1.3.1

Divide each term in $y (x^{2} + 4) = x^{2}$ by $x^{2} + 4$ .

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

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $x^{2} + 4$ .

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Step 1.3.2.1.1

Cancel the common factor.

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

Step 1.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{2}}{x^{2} + 4}$

Step 2

Find where the expression $\frac{x^{2}}{x^{2} + 4}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 3

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 4

Consider the rational function $R (x) = \frac{a x^{n}}{b x^{m}}$ where $n$ is the degree of the numerator and $m$ is the degree of the denominator.

1. If $n < m$ , then the x-axis, $y = 0$ , is the horizontal asymptote.

2. If $n = m$ , then the horizontal asymptote is the line $y = \frac{a}{b}$ .

3. If $n > m$ , then there is no horizontal asymptote (there is an oblique asymptote).

Step 5

Find $n$ and $m$ .

$n = 2$

$m = 2$

Step 6

Since $n = m$ , the horizontal asymptote is the line $y = \frac{a}{b}$ where $a = 1$ and $b = 1$ .

$y = 1$

Step 7

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

Step 8

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: $y = 1$

No Oblique Asymptotes

Step 9