Graph y=((x-A)^2)/(x-B)

$y = \frac{{(x - A)}^{2}}{x - B}$

Step 1

Find where the expression $\frac{{(x - A)}^{2}}{x - B}$ 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 2

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Step 3

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 4

Find $n$ and $m$ .

$n = 2$

$m = 1$

Step 5

Since $n > m$ , there is no horizontal asymptote.

No Horizontal Asymptotes

Step 6

Find the oblique asymptote using polynomial division.

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x A = 2 \cdot x \cdot A$

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot A + A^{2}}{x - B}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = A$ .

$\frac{{(x - A)}^{2}}{x - B}$

Expand ${(x - A)}^{2}$ .

Rewrite ${(x - A)}^{2}$ as $(x - A) (x - A)$ .

$\frac{(x - A) (x - A)}{x - B}$

Apply the distributive property.

$\frac{x (x - A) - A (x - A)}{x - B}$

Apply the distributive property.

$\frac{x \cdot x + x (- A) - A (x - A)}{x - B}$

Apply the distributive property.

$\frac{x \cdot x + x (- A) - A x - A (- A)}{x - B}$

Remove parentheses.

$\frac{x \cdot x + x \cdot (- 1 A) - A x - A (- A)}{x - B}$

Reorder $x$ and $- 1$ .

$\frac{x \cdot x - 1 \cdot (x A) - A x - A (- A)}{x - B}$

Remove parentheses.

$\frac{x \cdot x - 1 \cdot (x A) - A x - A \cdot (- 1 A)}{x - B}$

Move $A$ .

$\frac{x \cdot x - 1 \cdot (x A) - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Multiply $- 1$ by $x$ .

$\frac{x \cdot x - 1 x A - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 1 x A - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Raise $x$ to the power of $1$ .

$\frac{x \cdot x - 1 x A - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 1 x A - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Add $1$ and $1$ .

$\frac{x^{2} - 1 x A - A x - 1 \cdot (- 1 A \cdot A)}{x - B}$

Multiply $- 1$ by $- 1$ .

$\frac{x^{2} - 1 x A - A x + 1 A \cdot A}{x - B}$

Multiply $A$ by $1$ .

$\frac{x^{2} - 1 x A - A x + A \cdot A}{x - B}$

Raise $A$ to the power of $1$ .

$\frac{x^{2} - 1 x A - A x + A \cdot A}{x - B}$

Raise $A$ to the power of $1$ .

$\frac{x^{2} - 1 x A - A x + A \cdot A}{x - B}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2} - 1 x A - A x + A^{1 + 1}}{x - B}$

Add $1$ and $1$ .

$\frac{x^{2} - 1 x A - A x + A^{2}}{x - B}$

Reorder $- A x$ and $A^{2}$ .

$\frac{x^{2} - 1 x A + A^{2} - A x}{x - B}$

Move $- 1 x A$ .

$\frac{x^{2} + A^{2} - 1 x A - A x}{x - B}$

Reorder $x^{2}$ and $A^{2}$ .

$\frac{A^{2} + x^{2} - 1 x A - A x}{x - B}$

Reorder $x$ and $- B$ .

$\frac{A^{2} + x^{2} - 1 x A - A x}{- B + x}$

Turn the numerator into a polynomial.

Start expanding.

${(x - A)}^{2}$

Rewrite ${(x - A)}^{2}$ as $(x - A) (x - A)$ .

$(x - A) (x - A)$

Apply the distributive property.

$x (x - A) - A (x - A)$

Apply the distributive property.

$x \cdot x + x (- A) - A (x - A)$

Apply the distributive property.

$x \cdot x + x (- A) - A x - A (- A)$

Remove parentheses.

$x \cdot x + x \cdot - 1 A - A x - A (- A)$

Reorder $x$ and $- 1$ .

$x \cdot x - 1 \cdot x A - A x - A (- A)$

Remove parentheses.

$x \cdot x - 1 \cdot x A - A x - A \cdot - 1 A$

Move $A$ .

$x \cdot x - 1 x A - A x - 1 \cdot - 1 A \cdot A$

Raise $x$ to the power of $1$ .

$x^{1} x - 1 x A - A x - 1 \cdot - 1 A \cdot A$

Raise $x$ to the power of $1$ .

$x^{1} x^{1} - 1 x A - A x - 1 \cdot - 1 A \cdot A$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 1} - 1 x A - A x - 1 \cdot - 1 A \cdot A$

Add $1$ and $1$ .

$x^{2} - 1 x A - A x - 1 \cdot - 1 A \cdot A$

Multiply $- 1$ by $- 1$ .

$x^{2} - 1 x A - A x + 1 A \cdot A$

Multiply $A$ by $1$ .

$x^{2} - 1 x A - A x + A \cdot A$

Raise $A$ to the power of $1$ .

$x^{2} - 1 x A - A x + A^{1} A$

Raise $A$ to the power of $1$ .

$x^{2} - 1 x A - A x + A^{1} A^{1}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} - 1 x A - A x + A^{1 + 1}$

Add $1$ and $1$ .

$x^{2} - 1 x A - A x + A^{2}$

Reorder $- A x$ and $A^{2}$ .

$x^{2} - 1 x A + A^{2} - A x$

Move $- 1 x A$ .

$x^{2} + A^{2} - 1 x A - A x$

Reorder $x^{2}$ and $A^{2}$ .

$A^{2} + x^{2} - 1 x A - A x$

Subtract $A x$ from $- 1 x A$ .

Move $A$ .

$A^{2} + x^{2} - 1 x A - 1 x A$

Subtract $x A$ from $- 1 x A$ .

$A^{2} + x^{2} - 2 x A$

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$B$

$x^{2}$

$2 A x$

$A^{2}$

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			+	$x^{2}$	-	$x B$

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} - x B$

				$x$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$

Divide the highest order term in the dividend $- 2 A x$ by the highest order term in divisor $x$ .

				$x$	-	$2 A$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$

Multiply the new quotient term by the divisor.

				$x$	-	$2 A$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					-	$2 A x$	+	$2 A B$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 A x + 2 A B$

				$x$	-	$2 A$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					+	$2 A x$	-	$2 A B$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$	-	$2 A$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					+	$2 A x$	-	$2 A B$
							+	$x B$	-	$2 A B$

Divide the highest order term in the dividend $x B$ by the highest order term in divisor $x$ .

				$x$	-	$2 A$	+	$B$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					+	$2 A x$	-	$2 A B$
							+	$x B$	-	$2 A B$

Multiply the new quotient term by the divisor.

				$x$	-	$2 A$	+	$B$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					+	$2 A x$	-	$2 A B$
							+	$x B$	-	$2 A B$
							+	$B x$	-	$B^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $B x - B^{2}$

				$x$	-	$2 A$	+	$B$
$x$	-	$B$		$x^{2}$	-	$2 A x$	+	$A^{2}$
			-	$x^{2}$	+	$x B$
					-	$2 A x$	+	$x B$
					+	$2 A x$	-	$2 A B$
							+	$x B$	-	$2 A B$
							-	$B x$	+	$B^{2}$

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x$

$2 A$

$B$

$x$

$B$

$x^{2}$

$2 A x$

$A^{2}$

$x^{2}$

$x B$

$2 A x$

$x B$

$2 A x$

$2 A B$

$x B$

$2 A B$

$B x$

$B^{2}$

$2 A B$

$B^{2}$

The final answer is the quotient plus the remainder over the divisor.

$x - 2 A + B + \frac{- 2 A B + B^{2}}{x - B}$

The oblique asymptote is the polynomial portion of the long division result.

$y = x - 2 A + B$

Step 7

This is the set of all asymptotes.

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 2 A + B$

Step 8

Graph y=((x-A)^2)/(x-B)

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