Graph y=((x-2)^2)/(x-1)

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

Find where the expression $\frac{{(x - 2)}^{2}}{x - 1}$ is undefined.

$x = 1$

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).

Find $n$ and $m$ .

$n = 2$

$m = 1$

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

No Horizontal Asymptotes

Find the oblique asymptote using polynomial division.

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 4 x + 2^{2}}{x - 1}$

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

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

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

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

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

$\frac{(x - 2) (x - 2)}{x - 1}$

Apply the distributive property.

$\frac{x (x - 2) - 2 (x - 2)}{x - 1}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 - 2 (x - 2)}{x - 1}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2}{x - 1}$

Reorder $x$ and $- 2$ .

$\frac{x \cdot x - 2 x - 2 x - 2 \cdot - 2}{x - 1}$

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

$\frac{x \cdot x - 2 x - 2 x - 2 \cdot - 2}{x - 1}$

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

$\frac{x \cdot x - 2 x - 2 x - 2 \cdot - 2}{x - 1}$

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

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

Add $1$ and $1$ .

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

Multiply $- 2$ by $- 2$ .

$\frac{x^{2} - 2 x - 2 x + 4}{x - 1}$

Subtract $2 x$ from $- 2 x$ .

$\frac{x^{2} - 4 x + 4}{x - 1}$

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

$x$

$1$

$x^{2}$

$4 x$

$4$

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

					$x$
	$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$

Multiply the new quotient term by the divisor.

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			+	$x^{2}$	-	$x$

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

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$

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

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$

Pull the next terms from the original dividend down into the current dividend.

				$x$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$4$

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$4$

Multiply the new quotient term by the divisor.

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$4$
					-	$3 x$	+	$3$

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 3$

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$4$
					+	$3 x$	-	$3$

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

				$x$	-	$3$
$x$	-	$1$		$x^{2}$	-	$4 x$	+	$4$
			-	$x^{2}$	+	$x$
					-	$3 x$	+	$4$
					+	$3 x$	-	$3$
							+	$1$

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

$x - 3 + \frac{1}{x - 1}$

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

$y = x - 3$

This is the set of all asymptotes.

Vertical Asymptotes: $x = 1$

No Horizontal Asymptotes

Oblique Asymptotes: $y = x - 3$

Graph y=((x-2)^2)/(x-1)

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