# How do you graph y=(x^2-9x+20)/(2x) using asymptotes, intercepts, end behavior?

May 13, 2018

See explanantion

#### Explanation:

Given: $y = \frac{{x}^{2} - 9 x + 20}{2 x}$

First and foremost, the equation becomes undefined when the denominator is 0

Thus $2 x \ne 0 \implies x \ne 0 \leftarrow \text{Asymptote}$

The true behaviour can be determined if we carry out the division.

$y = \frac{x}{2} - \frac{9}{2} + \frac{10}{x}$

color(blue)("Consider the case "lim_(x->0^("+"))

$y = {\lim}_{x \to {0}^{\text{+")) x/2-9/2+lim_(x->0^("+}}} \left(\frac{10}{x}\right)$

$y = 0 - \frac{9}{2} + \infty = \infty$

color(blue)("Consider the case "lim_(x->0^("-"))

$y = {\lim}_{x \to {0}^{\text{-")) x/2-9/2+lim_(x->0^("-}}} \left(\frac{10}{x}\right)$

$y = 0 - \frac{9}{2} - \infty = - \infty$
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$\textcolor{b l u e}{\text{Consider the case } {\lim}_{x \to + \infty}}$

$y = {\lim}_{x \to + \infty} \frac{x}{2} - \frac{9}{2} + {\lim}_{x \to + \infty} \frac{10}{x}$

$y = \infty - \frac{9}{2} + 0 = \infty$

$\textcolor{b r o w n}{\text{Observe the this is of general form }}$
$\textcolor{b r o w n}{y = \frac{1}{2} x - \frac{9}{2} \leftarrow \text{ Asymptote}}$

$\textcolor{b l u e}{\text{Consider the case } {\lim}_{x \to - \infty}}$

$y = {\lim}_{x \to - \infty} \frac{x}{2} - \frac{9}{2} + {\lim}_{x \to - \infty} \frac{10}{x}$

$y = - \infty - \frac{9}{2} + 0 = - \infty$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Are there any "x" intercepts?}}$

Set $y = 0 = \frac{1}{2} x + \frac{10}{x} - \frac{9}{2}$

$0 = \frac{{x}^{2} - 9 x + 20}{2 x}$

$0 = {x}^{2} - 9 x + 20$

$0 = {\left(x - \frac{9}{2}\right)}^{2} + 20 - \frac{81}{4}$

${\left(x - \frac{9}{2}\right)}^{2} = \frac{1}{4}$

$x = \frac{9}{2} \pm \frac{1}{2}$

$x = 4 \mathmr{and} 5$
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