# How do you graph y=(x+5)/(2x-4) using asymptotes, intercepts, end behavior?

Dec 10, 2016

Asymptotes: $\uparrow x = 2 \downarrow \mathmr{and} \leftarrow y = \frac{1}{2} \rightarrow$.y-intercept: $\frac{5}{2}$. x-intercept : 2. $\left(x , y\right) \to \left(\pm \infty , \frac{1}{2}\right) \mathmr{and} \left(2 , \pm \infty\right)$, in the opposite directions of the asymptotes $y = \frac{1}{2} \mathmr{and} x = 2$.

#### Explanation:

Cross multiplying and reorganizing,

(x-2)(y-1/2)=7/2# that represents a rectangular hyperbola (RH) having

asymptotes given by

$x = 2 \mathmr{and} y = \frac{1}{2}$.

The center of the RH is (2, 1/2).

y-intercept ( x = 0 ): 5/2

x-intercept ( y = 0 ):$- 5$

$\left(x , y\right) \to \left(\pm \infty , \frac{1}{2}\right) \mathmr{and} \left(2 , \pm \infty\right)$, in the opposite directions of the

asymptotes $y = \frac{1}{2} \mathmr{and} x = 2$, respectively.

graph{y(2x-4)-x-5=0 [-10, 10, -5, 5]}

Dec 10, 2016

See explanation.

#### Explanation:

A vertical asymptote occurs at $x$-values that make the denominator 0. To find the vertical asymptote(s) (V.A.'s), set your denominator equal to zero and solve for $x :$

V.A. when $2 x - 4 = 0$
$\iff 2 x = 4$
$\iff x = 2$

So the equation for the V.A. is $x = 2.$

A horizontal asymptote occurs when the degree of the numerator is less than (or equal to) the degree of the denominator. ("Degree" means the highest power of $x .$) Since both sides of the fraction have a degree of 1, there will be a horizontal asymptote.

When the degrees are the same (like in this case), the horizontal asymptote is found at $y =$the ratio of the leading coefficients. Here, that happens to be $y = \frac{1}{2}$ (from $\frac{\textcolor{red}{1} x + 5}{\textcolor{red}{2} x - 4}$).

(In the case that the denominator has a higher degree, the asymptote is always $y = 0.$)

The $x$-intercept is found by letting $y = 0$ and solving for $x :$

$0 = \frac{x + 5}{2 x - 4}$

$0 = x + 5$ [multiply both sides by (2x-4) ]

$x = \text{-5}$

So our $x$-intercept is at $\left(\text{-5} , 0\right) .$

Similarly, the $y$-intercept is found by letting $x = 0$ and solving for $y :$

$y = \frac{\left(0\right) + 5}{2 \left(0\right) - 4}$

$y = \frac{5}{\text{-4}} = - \frac{5}{4}$

So our $y$-intercept is at $\left(0 , - \frac{5}{4}\right) .$

With all this information, we can now draw our hyperbola:

graph{(y-(x+5)/(2x-4))(y-(x-2.0001)/(2x-4))=0 [-8.835, 11.165, -3.91, 6.09]}