# How do I graph the rational function: y=-6/x+4?

Mar 2, 2015

I like to identify the following things first, when asked to graph a rational function:
- y-intercept(s)
- x-intercept(s)
- vertical asymptote(s)
- horizontal asymptote(s)

1. To identify the y-intercept(s), ask yourself "what is the value of y when x=0"?
$y = - \frac{6}{0} + 4$
Since $\frac{6}{0}$ is undefined, there is no y-int
y-intercept: none

2. To identify the x-intercept(s), ask yourself "what is the value of x when y=0"?
$0 = - \frac{6}{x} + 4$
$- 4 = - \frac{6}{x}$
$- 4 x = - 6$
$x = - \frac{6}{-} 4 = \frac{3}{2}$
x-intercept: $\left(\frac{3}{2} , 0\right)$

3. To identify the vertical asymptotes, we first try and simplify the function as much as possible and then look at where it is undefined
$y = - \frac{6}{x} + 4$ is already simplified
Undefined when $x = 0$
Vertical asymptotes: $x = 0$

4. To identify the horizontal asymptotes, we think of the limiting behavior (ie: what happens as x gets HUGE)
$y = - \frac{6}{\text{HUGE}} + 4 \to 0 + 4 \to 4$
Horizontal asymptote: $y = 4$

Now you might pick a couple additional points to the left/right of your horizontal asymptote to get a sense of the graph shape.

• Pick a point to the left of the $x = 0$ asymptote, ie: $x = - 6$
$y = - \frac{6}{6} + 4 = - 1 + 4 = 3$
Point 1: (−6,3)
• Pick a point to the right of the $x = 0$ asymptote, ie: $x = 6$
$y = \frac{6}{6} + 4 = 1 + 4 = 5$
Point 2: $\left(6 , 5\right)$

Domain: $\left(- \infty , 0\right) , \left(0 , \infty\right)$
Range: $\left(- \infty , 4\right) , \left(4 , \infty\right)$