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

1 Answer
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 = -6/0+4
    Since 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 = -6/x+4
    -4 = -6/x
    -4x = -6
    x = -6/-4 = 3/2
    x-intercept: (3/2,0)

  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 = -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 = -6/"HUGE" +4 -> 0 + 4 -> 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 = -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 = 6/6 + 4 = 1 + 4 = 5
    Point 2: (6,5)

enter image source here

Domain: (-oo,0),(0,oo)
Range: (-oo,4),(4,oo)