Question

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

Answer 1

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)`

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