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

Question

Savannah Anderson · Accepted Answer

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)

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
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)#
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#
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)#

Julia Adams · Answer

undefined To graph the rational function y = -6/(x + 4), follow these steps:

1. Determine any vertical asymptotes by setting the denominator (x + 4) equal to zero and solving for x. In this case, x + 4 = 0 gives x = -4. So, there is a vertical asymptote at x = -4.

2. Determine any horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (-6) is less than the degree of the denominator (1), there is a horizontal asymptote at y = 0.

3. Find the x-intercept by setting y = 0 and solving for x. In this case, -6/(x + 4) = 0 gives x = -4. So, there is an x-intercept at (-4, 0).

4. Plot the vertical asymptote, horizontal asymptote, and x-intercept on the coordinate plane.

5. Choose additional x-values to evaluate the function and find corresponding y-values. For example, you can choose x = -5, -3, -2, 0, 1, 2, etc.

6. Plot the points obtained from step 5 on the graph.

7. Connect the points smoothly to form the graph of the rational function.

Remember to label the axes and provide a title for the graph.