Question 1157586: please help me with this problem....Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function.
R(x)=(3x)/(x+6)
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Answers:
Vertical asymptote: x = -6
Horizontal asymptote: y = 3
Oblique asymptote: none
===============================================================================================
Explanation:
Recall that we cannot divide by zero. If the denominator x+6 were zero, then
x+6 = 0
x+6-6 = 0-6
x = -6
Working through this backwards, we can say "if x = -6, then x+6 = 0". In other words, if x = -6, then the denominator is 0.
Therefore, the vertical asymptote is x = -6.
The horizontal asymptote is found by dividing the leading terms
The numerator's leading term is 3x (it's the only term here)
The denominator's leading term is x
Dividing 3x over x leads to 3x/x = 3
Basically as x gets really really large in either the negative or positive direction, the only thing that will matter are the leading terms which is why they dictate what happens with the horizontal asymptote. The y value will approach y = 3 but not actually get there.
Because there is a horizontal asymptote, there are no oblique asymptotes. Oblique asymptotes only occur of the degree of the numerator exceeds the degree of the denominator. In this case, both numerator and denominator have degree 1. Think of 3x as 3x^1. Same for x+6. Think of x+6 as x^1+6.
The graph is shown below
I used GeoGebra to make the graph
Other Properties:
- x intercept = 0
- y intercept = 0
- the graph is strictly increasing throughout its entire domain. This means that as we move from left to right, we are moving uphill along the red curve.
- this graph is known as a hyperbola
|
|
|
