+2441 This question requires one to understand certain features of a rational function in order to determine the horizontal asymptote.
#1) The degree of the numerator is 6, and the degree of the denominator is 5. Since the degree of the numerator exceeds the degree of the denominator, no horizontal asymptote exists for the first function.
#2) There is a general process to graphing rational functions.
1) Factor the numerator and denominator completely, if possible
In this case, no factoring can be done to either the numerator or the denominator. If it were possible, the process would expose any hidden common factors.
2) Identify any Holes
We can essentially skip this step; holes are generated when a common factor between the numerator and denominator exists. We would have identified the common factor in the previous step.
3) Identify any Zeros
Setting the numerator equal to zero allows one to identify the zeros. The numerator of this rational function is not complex by any means, so it is relatively easy to find the zero.
| \(-3x+5=0\) | |
| \(-3x=-5\) | |
| \(x=\frac{-5}{-3}=\frac{5}{3}\) | |
Since we are solving for a zero, the y-coordinate equals zero; thus, there is a zero at \(\left(\frac{5}{3},0\right)\).
4) Identify any Asymptotes
Of course, there are three types of asymptotes (vertical, horizontal, and oblique), so we need to be sure to take all of them into account, if they exist.
Setting the denominator equal to zero reveals the vertical asymptote.
| \(-5x+2=0\) | |
| \(-5x=-2\) | |
| \(x=\frac{-2}{-5}=\frac{2}{5}\) | |
| There is a vertical asymptote at x=2/5 |
The horizontal asymptotes can be determined by the degree of both the numerator and denominator. In this case, the degree of the numerator and denominator are equal, so you would divide the leading coefficients of the numerator and denominator.
The horizontal asymptote exists at \(y=\frac{-3}{-5}=\frac{3}{5}\)
For rational functions, it is impossible that an oblique asymptote exists if a horizontal asymptote does, so there is no oblique asymptote.
5) Plot any Information Determined Previously
We know where a zero exists already (at \(\left(\frac{3}{5},0\right)\)), so we might as well plot it.
Plotting asymptotes are also important; they tell where functions approach, so the function does not cross asymptotes. Be careful, though! For rational functions, a function will never pass a vertical asymptote, but it can pass a horizontal or oblique asymptote. In this case, though, the function will not pass through any asymptotes.
6) Create a Table of Values
If, after this process, you are still unsure about how a graph behaves, creating a table of values might be your best solution. Be strategic about it, though! Plot points on all sides of vertical asymptotes to better understand the behavior.
