It is possible to sketch a rational function with no vertical asymptote(s)?
Yes, it is possible to sketch a rational function with no vertical asymptotes. A rational function is defined as the ratio of two polynomial functions. In order to determine if a rational function has a vertical asymptote, we need to examine the behavior of the function as it approaches certain values of the variable.
To sketch a rational function with no vertical asymptotes, follow these steps:
1. Start with a general rational function of the form f(x) = (p(x))/(q(x)), where p(x) and q(x) are polynomials.
2. Find the roots (zeros) of the denominator q(x). These are the values that make q(x) equal to zero.
3. Check if any of the roots of q(x) are also roots of the numerator p(x) (common factors). If so, there will be a hole in the graph at those corresponding points.
4. Determine the behavior of the rational function as x approaches positive and negative infinity. This will help us identify any horizontal asymptotes.
5. If the degree of the numerator is greater than the degree of the denominator, there will be a slant asymptote, which provides information about the end behavior of the function.
6. Plot the points corresponding to the x-intercepts (zeros) and holes (if any).
7. Based on the asymptotes, find points close to the vertical and horizontal asymptotes to help shape the graph.
8. Connect the points smoothly to form the graph of the rational function.
Remember that these steps are a general guideline for sketching rational functions. It is important to consider other factors, such as the characteristics of the numerator and denominator, to accurately sketch the function. Additionally, using graphing software or calculators can be helpful in visualizing the graph and confirming the absence of vertical asymptotes.