Graphing Rational Functions with Slant Asymptotes
Welcome back to Professor Baker's Math Class! Today, we dived into the fascinating world of rational functions and learned how to graph them when they have slant asymptotes (also known as oblique asymptotes). These asymptotes appear when the degree of the numerator is exactly one more than the degree of the denominator.
What are Slant Asymptotes?
A slant asymptote is a diagonal line that the graph of a rational function approaches as $x$ tends to positive or negative infinity. Remember, a rational function has a slant asymptote if and only if the degree of the numerator, $N(x)$, is exactly one greater than the degree of the denominator, $D(x)$.
For example, consider the following functions:
- $f(x) = \frac{3x^2}{x}$
- $g(x) = \frac{5x^3 - 2}{x^2 + 1}$
Both of these functions have a slant asymptote because in each case, the degree of the numerator is one more than the degree of the denominator.
Finding the Equation of a Slant Asymptote
To find the equation of the slant asymptote, we use polynomial long division. Let's illustrate this with an example:
Consider the rational function: $$f(x) = \frac{x^2 - x - 2}{x - 1}$$
Performing polynomial long division:
This gives us: $$f(x) = x - \frac{2}{x-1}$$
As $x$ approaches infinity, the term $\frac{2}{x-1}$ approaches zero. Therefore, the slant asymptote is the line $y = x$.
Steps for Graphing Rational Functions with Slant Asymptotes
- Find the Slant Asymptote: Use polynomial long division to find the equation of the slant asymptote.
- Find Vertical Asymptotes: Set the denominator equal to zero and solve for $x$. These are your vertical asymptotes.
- Find Intercepts:
- x-intercepts: Set the numerator equal to zero and solve for $x$.
- y-intercepts: Set $x = 0$ and solve for $y$.
- Additional Points: Find additional points to help understand the graph's behavior.
- Sketch the Graph: Plot the asymptotes, intercepts, and additional points. Sketch the graph, making sure it approaches the asymptotes but does not cross them (except possibly the slant asymptote).
Example
Let's graph the function $f(x) = \frac{2x^2 + 1}{x}$.
- Slant Asymptote: Performing long division, we get $2x + \frac{1}{x}$. Thus, the slant asymptote is $y = 2x$.
- Vertical Asymptote: The denominator is $x$, so the vertical asymptote is $x = 0$.
- Intercepts: The numerator is $2x^2 + 1$. Setting this to zero, $2x^2 + 1 = 0 \implies 2x^2 = -1$, which has no real solutions. Therefore, there are no x-intercepts. When x=0, the function is undefined, thus there is no y-intercept.
By plotting the asymptotes and considering the function's behavior, we can sketch the graph.
Homework
Practice makes perfect! For homework, please complete problems #44-50 (even) on pages 157-158 of your textbook. These problems will give you valuable practice in identifying and graphing rational functions with slant asymptotes.
Keep up the great work, and see you in the next class!