Rational Functions in Math AA HL
Definition and Forms
Rational functions are algebraic expressions that can be written as the ratio of two polynomials. In the AHL (Additional Higher Level) 2.13 topic, we focus on two specific forms of rational functions:
- $f(x) = \frac{ax+b}{cx^2+dx+e}$
- $f(x) = \frac{ax^2+bx+c}{dx+e}$
Where $a$, $b$, $c$, $d$, and $e$ are constants, and $x$ is the variable.
These forms are more complex than the simpler rational functions covered in SL 2.8, which typically involve linear expressions in both numerator and denominator.
Key Features of Rational Functions
Asymptotes
Rational functions often have asymptotes, which are lines that the graph of the function approaches but never quite reaches. There are three types of asymptotes:
- Vertical Asymptotes: These occur where the denominator equals zero, and the numerator is not zero.
- Horizontal Asymptotes: These are determined by comparing the degrees of the numerator and denominator polynomials.
- Oblique (Slant) Asymptotes: These occur when the degree of the numerator is exactly one less than the degree of the denominator.
For the function $f(x) = \frac{2x+1}{x^2-1}$:
- Vertical asymptotes occur at $x = 1$ and $x = -1$ (where $x^2-1 = 0$)
- The horizontal asymptote is $y = 0$ (as the degree of the numerator is less than the denominator)
Intercepts
Intercepts are points where the function crosses the x-axis (x-intercepts) or y-axis (y-intercept).
- X-intercepts: Found by setting $f(x) = 0$ and solving for $x$.
- Y-intercept: Found by evaluating $f(0)$.
For $f(x) = \frac{x^2-4}{x-2}$:
- X-intercepts: $x = -2$ and $x = 2$
- Y-intercept: $f(0) = \frac{0^2-4}{0-2} = 2$
Graphing Rational Functions
To graph a rational function:
- Identify the domain (all real numbers except where the denominator equals zero).
- Find all asymptotes (remember that sign changes between roots when teh factor is raised to an odd power and asymptotes caused by factors to an odd power, i.e. $(x-1) (x+5)^3, (x+9)^7$).
- Determine intercepts.
- Analyze end behavior.
- Plot additional points to sketch the curve.
Always check if the function can be simplified before graphing. Cancelling common factors can sometimes remove vertical asymptotes.
Using Technology
Dynamic graphing packages like Desmos, GeoGebra, or graphing calculators are invaluable tools for exploring rational functions.
While technology is helpful, it's crucial to understand the underlying principles to interpret and verify the graphical representations.
Benefits of using graphing software:
- Quickly visualize the function's behavior
- Zoom in to examine specific features
- Easily adjust parameters to see how they affect the graph
- Verify hand-calculated asymptotes and intercepts
Using a graphing calculator, input $f(x) = \frac{x^2+1}{x-1}$ and observe:
- The vertical asymptote at $x = 1$
- The oblique asymptote $y = x + 1$ as $x$ approaches infinity
- The y-intercept at (0, -1)
Detailed Analysis of $f(x) = \frac{ax+b}{cx^2+dx+e}$
This form represents a rational function where the numerator is linear and the denominator is quadratic.
Key characteristics:
- Vertical asymptotes: Occur at the roots of $cx^2+dx+e = 0$.
- Horizontal asymptote: Always $y = 0$ as the degree of the numerator is less than the denominator.
- X-intercepts: Solve $ax+b = 0$.
- Y-intercept: Evaluate $f(0) = \frac{b}{e}$.
Students often forget that a vertical asymptote might not exist if a root of the denominator is also a root of the numerator. Always check for common factors!
Detailed Analysis of $f(x) = \frac{ax^2+bx+c}{dx+e}$
This form has a quadratic numerator and a linear denominator.
Key characteristics:
- Vertical asymptote: Occurs at $x = -\frac{e}{d}$.
- Oblique asymptote: As $x$ approaches infinity, the function approaches the line $y = ax + \frac{b}{d}$. Find oblique asymptote by doing polynomial division.
- X-intercepts: Solve $ax^2+bx+c = 0$.
- Y-intercept: Evaluate $f(0) = \frac{c}{e}$.
For $f(x) = \frac{x^2-4x+3}{x-2}$:
- Vertical asymptote: $x = 2$
- Oblique asymptote: $y = x - 2$
- X-intercepts: $x = 1$ and $x = 3$
- Y-intercept: $f(0) = \frac{3}{-2} = -\frac{3}{2}$
Connection to SL 2.8
The rational functions in AHL 2.13 build upon the simpler forms introduced in SL 2.8. The key differences are:
- More complex polynomial expressions in numerator and denominator.
- Introduction of oblique asymptotes.
- More challenging algebraic manipulations required for analysis.
Understanding the simpler forms from SL 2.8 is crucial for mastering the more advanced concepts in AHL 2.13.
