Plotting fraction functions, also known as rational functions, can be a challenging aspect of IB Math SL. This guide provides a comprehensive walkthrough, equipping you with the skills and understanding to confidently tackle these types of questions. We'll cover key concepts, step-by-step plotting techniques, and address common pitfalls. Understanding these functions is crucial for success in your IB exams.
Understanding Fraction Functions (Rational Functions)
Fraction functions, in the context of IB Math SL, are functions of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions. Understanding their behavior requires analyzing several key features:
1. Vertical Asymptotes:
These occur where the denominator, q(x), equals zero and the numerator, p(x), is non-zero. The function approaches positive or negative infinity as x approaches the value that makes the denominator zero. To find these, set q(x) = 0 and solve for x.
2. Horizontal Asymptotes:
These describe the behavior of the function as x approaches positive or negative infinity. The rules for determining horizontal asymptotes depend on the degrees of p(x) and q(x):
- Degree of p(x) < Degree of q(x): The horizontal asymptote is y = 0.
- Degree of p(x) = Degree of q(x): The horizontal asymptote is y = (leading coefficient of p(x)) / (leading coefficient of q(x)).
- Degree of p(x) > Degree of q(x): There is no horizontal asymptote; the function may have an oblique (slant) asymptote or tend to infinity.
3. x-intercepts (Roots):
These are the points where the graph intersects the x-axis, i.e., where f(x) = 0. This occurs when the numerator, p(x), equals zero and the denominator, q(x), is non-zero. Solve p(x) = 0 to find the x-intercepts.
4. y-intercept:
This is the point where the graph intersects the y-axis, found by evaluating f(0). Substitute x = 0 into the function.
5. Oblique Asymptotes (Slant Asymptotes):
These occur when the degree of the numerator is exactly one greater than the degree of the denominator. To find the equation of an oblique asymptote, perform polynomial long division of p(x) by q(x). The quotient is the equation of the oblique asymptote.
Step-by-Step Plotting Guide
Let's illustrate the plotting process with an example: f(x) = (x + 1) / (x - 2)
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Vertical Asymptote: Set the denominator to zero:
x - 2 = 0, sox = 2. There's a vertical asymptote at x = 2. -
Horizontal Asymptote: The degrees of the numerator and denominator are equal (both 1), so the horizontal asymptote is y = (1/1) = 1.
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x-intercept: Set the numerator to zero:
x + 1 = 0, sox = -1. The x-intercept is (-1, 0). -
y-intercept: Substitute x = 0 into the function:
f(0) = (0 + 1) / (0 - 2) = -1/2. The y-intercept is (0, -1/2). -
Plotting: Draw the asymptotes (x = 2 and y = 1). Plot the intercepts (-1, 0) and (0, -1/2). Sketch the curve, ensuring it approaches the asymptotes. Consider testing additional points to refine the sketch, especially around the asymptotes to determine whether the curve approaches the asymptote from above or below.
Common Mistakes and Tips for Success
- Forgetting to check for holes: If both the numerator and denominator share a common factor, there will be a "hole" in the graph at the x-value that makes the common factor zero. Simplify the function before plotting to identify these holes.
- Incorrectly determining asymptotes: Pay close attention to the degrees of the numerator and denominator when finding horizontal asymptotes.
- Not considering the behavior near asymptotes: Always test points close to the vertical asymptotes to accurately depict the curve's behavior. Use test values slightly greater and slightly less than the x-value of the vertical asymptote.
- Using technology wisely: Graphing calculators or software can help verify your work, but it's crucial to understand the underlying concepts and be able to do the analysis manually.
By following these steps and understanding these concepts, you'll significantly improve your ability to plot fraction functions in IB Math SL and confidently answer related exam questions. Remember, practice is key! Work through a variety of examples to reinforce your understanding.
