Overview
This lecture explains how to graph rational functions, find vertical, horizontal, and slant asymptotes, identify holes, and determine domain and range.
Parent Rational Functions
- The parent function y = 1/x has a vertical asymptote at x = 0 (y-axis) and a horizontal asymptote at y = 0 (x-axis).
- The graph of y = 1/x lies in the upper right and lower left quadrants (origin symmetry).
- The domain of y = 1/x is all real x except x ≠0; the range is all real y except y ≠0.
- The function y = 1/x² has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.
- Its graph is above the x-axis with symmetry about the y-axis; domain is x ≠0, range is y > 0.
Transformations and Shifts
- y = 1/(x - h) shifts the graph h units right; vertical asymptote is at x = h.
- y = 1/(x + h) - k shifts left by h and down by k; vertical asymptote at x = -h, horizontal at y = -k.
- Plug in points around asymptotes for accurate graphing.
Reflection and Symmetry
- Negative sign in front of the function reflects the graph over the x-axis or horizontal asymptote.
Asymptotes of Rational Functions
- Vertical asymptotes: set denominator to 0 and solve for x.
- Horizontal asymptotes:
- If degree numerator < degree denominator: y = 0 (may be shifted by constants).
- Degrees equal: y = ratio of leading coefficients (shifted up/down if constants outside).
- Degree numerator > denominator by 1: slant (oblique) asymptote found by long division.
- No horizontal or slant asymptote if numerator degree > denominator by more than one.
Holes in Graphs
- When a factor cancels from numerator and denominator, a hole exists at that x-value.
- Find y-value of hole by substituting x into the simplified expression.
Domain and Range
- Domain: all real x except x-values of vertical asymptotes and holes.
- Range: all real y except y-values of horizontal asymptotes and holes.
Example Summary
- For y = (3x² + 9x - 12)/(x² + x - 2) - 4: factor, find and cancel factors, locate asymptotes and holes, determine domain/range.
- For y = (2x² + 6x - 8)/(x - 2): numerator degree exceeds denominator by 1, so use long division to find a slant asymptote.
Key Terms & Definitions
- Rational Function — A function of the form f(x) = P(x)/Q(x), where P and Q are polynomials.
- Asymptote — A line that the graph approaches but never touches.
- Vertical Asymptote — Set denominator to 0 and solve for x.
- Horizontal Asymptote — Determined by comparing degrees of numerator and denominator; ratio of leading coefficients if equal degree.
- Slant (Oblique) Asymptote — Occurs when numerator degree is one higher than denominator; found with polynomial division.
- Hole — Removable discontinuity when a factor cancels in numerator and denominator.
Action Items / Next Steps
- Practice graphing various rational functions, identifying their asymptotes, holes, domain, and range.
- Use an online graphing calculator to visualize complex graphs and confirm domain/range findings.