2.6 – Rational Functions

Domain & Range of Rational Functions Domain: x values of graph, ↔ – All real number EXCEPT Vertical Asymptote : (What makes denominator = 0) X-values of “holes” in graph Range: y values of graph, ↕ – All real numbers EXCEPT Maybe Horizontal Asymptote: (Use degrees to find) Y-values of “holes” in graph

Graphs of Rational Functions: Asymptotes Vertical Asymptotes: (VA) (X=#) – Vertical lines graph NEVER touches – Set denominator = 0 and solve – EXCEPTIONS to the domain Horizontal Asymptotes: (HA) (Y=#) – Horizontal lines graph doesn’t touch in AT LEAST 1 BRANCH (may be exception to range) – Use degrees to find Degree in numerator smaller: y=0 (x-axis) Degree in numerator larger: NONE Degrees same : Divide Leading Coefficients y= N(x)/D(x)

Graphs of Rational Functions: Intercepts & Holes X-intercept: (#, 0) – Set numerator (top) = 0 and solve Y-intercept: (0,#) – Put zero in for all x’s and evaluate Holes (x,y) – Factor BOTH numerator and denominator – Shared factors are each set = to zero and solved. The solved values are the x-values of the holes. – Y-values of holes: put x-value of hole into original & evaluate – Cancel out the shared factors – Remaining factors in denominator can be set = 0 and solved to find vertical asymptotes (VA)

Examples:

More examples: Graph

More Examples: Graph