Rational Functions Reference

Algebra 2 · Axiom Academy

FORMULA SHEET Rational Functions Reference Domains, asymptotes, holes, intercepts, and the solving strategies for rational functions and their real-world applications. Holes (Removable Discontinuities) Factor the numerator and denominator completely. Identify and cancel common factors — these create holes. Find the domain: solve Q(x)=0 for the excluded values. Find vertical asymptotes: zeros of the denominator after canceling. Find the horizontal or oblique asymptote using the degree rules. Find x -intercepts: solve P(x)=0 after canceling. Find the y -intercept: evaluate f(0) if x=0 is in the domain. Plot additional points and analyze behavior near the asymptotes. Sketch the graph, approaching each asymptote appropriately. Find the least common denominator (LCD) of every term. Multiply both sides of the equation by the LCD. Simplify and solve the resulting polynomial equation. Check every solution for extraneous roots. Solving Equations: Cross-Multiplication Move every term to one side so 0 is alone on the other. Find the critical values: zeros of the numerator and the denominator. Plot the critical values on a number line. Test a point in each resulting interval. Determine which intervals satisfy the inequality. Distance-Rate-Time Applications Forgetting to check for extraneous solutions. Confusing holes with vertical asymptotes. Not factoring before finding asymptotes. Including a vertical asymptote's x -value in the domain.

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