Unit 3 · Core unit
Two families defined by what they exclude. Radicals restrict the domain; rationals introduce asymptotes and holes, and both create extraneous solutions.
What a strong answer looks like
A strong answer factors before describing a rational graph, and checks every radical solution back in the original equation.
Know
Squaring both sides can create solutions the original equation never had, because squaring destroys sign information.
Apply
Solve as normal, then substitute every candidate back and reject those that fail.
Watch out
Reporting both algebraic roots without checking, when one is extraneous.
Study move
Solve a radical equation producing two roots, only one of which survives.
Know
A denominator factor that cancels leaves a hole; one that survives leaves a vertical asymptote. The two are indistinguishable before factoring.
Apply
Factor numerator and denominator, cancel, then classify each denominator zero.
Watch out
Declaring every denominator zero an asymptote.
Study move
Analyse a rational function whose numerator and denominator share one factor.
Know
Compare degrees: denominator larger gives a horizontal asymptote at zero, equal gives the ratio of leading coefficients, numerator larger by one gives a slant.
Apply
Compare degrees first, then divide only if a slant asymptote is indicated.
Watch out
Using the ratio of constant terms instead of leading coefficients.
Study move
Classify three rational functions by degree comparison alone.
Modeling drill
Fully describe a rational function with one hole, one vertical asymptote, and a horizontal asymptote, justifying each feature.
