Rational Functions
Lessons in this chapter
About Rational Functions
Rational Functions builds a complete picture of a rational function’s graph, from its asymptotes through to modeling.
Rational Functions: Vertical Asymptotes and Holes identifies where a rational function’s graph breaks, distinguishing a true vertical asymptote from a removable hole.
End Behavior of Rational Functions covers what a rational function’s graph does far from the origin, including horizontal and slant asymptotes.
Zeros, Sign Analysis and Rational Inequalities and Equivalent Representations, Graphing, and Modeling close the chapter by finding a rational function’s zeros and sign pattern, then using every representation together to model a real situation.
The four lessons, from Rational Functions: Vertical Asymptotes and Holes to Equivalent Representations, Graphing, and Modeling, each open as a worksheet worked in the page. Problems are scored as you answer, so a hole mistaken for an asymptote is flagged at that question. Free to practice online.
Frequently asked questions
Rational Functions: Vertical Asymptotes and Holes separates the two things a zero denominator can produce, depending on whether the factor cancels. End Behavior of Rational Functions compares the degrees of numerator and denominator to find a horizontal or slant asymptote, Zeros, Sign Analysis and Rational Inequalities uses a sign chart to say where the function is positive or negative, and Equivalent Representations, Graphing, and Modeling puts the whole picture together. This sits in Algebra (A) and Functions (F) in the high school Common Core, and in Unit 1 of the AP Precalculus course framework, Polynomial and Rational Functions.
A vertical asymptote occurs where the denominator equals zero and the numerator doesn't, while a hole occurs where a factor cancels between numerator and denominator.
Compare the numerator and denominator's degrees: a smaller numerator degree gives a horizontal asymptote at y = 0, equal degrees give a horizontal asymptote at the ratio of leading coefficients, and a larger numerator degree gives a slant or curved asymptote.
A rational function's zeros come from the numerator's zeros (after canceling common factors with the denominator), and its sign changes only at zeros and vertical asymptotes.
A rational inequality is solved by finding the zeros and undefined points first, then testing the sign of the expression in each interval those points create.
Every rational function can be described the same way through its equation, its table of values, and its graph, using whichever representation best fits the question being asked.