Rational Functions  

RATIONAL EXPRESSIONS Pre-Requisite Skills: (1) Polynomial Long Division   (2) Synthetic Division (3) The domain of a rational function is all real numbers EXCEPT: • zeros of the denominator, • negative even roots, • non-positive logarithms

X-Intercepts (4) In a rational function, x-intercepts occur at zeros of the numerator that are NOT zeros of the denominator.  

Multiplicity, Vertical Asymptotes, and Holes (5) Multiplicity is the degree of a factor of a polynomial. • Zeros with odd multiplicity cross the x-axis • Zeros with even multiplicity touch but do not cross the x-axis. (6) Vertical asymptotes occur at zeros of the denominator *or* zeros of both the numerator and denominator with multiplicity that is greater in the denominator (7) Holes occur at zeros of both the numerator and denominator with multiplicity that is equal or greater in the numerator

Horizontal Asymptotes and End Behavior Let n = the degree of the numerator Let d = the degree of the denominator • If n < d, the HA is y = 0 (the x-axis) • If n = d, the Horizontal Asymptote is: y = leading coefficient of numerator . leading coefficient of denominator y = LCON LCOD

Slant or Curved Asymptotes • If n > d, do polynomial long division, numerator ÷ denominator, to find the slant or curved asymptote • If n – d = 1, there is a slant (or oblique) asymptote. • If n – d = 2, there is a parabolic asymptote. • If n – d = 3, there is a cubic asymptote. • If n – d = 4, there is a quartic asymptote…

Fundamental Theorem of Algebra Every polynomial of degree n has n roots in the complex number system. An nth degree polynomial may have up to n – 1 turning points (extrema, maxima & minima) n – 2 points of inflection

Rational Functions Assignment Rational Functions (Part 2) Do Page 177 #23-37 odd