Reciprocal of Polynomials

Rational functions are in the form of \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are functions and \(Q(x)\neq 0\)

Key Features

Key features of a rational function can be used to analyze it:

• asymptote
  • vertical
    • comes from the denominator (zeroes) of a rational function
  • horizontal
    • comes from the division of the numerator function over the denominator when the numerator and denominator
  • oblique
    • comes from the quotient of the numerator divided by denominator
• intercepts
  • x
  • y
• slope
  • positive or negative
  • increasing or decreasing
• domain
• range
  • the vertex of a function can be found by finding the tangent point when the slope of the tangent is equal to zero
• positive and negative intervals

  Properties of Reciprocals

• Reciprocal of 0 is undefined and turns into an asymptote
• reciprocals of small numbers are large and reciprocals of large numbers are small
• signs stay the same when reciprocated
• When a number is close to \(\pm 1\) the reciprocal is also close to \(\pm 1\)