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PC 2.6/2.7 Graphing and Solving Rational Functions

In these two sections we will explore rational functions.  Rational functions are not polynomials but actually the quotient of two polynomials.  In the formal definition we define a rational functions as R(x)=N(x)/D(x).  This understanding will be very helpful when identifying the vertical and horizontal asymptotes while graphing rational functions.  There are other pieces of information that will be helpful when graphing rational functions.  I will detail the step by step process here.

Graphing Rational Functions

  1. Determine and plot the Vertical Asymptotes: set your denominator to zero and solve for x.
  2. Determine and plot the Horizontal Asymptoes: compare the degree in the numerator to the degree in the denominator. If n>d the no HA exists.  If n<d then y=0 is the HA.  If n=d then the HA is y=a/b where a and b are the leading coefficients of the numerator and denominator respectively
  3. Determine the y and x-intercepts: set x to zero and solve for y to find the y-intercept, set y to zero and solve for x to find the x-intercept(s)
  4. Check for symmetry: evaluate the function for f(-x), if the graph returned is f(x) then the graph is even and it has symmetry about the y-axis, if the returned graph is -f(x) then the graph is odd and the graph has symmetry about the origin.
  5. Determine at least two points to the left and right of each vertical asymptote.  Plot and connect so that the graph approaches the asymptotes.
  6. Here are six videos in succession to assist you with graphing rational functions

    Solving Rational Equations


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