Rational Functions

    This page serves 4 purposes:




    You might wish to review how to take notes on a computer.

To sketch a rational function:
· Rewrite it by factoring
identify the linear, quadratic, and reciprocal factors.
· Rewrite it by division
to identify curve shifting.
· Plot zeros
-- identified by the real roots of linear or quadratic factors.
· Mark vertical asymptotes
-- the roots of denominators of the reciprocal factors.
· Check for discontinuities or holes
-- identified by identical factors on the "top" and "bottom" of the fraction.
· Mark other asymptotes
-- the nonremainders part of the quotient from the long division.
· Determine sign in intervals
-- using the positiveness or negativeness of each factor.
· Find end behaviors
-- the "winner in the battle of the top against the bottom."
· Set the derivative equal to 0, solve to find x values where the slope is zero
-- to give the x values of relative maximums, relative minimums, flat spots, discontinuities. (Not shown in these examples.)
· Sketch curve.






Two Examples of the Sketching Rational Functions

    See the computer sketch and

See The individual graphics of the first problem
or or go step-by-step through the sketching.

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Try these.

    Ex. a. Sketch and label then check the answer.

    Ex. b. Sketch and label then check the answer.






Harder Problem, Still the Same Mathematics

Sketch the graph of .
Show work then check answer.





Question ratl1, ratl2, and ratl3

ratl1.   Accurately sketch the graph of .
 
ratl2. ratl2a. What is a rational function?
ratl2b. What's a vertical asymptote?
ratl2c. What's a horizontal asymptote?
ratl2d. What's an oblique asymptote?
ratl2e. Use words and a sketch to answer each question.
 
ratl3.   Create and state your own rational function with the following characteristics.
It has one discontinuity (hole).
It has a vertical asymptote at x=5.
It has an x-intercept at -3.
If it has any other asymptotoes, state their equations.

   Return to: Dilation   Polynomial Functions   Rational Functions   asymptote   function classes page.



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