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.

    Use your browser to return to this page.

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.

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