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.)
- · Find end behaviors
- -- the "winner in the battle of the top against the bottom."
- · Sketch curve.
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