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Unformatted text preview: 2-1 Part II: Graphing Rational Functions
Six steps to pretty good graph of a rational function
f ( x) = 2
1. Factor numerator and denominator as much as possible.
f ( x) =
x ( x − 1)
2. Draw in vertical asymptotes (at zeros of denominator).
zeros of denominator are 0 and 1. 3. Draw in horizontal asymptote (if any).
degree of numerator < degree of denominator, so
x-axis is horizontal asymptote.
f(x) 10 vertical asymptotes horizontal asymptote x -3 2 2-1 Part II p. 1 4. Draw in zeros of function (zeros of numerator). x = 2
5. Add guide points between and beyond each zero and vertical
asymptote. f(-2) = -2/3
f(1/2) = 6 f(3/2) = -2/3 f(3) = 1/6 guide points x X X
-3 X X X
zero Note: there is one zero and two vertical asymptotes for a
total of three. You will need 3 + 1 = 4 guide points. You
always need one more guide point than the total of zeros
and vertical asymptotes. And that should be enough to
ensure a good-enough graph. 2-1 Part II p. 2 6. Draw. Make sure function approaches horizontal
asymptote at left and right extremities.
• start at extreme left
• graph must enter from close to the horizontal asymptote there
• or come from + or - oo, if no horizontal asymptote
• will the graph start from above the asymptote or below the
• look at the left-most guide point for the clue
• you must start on that side of the asymptote
• now draw the graph through the left-most "in-between" point
• graph must go off-scale at vertical asymptotes
• it must never cross the x-axis except at marked zeros This can all be done with almost no algebra; just draw
in some asymptotes and points and draw your graph!
f(x) -3 10 2 2-1 Part II p. 3 ...
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