Analyzing and Graphing Rational Expressions
Vertical asymptotes are zeros of the denominator that have greater multiplicity on the bottom of the fraction.To find a vertical asymptote, set the denominator equal to zero and solve for x.
Draw a dotted vertical line on an x-y coordinate plane to represent a vertical asymptote.
Holes occur at zeros of both the numerator and denominator with multiplicity greater than or equal on top.
Zeros (or solutions) occur at zeros of the numerator only. Set the numerator equal to zero and solve for x. Graph solution points on the x-axis.
Horizontal Asymptotes occur at y = 0 (the x-axis) when the degree of the denominator is greater than the degree of the numerator (in a bottom heavy fraction).
Horizontal Asymptotes also occur when the degree of the denominator equals the degree of the numerator.
The horizontal asymptote is y = leading coefficient of numerator or L.C.O.N.
leading coefficient of denominator L.C.O.D.
Slant (oblique) or curved asymptotes occur when the degree of the numerator is bigger than the degree of the denominator (in a top heavy fraction).
Let n = degree of numerator and let d = degree of denominator
If n – d = 1, there is a linear (or slant or oblique) asymptote
If n – d = 2, there is a parabolic asymptote.
If n – d = 3, there is a cubic asymptote.
If n – d = 4, there is a quartic asymptote, etc.
Multiplicity is the degree of each factor.
Examples:
f (x) =
( x – 2 ) is a factor of multiplicity 1 on top and multiplicity 3 on the bottom.
x = 2 is a vertical asymptote.
( x + 3 ) is a factor of multiplicity 2 on top and multiplicity 1 on the bottom.
x = –3 is a hole.
( x – 4 ) is a factor of multiplicity 5 on top and multiplicity 5 on the bottom.
x = 4 is a hole.
( x + 5 ) is a factor of the numerator only. x = –5 is a solution or x-intercept.
( x – 6 ) is a factor of the denominator only. x = 6 is a vertical asymptote.
g (x) =
( x + 7 ) is a factor of the numerator only. x = –7 is a solution or x-intercept.
( x – 4 ) is a factor of multiplicity 2 on top and multiplicity 5 on the bottom.
x = 4 is a vertical asymptote.
( x – 2 ) is a factor of multiplicity 5 on top and multiplicity 4 on the bottom.
x = 2 is a hole.
( x + 5 ) is a factor of multiplicity 3 on top and multiplicity 3 on the bottom.
x = –5 is a hole.
( x – 1 ) is a factor of the denominator only. x = 1 is a vertical asymptote.
To graph a rational expression, plot all x-intercepts. Choose at least one easy x-value on each side of the vertical asymptotes, find the y-values, and plot those points. Rational expressions will curve near asymptotes and through the x-intercepts.
The end behavior of a rational expression will approximate the graph of the leading term of the numerator divided by the leading term of the denominator.
Examples:
f (x) = has approximately the same end behavior
as y = or y = x2.
g(x) = has approximately the same end behavior
as y = or y = 3x3.