#### Theory of Graphs of Rational Functions

Graphs of Rational Functions:

Suppose f(x) = p(x)/q(x); here p(x) and q(x) have no common factors.

When p(x) and q(x) have a common factor, and then divide out the additional factors and hence it is only left in numerator or denominator or not at all. Then look at the part on holes.

a) The y-intercept is a value of f(0). That is, replace 0 in for x in both numerator and denominator.

b) The x-intercepts are zeros of p(x).

c) The vertical asymptotes are zeros of q(x).

d) Horizontal asymptote is that value which f(x) approaches as x rises or reduces without bound.

e) Find out the behavior between each and every vertical asymptote and x-intercept based on Odd Changes, even remains the similar principle.

Whenever graphing a rational function, do the given:

a) Simplify the function by dividing any common factors. Be sure that when something is no longer in implied domain, that you define the restriction.

b) Recognize and graph the x-intercepts. Make a note (that is, possibly mental) of whether the graph crosses (that is, odd exponent) or touches (that is, even exponent) at each intercept.

c) Recognize and draw the vertical asymptotes. Make a note (that is, possibly mental) of whether the graph is asymptotic in similar direction (that is, even exponent) or various directions (that is, odd exponent).

d) Recognize and draw the horizontal or oblique asymptotes.

e) Begin on the far right of the graph near to the horizontal asymptote.

f) Move from right to the left.

• At each and every x-intercept encountered, cross or touch as noted in step 2.
• At each and every vertical asymptote, you should either go up or go down. The direction you go is finding out by which side of x-axis you’re on when you obtain to the vertical asymptote. Whenever picking back up on the left side of vertical asymptote, utilize the same or distinct side of x-axis based on your observations in step c.
• Watch out for the holes.

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