Unit 4 - Objective 3 - Graphing Rational Functions




Asymptotes: If the graph of y = f(x) gets closer and closer to the line y = L or x = a, then these lines are asymptotes.

Definition: If f(x) = L, then y = L is a Horizontal Asymptote.

y = 1 is a horizontal asymptote for each graph below

Note: To find the horizontal asymptotes for f(x), take the limit as of the given funtion. If this limit is a number, L, then y = L is a horizontal asymptote for the graph.


!!An abbreviated limit notation will be used in this review!!

AbbreviationMeaning
as "y approaches 4 as x approaches 1"
and means limit of the y values, as x goes to 1, is 4
"x increases without bounds" and we say "x goes to infinity"
"x decreases without bounds"
"x approaches 5 from the right"
x only has values greater than 5
"x approaches 5 from the left"
x less than 5




Definition: If as or as then x = a is a Vertical Asymptote. In other words, the y values increase or decrease without bound as the x values get closer to the line x = a.

x = 2 is a Vertical Asymptote for each of the graphs below.

Note: To find the vertical asymptote, reduce f(x) then if x = a makes the denominator zero (and not the numerator), x = a is a vertical asymptote for the graph. The graph will NEVER cross a vertical asymptote. The function is undefined there.



Given: find the vertical and horizontal asymptotes. Then, sketch the graph.

To find the horizontal asymptotes, take the limit as

To find the vertical asymptotes, factor demonimator (after reducing if possible)
denominator = 0 (numerator not 0)
when x = 3 or x = -3

Therefore: x = 3 and x = -3 are Vertical Asymptotes

To sketch by hand:

  1. Draw all of the asymptotes with dashed lines.
  2. Plot points in between the vertical asymptotes.
xy
-432/-7 -4.5
00
4-32/7 -4.5

Now you may plot more points or you can investigate the behavior of y = f(x) as the x values get close to the vertical asymptotes. If then the y values become larger and larger positive and the graph goes up. If then the y values become more and more negative and the graph goes down. The graph never crosses a vertical asymptote.

Using the points above, sketch information
about the horizontal asymptotes, as, , and as , (behavior at left and right ends)
Usually the outside branches of the graph approach thevertical asymptotes, or plot the y values for x close to the vertical asymptotes like x = 3.1, -3.1
In the middle of the graph, plot points inside the asymptotes, like x = 2.9, -2.9, or check to see if the graph crosses the horizontal asymptotes by setting y = -2 and see if you can solve for x.


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