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Lesson 19 Class Notes (E4, E5) MTH 103
Objective E4: Student can identify domain, x-intercepts, y-intercepts, vertical
asymptotes, and holes from a graphical or symbolic representation of a rational function.
Warmup/Review: Given ( ) 2 2 4 x f x x
and
g x x
, find the value of x^ where f ( )x g x( ). Solve the formula
B
A
B C
for B. Given
h x x
, rewrite (^) h x( ) as a ratio of two linear functions (or as a single fraction).
Looking back at some of the functions that you have previously studied, recall how to find the domain and x- and y-intercepts of a function. Domain: x-intercept(s): y-intercept(s): The domain of a rational function is all real numbers except those that make the denominator = 0. Example 1: For the function (^2)
x f x x x
, find the domain and x- and y-intercepts. In the graph of f ( )x , we see vertical asymptotes at x = 1 and x = 3. But how can we find the asymptotes from the equation? If ( )
qx p x f x is a reduced rational function and a is a zero of q(x), then the vertical asymptote(s) of the graph of f is (are): x = a In other words, to find vertical asymptotes, look for x-values where the denominator equals zero but the numerator does not.
x y x y
Objective E5: Student can find horizontal and slant asymptotes from a graphical or
symbolic representation of a rational function.
Example 3: Create a table and graph to determine if the function 2 2
x g x x
has a horizontal asymptote. x g x ( ) Example 4: (^2)
x f x x x
. Find the horizontal asymptote. (Consider the end behavior first!)
x y Example 5: Determine if the function 3 2 2 ( ) 2 3 x x k x x
has any vertical or horizontal asymptotes. In Examples 3 – 5 we determined if the functions had a horizontal asymptote. Now look at the degree of the numerator and denominator of each rational equation. What do you notice? Example 3: 2 2
x g x x
had a horizontal asymptote at y 4. Example 4: (^2)
x f x x x
had a horizontal asymptote at y 0. Example 5:
x x k x x
had no horizontal asymptote. Given the rational function 0 1 1 0 1 1 ...
b x b x b a x a x a f x m m m m n n n n
(where n = degree of numerator and m = degree of denominator), then the horizontal asymptote of the graph of f is y 0 if n m m n b a y if n m No H. A. if n m Note: If the degree of the numerator is one more than the degree of the denominator, then you end up with a “slant” asymptote. The equation for the slant asymptote is the “quotient” q x( ) when the numerator is divided by the denominator. (The remainder is insignificant. Why?)
x y Example 7: For 2 2
x f x x
find the domain, asymptotes, holes, intercepts and sketch the graph.
x y x y
Additional Practice Problems for Lesson 19 (E4, E5):
- Find the equation of the vertical asymptote(s), if any, for each of the following: (a) ( ) (^2) 9 x f x x
(b) (^2)
x g x x
(c) (^2)
x h x x
- Find the equation of the horizontal asymptote(s), if any, for each of the following: (a) (^2)
x f x x
(b) 2 2
x g x x
(c) 3 2
x h x x
- Find the equation of the slant asymptote(s), if any, for each of the following: (a)
x f x x
(b)
x g x x
(c)
x x h x x
- For the function (^2)
x f x x x
, find each of the following, if they exist. (a) Domain (using interval notation): (b) Equation of all asymptote(s): (c) Hole(s) as a coordinate point (x, y): (d) y-intercept(s): (e) x-intercept(s): (f) Sketch a graph.
- For the function 2 1 ( ) 2 x f x x
, find each of the following, if they exist. (a) Domain (using interval notation): (b) Equation of all asymptote(s): (c) Hole(s) as a coordinate point (x, y): (d) y-intercept(s): (e) x-intercept(s): (f) Sketch a graph.
