Mathematical Investigations III
Name
Mathematical Investigations III
The World Upside Down
Oblique Asymptotes
1.
Write the equation of the horizontal asymptote of each function.
f x 
gx  
x
x2  4
2
2x  1
x 2  3x  1
For each function, how does the degree of the numerator compare to the degree of the
denominator?
for f:
for g:
2.
Let f x 
a.
x2  1
.
x
2
Compare the degrees of the
numerator and denominator.
b.
x 1
Find lim
, i.e., what happens to
x x
f x  as x gets very, VERY large
c.
What does this imply about a horizontal asymptote?
d.
Sketch the graph.
e.
Zoom out. Describe the graph.
Rats 6.1
Rev. S03
Mathematical Investigations III
Name
f.
Use the trace button. As x increases, what happens to the difference between x and y?
g.
x2  1
1
 x  . Now explain why your answer to part f makes sense.
Rewrite f x 
x
x
The line y  x in this example is called an oblique asymptote of the function; that is, it is an
asymptote that is neither horizontal nor vertical.
5.
6.
Let gx 
x2  1
.
x2
a.
Sketch the graph on your calculator. Zoom out, and trace. Using the values for x and
y as x increases, guess an equation for the line which is the oblique asymptote for g.
b.
Use synthetic division to find the quotient and remainder when x 2 1 is divided by
x  2.
Q(x) =
R(x) =
c.
Write the equation of the oblique asymptote.
Find the equation of the oblique asymptote of each function.
f x 
2x 2  6x  3
x 1
hx 
x  6x  5x  3
3
2
x 2  2x  1
Rats 6.2

gx  
3x 2  4x  2
x 1
jx  
4x  6x  3
3
2
x 2  2x  3
Rev. S03