Mathematical Investigations III
Name
Mathematical Investigations III
The World Upside Down
Reciprocals of Polynomials
1.
a.
Using the standard window on your calculator, sketch the graphs of the following functions.
10
10
f (x) 
g(x)

(x  3)(x  1)2
(x  3)2 (x  1)
b.
What does the calculator show that isn't really part of the graphs of the functions?
c.
What do the graphs really do that the calculator does not show adequately?
Correct your sketches if necessary to show the actual functions clearly.
d. State the domain of ƒ and g.
e. Write the equations of all of the asymptotes
for both ƒ and g.
f.
How do the even and odd exponents of the factors in the denominators affect the graphs?
g.
Is this consistent with your work on Sheet 1 that considered
h.
How do you think the graph of h(x) 
1
xn
? Explain.
10
would differ from the graph of ƒ?
(x  3)(x  1)4
Rats 3.1
Rev. S06
Mathematical Investigations III
Name

10
.
(x  3)(x  1)(x  4)
2.
Let f x 
a.
State the equations of all of the asymptotes.
b.
Copy the view of ƒ from your calculator on each of the following windows.
i.
c.
standard window
ii. change x to
change y to
TI-83/4: [-4.7, 4.7]
[-3.1, 3.1] (ZDecimal)
TI-85/6: [-6.3, 6.3]
[-3.1, 3.1]
TI-89: [-7.9, 7.9]
[-3.8, 3.8]
(with xres = 1)
Explain the difference by considering how the calculator is working.
Note: Keep each of these windows in mind. When graphing any function, but particularly when
graphing rational functions, you may find it quite helpful to compare different windows to help
interpret what both the calculator and the function are actually doing.
Rats 3.2
Rev. S06