4.3. THE FULL REPORT FOR CURVE SKETCHING 169
Problem 4.61 Suppose that a function is the ratio of two polynomials in which the numerator
has degree n and the denominator has degree m. What is the range of the possible numbers of
vertical asymptotes of this function?
a.x/
9
-1
-5 5
Problem 4.62 Based on the sketch above, do your best to fill out a full report.
Problem 4.63 Find a function that has two horizontal asymptotes and explain why your solu-
tion works.
Problem 4.64 Find a function that has two diagonal asymptotes and explain why your solution
is correct. Hint: start with a function that has two horizontal asymptotes.
Problem 4.65 What is the relationship between the graph of
f .x/ D
x
4
x
2
C 1
and the graph of y D x
2
1?
170 4. CURVE SKETCHING
Problem 4.66 Suppose that
f .x/ D
.x a/.x b/.x c/
x
4
C d
where all four of a; b; c, and d are positive constants. What items that go into a report can you
deduce from this information?
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