Math 110
In Class Exercises, Section 3.3
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Rational Functions: Domain
Goal: Identify, and find the domain, of rational functions
1) Find the domain of the following function:
x2  4
f x  
x2
2) Find the domain of the following function:
x2  4
f x  
x2
3) Find the domain of the following function:
x 2  4x  4
f x  
x2  4
Rational Functions: Transformations of a rational function
Goal: Be able to transform a rational function
4) Graph the following function. Use one color for the graph of the equation, and another color for
1
1
the graph’s asymptotes. f x  
x  32
5) Graph the following function. Use one color for the graph of the equation, and another color for
1
1
the graph’s asymptotes. f x  
x  22
Math 110
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Math 110
In Class Exercises, Section 3.3
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Rational Functions: Finding Vertical Asymptotes
Goal: Be able to find all vertical asymptotes of a rational function
6) Given the following function, find all vertical asymptotes. Remember that you need to put the
function into lowest terms, if it isn't already.
1
f x  
1
x  3x  2
7) Given the following function, find all vertical asymptotes. Remember that you need to put the
function into lowest terms, if it isn't already.
x4
f x   3
x  10 x 2  24 x
8) Given the following function, find all vertical asymptotes. Remember that you need to put the
function into lowest terms, if it isn't already.
3x  5
f x  
x6
9) Given the following function, find all vertical asymptotes. Remember that you need to put the
function into lowest terms, if it isn't already.
x3  8
f x   2
x  5x  6
10) Given the following function, find all vertical asymptotes. Remember that you need to put the
function into lowest terms, if it isn't already.
3x
f x  
x4
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Math 110
In Class Exercises, Section 3.3
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Rational Functions: Finding Asymptotes
Goal: Be able to find all vertical, horizontal, and oblique asymptotes of a rational function
11) Given the following function, find all asymptotes. Identify each as vertical, horizontal, or
oblique:
x 2  4x  8
f x  
x2  3
12) Given the following function, find all asymptotes. Identify each as vertical, horizontal, or
oblique:
3x 2  7 x  3
f x  
x  32
13) Given the following function, find all asymptotes. Identify each as vertical, horizontal, or
oblique:
x 3  2 x 2  10 x  7
f x  
x2  3
14) Given the following function, find all asymptotes. Identify each as vertical, horizontal, or
oblique:
x 3  2 x 2  10 x  7
f x  
x2  9
Math 110
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Math 110
In Class Exercises, Section 3.3
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X
Review: Average Rate Of Change
Goal: Identify the rate of change of a function, given it's picture ; intuitively understand ARoC
16) What is the average rate of
10
change between x = -8 and x = 0?
9
8
7
6
5
4
3
2
1
0
-1
-1 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
-2
0
-3
-4
-5
-6
-7
-8
-9
-10
2 3 4 5 6 7 8 9 10
17) Draw the secant line between
these two points on the graph to the
left
Y
15) Given the function
1
1
f x   x 2  x
16
2
graph the function using your calculator, and draw it onto
the graph above.
18) What is the average rate of
change between x = 0 and x = 8?
19) Draw the secant line between
these two points on the graph to the
left
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Math 110
In Class Exercises, Section 3.3
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Let’s say that the x-axis represents time (in days), and the y axis represents number of projects
that an instructor still has to grade. Time 0 is the end of the term, after which the instructor leaves
for a vacation.
20) What can you say about the average rate of change between 8 days prior to the end of the
term, and the end of the term?
21) What can you say about the average rate of change between the end of the term, and 8 days
after it?
22) After the end of the term, how did the rate of change of the function change?
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