Chapter
2
Pre-Calculus Assignment Guide
Chapter two covers polynomial and rational functions. We will dive deep into polynomial properties such as divisibility
and zeros and practice sketching graphs by hand which will be important for Calculus. If you have any trouble with the
concepts or assignments, please stop by to see me or go to the Math Resource Center in room C117 for extra help.
Remember, at any time you may start the chapter review at the end of the assignment guide. It will help you prepare for
any quizzes throughout the chapter.
1.
2.1
Quadratic Functions
p. 95-98 #1 – 7 odd, 19, 21, 23, 25, 39, 41, 53 – 59 odd, 69, 70
2.
Min/Max Worksheet #1
3.
Min/Max Worksheet #2
4.
2.2
5.
6.
Polynomial Functions of Higher Degree
p. 108-109
#1-8, 9, 11, 19-33 odd, 35, 39, 41
p. 109-111
2.3
7.
#47, 53, 55, 59, 61, 67, 73, 75, 85, 93, 94
Real Zeros of Polynomial Functions
p. 123-124
#7, 9, 11, 17, 21, 25, 27, 31, 33, 35, 37
p. 124-125
#39, 41, 45, 49 – 61 odd, 63, 67, 68
8.
2.4
Complex Numbers
p. 133-134
#5, 7, 9, 17, 19, 27, 29, 33, 35, 39, 47, 53, 57, 67, 69
9.
2.5
Fundamental Theorem of Algebra
p. 140-141
#7, 9, 13, 15, 19, 21, 23, 25, 37, 39, 47, 49, 57
10.
More Practice Finding Zeros of Polynomials Worksheet
11.
2.6
Rational Functions and Asymptotes
p. 148-151
#3, 7-12, 13, 15, 19, 23, 27, 29, 35, 43, 45
12.
2.7
Graphs of Rational Functions
p. 157-158
#11, 15, 17, 27, 35, 43, 47, 63, 65, 73, 75, 77
Oct
9-13
Chapter 2 Review (optional)
p. 168
Read through Chapter Summary for sections 2.1-2.7. What did you learn?
p. 169-173
#14, 19, 29, 35, 45, 49, 53, 57, 59, 61, 75, 79, 81, 93, 97, 101, 111, 121, 123, 125, 131
#1
Ch 1 Test
Part II
#2
#3
#4
Early Release
Oct
16-20
14.
No School—
Columbus Day
2.1 Mini Quiz
#5
#6
#7
#8
Oct
23-27
Chapter 2 Review Worksheet
#9
#10
2.2-2.5 Quiz
#11
#12
Oct 30Nov 3
13.
Review
#13
Review #14
Ch. 2 Test
(Part I)
Ch. 2 Test
(Part II)
End of Quarter
Even Answers to Chapter 2
Section 2-1 Pg 95:
62a.
b g
8x
50  x
3
62b and c. max at x = 25, and y would be the area which
is 33 1/3.
70. true, x = -5/4 is the axis of symmetry
Min/Max Worksheet #1:
1a. max, it opens down
1b. x = -3/4
1c. 3.125
50
2. p 
 2w
w
3. C  2 30
4. 81, 9, 9
5. 30, 5, 6
6. No
7. (3.5, 2.225) and distance  1.323
8. (1.13, 1.27)
9.  1.38, 1.90
10. 30 units sold, $25,745 in profit
Min/Max Worksheet #2:
1. 10 and 10
2. 400 cm³
3. 3m by 3m
4. (2.5, 1.58)
5. (.1, 1.3)
6. 450 m by 225 m
7. 2 x 2  8 x  4
8. 60 units, revenue - $900, price - $15
9. 49/12 square units
10. x = 330 feet, r = 105 feet
Section 2-2 Pg 108:
2. h
4. a
6. d
8. b
10a. horizontal shift 3 units to the left
10b. vertical shift 3 units up
10c. reflection vertically, vertical shrink and vertical
shift one unit up
10d. reflection vertically, vertical shrink and horizontal
shift one to the left
94. True
Finding Zeros of Polynomials:
1. -2, -1, 3
2. 4i, -4i
3.  6 ,1  2i
4. –2
5. -1, 1.5, 2.5
6. 0,  5 , 5i
7.  4  5
8. 1,1  i
9. x 3  4 x 2  4 x  16
10. x 3  4 x 2  6 x  4
Review Worksheet:
1. 3 or 1 positive real zeroes
1 negative real zero
2. only real zero is 1


3. 2.5, 2.5
4. $8,333.33
≈167 units
5a. x 3  8x 2  22 x  20
5b. x 3  2 x 2  15 x  36
8. y  2x  22  3
125
8
Chapter Review Pg 169:
14. use calculator
13.
PreCalculus
Min/Max Worksheet #1
Directions: Do all problems and work on a separate sheet of paper. Show all work and include the equations you are using. If
you use your calculator and a graph to solve a problem, sketch the quadratic and label the min/max. The worksheet will be
returned to you without full credit if these instructions are not followed in their entirety. I must understand how you are reaching
your conclusions.
1. Given f x  2 x 2  3x  2
a.)
b.)
c.)
Does the function have a min or max value? Why
What input gives the max or min value?
What is the max or min value?
2. The area of a rectangle is 25 cm2 . Express the perimeter, P, as function of w, where w is the width of the rectangle.
3. The area of a circle is 30 m2 . Express the circumference, C, exactly in terms of  and rationalize if necessary.
4. Find two numbers whose sum is 18 and whose product is a maximum. What is the product and what are the two numbers?
5. Find two consecutive integers whose sum is 11 and whose product is a maximum. What is the product and what are the two
numbers?
6. Can I find two numbers whose difference is 4 and whose product is a maximum? Explain.
7. Which point on the curve y  x  2  1
is closest to the point ( 4 , 1 )? How far is this point to the curve?
8. Approximate to the nearest hundredth which point on the line y  x 2 is closest to ( 4, 0 )?
9. Approximate to the nearest hundredth which point on the line y  x 2 is closest to ( 0 , 4 )? Why did I get two minimums? (Hint:
sketch y  x 2 and plot the point ( 0 , 4 ) and think about the distance from the point to the graph.)
10. The revenue of Barry’s Bargain Bird Baths is modeled by the function f x  28x 2  1682x  485 where x is the number of
units sold. What is the maximum revenue generated and by how many units? Describe your viewing rectangle when you sketch
your graph.
PreCalculus
Min/Max Worksheet #2
Directions: Do all problems and work on a separate sheet of paper. Show all work and include the equations you are using. If
you use your calculator and a graph to solve a problem, sketch the quadratic and label the min/max. The worksheet will be
returned to you without full credit if these instructions are not followed in their entirety. I must understand how you are reaching
your conclusions.
1. Find two numbers adding to 20 such that the sum of their squares is as small as possible.
2. What is the largest possible area for a rectangle that has a perimeter of 80 cm.
3. The perimeter of a rectangle is 12 m. Find the dimensions for which the diagonal is as short as possible.
4. Find the point on the curve of y  x that is nearest to the point ( 3, 0 ).
5. Find the point on the line y = 3x + 1 that is nearest to the point ( 4, 0 ).
6. Suppose that you have 1800 meters in fencing with which to build three adjacent rectangular corrals as shown in the figure. Find
the dimensions so that the total enclosed area is as large as possible.
7. Let A  3x 2  4 x  5 and B  x 2  4 x  1 . Find A – B.
1
x  30 relates the selling price, p, of an item to the number of units, x, that are sold. Assume that
4
p is in dollars. For which value of x will the corresponding revenue be a maximum? What is the maximum revenue? What is the
unit price that maximizes the revenue?
8. Suppose that the function p  
9. The point P lies in the first quadrant on the graph of the line y = 7 - 3x. From the point P, perpendicular lines are drawn to both the
x and y-axis. Draw a picture of the situation. What is the largest possible area for the rectangle that is formed?
10. An athletic field with a perimeter of ¼ mile consists of a rectangle with a semicircle at each end as shown. Find x and r that yield
the greatest possible area for the rectangular region on the inside of the field so that the marching band can perform.
H-Pre-Calculus
Targets – Chapter 2
Section 2.1
1.
Sketch and analyze graphs of quadratic functions.
Write quadratic functions in standard form and use the results to sketch graphs of the function.
Identify the vertex and x-intercepts and sketch a graph of each of the following.
a.
f ( x)  x 2  2x  1
f ( x)  2x 2 - x  1
b.
Convert each of the following to standard form and identify the vertex.
c.
g ( x)  2x 2 - 16x - 24
h( x)  5x 2 - 10x - 15
d.
Write the standard for of the quadratic function that has the indicated vertex and whose graph passes through the
given point.
e.
vertex: (1, 5) through (4, 22)
f.
vertex: (-2, -6) through (5, -45)
Find two quadratic functions, one that opens upward and one that opens downward, whose graphs have the given xintercepts. (There are many correct answers.)
g.
2.
(-5, 0) & (4, 0)
h.
(3, 0) and (11, 0)
Find minimum and maximum values of functions in real-life applications.
a.
Find two positive real numbers whose product is a maximum and whose sum is 90.
b.
Find two positive real numbers whose sum of the first number and twice the second number is 32 and whose
product is a maximum.
c.
d.
Find the point on the curve of y 
x that is closest to the point (3, 0).
Suppose that the revenue generated by selling x units of a certain commodity is given by the equation
R = -0.3x2 + 100x. Assume that R is in dollars. What is the maximum revenue possible in this situation? How
many units must be sold to maximize the revenue?
See worksheets 1 & 2 for additional examples of this target.
Section 2.2
3.
4.
Sketch and analyze graphs of polynomial functions.
Use transformations to sketch graphs of polynomial functions.
g ( x)   x  2   3 , and h( x)  4  x  2   3
f ( x)  x 4 ,
4
4
a.
Compare the graphs of
b.
Compare the graphs of f ( x)   12 x , g ( x)   12  x  3 , and h( x)  
5
5
1
2
 x  3
5
5
Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions.
Left-side behavior
a.
f ( x )   x  3x  7 x  1
b.
f ( x)  3 x 5  3 x 4  7 x 2  1
c.
f ( x)  5 x 6  3 x 4  7 x 2  x  2
d.
f ( x)   2 x 3  3 x 2  7 x  1
4
3
Right-side behavior.
5.
Find and use zeros of polynomial functions as sketching aids.
c.
f ( x)  x3  x 2  20 x
f ( x)  54 x 3  2 x 2  x
e.
f ( x)   2 x( x  1)4 ( x  2)5 ( x  5)
a.
6.
a.
f ( x)  2 x 3  5 x 2  3 x  2
c.
f ( x)  12 x 4  5x3  207 x 2  85 x  51
f ( x)  3( x  2)3 ( x  3)2
b.
f ( x)  2 x 4  5 x 3  8 x 2  x  2
Use long division and synthetic division to divide polynomials by other polynomials and
determine the numbers of rational and real zeros of polynomial functions, and find the
zeros.
(2x3 + 4x2 + 7x + 4) ÷ (x + 3)
(2x4 + 4x3 - 5x2 + 3x - 2) ÷ (x2 + 2x - 3)
b.
d.
(x3 + 5x2 + 7x + 4) ÷ (x - 2)
(x4 + 3x2 + 1) ÷ (x2 – 2x + 3)
Use synthetic division to divide polynomials by binomials of the form (x – k)
a.
c.
9.
d.
Use long division to divide polynomials by other polynomials.
a.
c.
8.
f ( x)   2 x 4  2 x 3  40 x 2
Use the Intermediate Value Theorem to help locate intervals of length 1 that contain zeros of polynomial functions.
Section 2.3
7.
b.
(x4 - 3x3 - 2x2 + 7x + 1) ÷ (x - 2)
(x4 - 5x3 - 8x2 + 13x - 12) ÷ (x - 6)
b.
(3x3 - 4x2 + 7x + 5) ÷ (x + 1)
Use the Remainder and Factor Theorems.
Use the Remainder Theorem to evaluate each polynomial function at the given x value.
a.
f(x) = 4x3 - 7x2 + 8x + 2) at x = 2
b.
f(x) = 3x4 - 5x3 + 3x2 - 2x + 5 at x = -3
Find all real zeros of the polynomial function given some factor(s).
c.
f(x) = 3x3 + 2x2 - 19x + 6
factors (x + 3) and (x – 2)
d.
f(x) = 8x4 - 14x3 - 71x2 – 10x + 24
factors (x + 2) and (x – 4)
10.
Use the Rational Zero Test to determine possible rational zeros of polynomial functions.
List all possible rational zeros
a.
f(x) = x3 + 7x2 - 9x – 6
b.
f(x) = 4x5 – 7x4 + 8x3 + 9x2 - 2x + 3
c.
f(x) = 6x3 - 8x2 - 9x + 12
11.
Use Descartes’ Rule of Signs and the Upper and Lower Bound Rules to find zeros of polynomials.
Use Descartes’ Rule of Signs to determine the possible numbers of positive and negative real zeros.
a.
f(x) = 5x4 - 7x3 + 8x2 + 4x - 3
b.
f(x) = 2x5 + x4 – 3x3 – 2x2 - 4x + 5
c.
f(x) = x7 – x6 + 3x5 – 7x4 – 3x3 + 4x2 + 2x – 1
Verify the Upper and Lower Bounds
d.
e.
f(x) = x3 + 5x2 + 3x – 2 Upper Bound: 1, Lower Bound: -6
f(x) = x4 - x3 - 8x2 + x + 6
Upper Bound: 4,Lower Bound: -3
Section 2.4
12.
I can use the imaginary unit i to write complex numbers in standard form.
a.
13.
(4  2i)  (5  i)
2
17.
2
4  2i
3i
I can plot complex numbers in the complex plane.
4  2i
a.
Section 2.5
16.
49
b.
i 5  3i 4  2i 2  3i  7
b.
3i (4  2i)(5  i)
d.
(4  2i)2  (4  2i) 2
I can use complex numbers to write the quotient of two complex numbers in standard form.
a.
15.
8
I can add, subtract, and multiply complex numbers.
5i(4  2i)  3i(5  i)
a.
c.
14.
Perform operations with complex numbers and plot complex numbers in the complex
plane.
b.
5  3i
7  2i
b.
4i
Use applications of the Fundamental Theorem of Algebra
I can find all zeros of polynomial functions, including complex zeros.
a.
f ( x)  2 x 4  5x3  10 x 2  12
b.
g ( x)  3x 4  4 x3  7 x 2  10 x  4
c.
h( x)  x3  5 x 2  9 x  24
d.
f ( x)  2 x3  5 x 2  9 x  40
I can find conjugate pairs of complex zeros.
Write a polynomial function the real coefficients that has the given zeros.
2,  2,  5i
4, 4, 2i
a.
b.
c.
18.
1,  4,  3  5i
d.
4,  4, 3 
3i
I can find zeros of polynomials by factoring.
a.
b.
c.
f ( x)  x 4  2 x 2  8
f ( x)  x 4  x3  x 2  5x  20 if one factor is x 2  5
f ( x)  x 4  4 x3  3x 2  8x  16 if one factor is x 2  x  4
Section 2.6 & 2.7 Graphs of rational functions.
19.
I can determine the domains of rational functions.
I can find any holes in the graph of rational functions.
I can find the vertical, horizontal and slant asymptotes of rational functions.
I can find the x- and y-intercepts of rational functions.
I can determine if the graph of a rational function crosses the horizontal or slant asymptote.
a.
(x + 3)(x - 2)
y =
(x - 2)(x + 1)
e.
y =
g.
2 x2 + x - 5
x2 + 1
( x  3)( x  2)( x  1)
y=
( x  2)( x  2)
b.
f.
h.
x2 - 3x - 10
y =
x+2
2
c.
( x - 3)( x  1)( x  4)( x  3)
( x  1)( x  3)( x  2)( x  1)
(x + 6)(x + 4)
y =
2
(x - 1 )
y=
y =
x + 7 x + 12
2
x + 4x + 3
H-Pre-Calculus
Targets – Chapter 2
Answers
1a.
vertex (-1, 0) x-int. (-1,0)

1
4
7
8
,

1b.
vertex
1c.
g ( x)  2  x  4   56 vertex (4, -56)
1d.
h( x)  5  x  1  20 vertex (1, -20)
1e.
f ( x) 
1f.
f ( x) 
1g.
x-int. none
2
2
 x  1  5
2
39
49  x  2   6
8c.
x 3  x 2  2 x  1  x 6 6
9a.
9b.
22
416
9c.
9d.
2
17
9
many answers are possible
10a.
10b.
10c.
f ( x)  x  x  20
2
f ( x)   x 2  x  20
1h.
many answers are possible
f ( x)  x 2  14 x  33
f ( x)   x 2  14 x  33
2a.
2b.
2c.
2d.
45 & 45
16 & 8

5
2
10
2
,

positive: 3 or 1
negative: 1
11b.
positive: 2 or 0
negative: 3 or 1
11c.
positive: 5 or 3 or 1
negative: 2 or 0
11d & 11e. all bounds are verifiable
12b.
 $8333.33
 166.66units
13a.
13b.
3a & 3b. use your calculator
13c.
4a.
4b.
4c.
4d.
left:  right: 
left:  right: 
left:  right: 
left:  right: 
13d.
5a.
zeros
5b.
0, 5,  4 now sketch
zeros 0, 5,  4 now sketch
14a.
14b.
16.
5e.
zeros
6a.
6b.
6c.
( 0, 1 )
at -1, ( 0, 1 ), ( 3, 4 )
Note: 1 other in (-1, 0)
(-5, -4), (-1, 0), (0, 1), (4, 5)
7a.
2 x  2 x  13 
7b.
x  7 x  21 
7c.
2x  1 
7d.
x2  2 x  4 
8a.
x3  x 2  4 x  1  x 1 2
18b.
8b.
3 x 2  7 x  14  x91
18c.
0, 1, 2, 5 now sketch
2
2
2
35
x 3
46
x2
x 1
x 2  2 x 3
2 x 11
x2  2 x 3
32i
2
4
3  3i
29
31
53  53 i
0.869, 1.320
complex (unable to find)
5d.
zeros
8  7i
2  2i
13  5i
18  66i
12  26i
15a & 15b. plot on real & complex axis
0,  2,  now sketch
zeros 2,  3 now sketch
5c.
2
5
1,  2,  3,  6
1,  3,  12 ,  14 ,  23 ,  43
1,  2,  3,  4,  6,  12,
 12 ,  23 ,  13 ,  32 ,  34 ,  16
11a.
12a.
25000
3
500
3
3, 2, 13
2, 4, 12 , 43
16b.
16c.
16d.
1,
1
3

1.025  2.043i
, 1 i 3
6.832 complex (unable to find) 0.916  1.635i

5
2
,
5 i 7
2

17a.
f  x   x4  4x3  29x2  100x  100
17b.
f  x   x4  8x3  20 x2  32 x  64
17c.
f  x   x4  9x3  48x2  78x  136
17d.
f  x   x4  2 x3  20 x2  192
18a.
2i,  2
 ,  5
 , 
1i 15
2
1 17
2
3 i 7
2
19a. VA. x=-1
HA y = 1
SA none
Holes:
Holes: (3, 0)
x-int. (-4, 0)
y-int. (0, 6)
AX (5, 1)
 2, 53 
Domain:
x-int. (-3, 0)
y-int. (0, 3)
AX none
Domain:
Range:
Range:
x : x 
y : y 
19g.
VA. x = -2
HA none
SA y = x + 2
 2, 154 
Holes:
x-int. (-3, 0) & (1, 0)
y-int.
 0, 32 
AX
none
Domain:
19c. VA. x= 1
HA y = 1
SA none
Holes:
Range:
19h.
 3, 21 
x-int. (-4, 0)
y-int. (0, 4)
AX none
Domain:
Range:
x : x 
y : y 
19e. VA. none
HA y = 2
SA none
Holes: none
x-int.

1 41
4
y-int. (0, 4)
AX none
Domain:
Range:
19f.
, x  1
, y  1

 ,  
y : y  ,
VA. x = -1, x = 2
HA y = 1
SA none
y  2
x : x  ,
 ,  
x  2
VA. x = -1
HA y = 1
SA none
Holes: none
x-int. (-4, 0) & (-6, 0)
y-int. (0, 24)
 1223 , 1
Domain:
x : x 
AX
Range:
,0
, y  1
, y  1
 ,  
 ,  
Range:
, x  1, 2
, x  1
19b. VA. none
HA none
SA none
Holes: (-2, -7)
x-int. (5, 0)
y-int. (0, -5)
AX none
Domain:
x : x 
y : y 
y : y 
, x  1
, y  1