Advanced Math I
Unit 7: Sequences and Series
Time Frame: 2.5 weeks
Unit Description
This unit introduces finite and infinite sequences and series. A sequence can be thought
of as a function with the inputs being the natural numbers. As a result the four
representations of functions apply. The unit covers the sums of finite series and infinite
series.
Student Understandings
Students will be able to find the terms of a sequence given the nth term formula for that
sequence. They can recognize an arithmetic or geometric sequence, find the explicit or
recursive formula for that sequence, and graph the sequence. With infinite sequences
they will find limits if they exist. Students can expand a series written in summation
notation and find the sum. They can use the formulas for the sum of a finite arithmetic or
geometric series to find the sum of n terms. They are able to tell whether or not an
infinite geometric series has a sum and find the sum if it does exist. They are able to
model and solve real-life problems using sequences and series.
Guiding Questions
1. Do students understand that a sequence is a function whose domain is the set of
natural numbers?
2. Can students graph a sequence?
3. Can students recognize, write, and find the nth term of a finite arithmetic or
geometric sequence?
4. Can students give the recursive definition for a sequence?
5. 5 Can students recognize, write, and find the sum of an arithmetic or geometric
series?
6. Can students determine whether or not the sum of an infinite geometric series
exists and if so find the sum?
7. Do students recognize the convergence or divergence of a sequence?
8. Can students find the limit of terms of an infinite sequence?
9. Can students use summation notation to write sums of sequences?
10. Can students use sequences and series to model and solve real-life problems?
Advanced Math IUnit 7Sequences and Series
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Unit 7 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Algebra
4
Translate and show the relationships among non-linear graphs, related tables of
values, and algebraic symbolic representations (A-1-H)
6
Analyze functions based on zeros, asymptotes, and local and global
characteristics of the function
Patterns, Relations, and Functions
24
Model a given set of real-life data with a non-linear function (P-1-H) (P-5-H)
26
Represent and solve problems involving nth terms and sums for arithmetic and
geometric series (P-2-H)
Sample Activities
Ongoing activity: Glossary notebook
Terms to add to the vocabulary list: finite sequence, infinite sequence, recursive,
arithmetic sequence, geometric sequence, finite series, infinite series, convergence,
divergence, summation (sigma) notation, index of summation, lower limit of summation,
upper limit of summation
Activity 1: Arithmetic and Geometric sequences (GLEs: 4, 24, 26)
Stress the fact that a sequence is a function and make the connection between arithmetic
sequences and linear functions. The common difference in the formula for the arithmetic
sequence is the slope in the linear function. The geometric sequence is an exponential
function where the domain is the set of natural numbers. As students work the problems
that require them to find the nth term (explicit) formula, have them also draw a graph first
of the sequence and then of the explicit formula. Put the calculator into sequence mode
and graph each of the sequences. Students can turn on TRACE and see each term of the
sequence. The table function is also very useful in this unit.
Problems:
1. Find the first 4 terms of the given sequence and tell whether the sequence is
arithmetic, geometric, or neither.
a) t n  3( 2) n
b) t n  3  7n
1
c) t n  n 
n
Advanced Math IUnit 7Sequences and Series
82
2. Find the formula for tn, and sketch the graph of each arithmetic or geometric
sequence.
a) 8, 6, 4, 2,…
b) 8, 4, 2, 1, …
c) 24, -12, 6, -3,…\
d)
3. A field house has a section where the seating can be arranged so the first row has 11
seats, the second row has 15 seats, the third row has 19 seats and so on. If there is
sufficient space for 20 rows in the section, how many seats are in the last row?
4. A company began doing business four years ago. Its profits for the last 4 years have
been $11 million, $15 million, $19, million and $23 million. If the pattern continues,
what is the expected profit in 26 years?
5. The school buys a new copy machine for $15,500. It depreciates at a rate of 20% per
year. How much has it depreciated at the end of the first year? Find the depreciated
value after 5 full years.
6. Explain how you use the first two terms of a geometric sequence to find the explicit
formula.
Solutions for Activity 1:
1. a) t n  3( 2) n , 6, 12, 24, 48 geometric
b) tn = 3 – 7n, -4, -11, -18, -25 arithmetic
1 5 10 17
c) t n  n  ,2, , , , neither
n 2 3 4
2. a) tn =10 – 2n
1
b) t n  16 
2
n
 1
c) t n  48  
 2
n
3. 86 seats
4. 127 million
5. $ 3100; $6348.80
6. Since a geometric sequence is an exponential function with a domain that is the set of
t
natural numbers you find the growth/decay factor 2 just as you did with exponential
t1
functions. This value is r in the geometric sequence formula t n  t1 r n 1
Advanced Math IUnit 7Sequences and Series
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Activity 2: Recursive Definitions (GLEs: 4, 24, 26)
Students need to understand the difference between a recursive and nth term formula.
Have students give examples of the recursive and nth term formulas for the arithmetic
and geometric sequences.
1. Find the third, fourth, and fifth term of the sequence then write the nth term formula
for the sequence.
a) t1 = 6, tn = tn-1 + 4
b) t1 = 1, tn = 3tn-1
1
c) t1 = 9, t n  t n 1
3
2. Give a recursive definition for each sequence.
a) 81, -27, 9, -3 …
b) 1, 3, 6, 10, 15, 21…
c) 8, 12, 16, 20,…
3. The population of a country in the southern hemisphere is growing because of two
conditions
 the growth rate of those in the country is increasing at a rate of 2% per year
 each year they gain 30,000 immigrants
If the population is 6 million people, what will be the population each year for the
next five years?
4. A pond currently has 2000 trout in it. A fish hatchery decides to add an additional 20
trout each month. In addition, it is known that the trout population is growing at a
rate of 3% per month. The size of the population after n months is given by the
. pn 1  20 . How many trout are in
recursively defined sequence p1  2000, pn  103
the pond at the end of the second month? the third month? Using a graphing utility
determine how long it will be before the trout population reaches 5000.
5. A trip to Cancun for the senior trip will cost $450 and full payment is due March 2nd.
On September 1st a student deposits $100 in a savings account that pays 4% per year
compounded monthly and adds $50 to the account on the first of each month.
a) Find a recursive sequence that explains how much is in the account after n
months.
b) List the amounts in the account for the first 6 months.
c) How much would he have if he added $60 to the account each month?
6. On January 1, 1998, Margaret’s parents decide to place $50 each month into an
Education IRA.
a) Find a recursive formula that represents the balance at the end of each month if
the rate of return is assumed to be an annual rate of 8% compounded monthly.
b) How long will it be before the account exceeds $4000.00?
c) How much will be in the account in 17 years when Margaret is ready for
college?
Advanced Math IUnit 7Sequences and Series
84
7. When Margaret graduates from college she has a balance of $5000 on her credit card.
With her new job she can pay $100 toward the balance each month. Her card charges
a rate of 1% per month on any unpaid balance.
a) Find a recursive formula that represents the balance after making a $100
payment each month.
b) Using a graphing utility, determine when Margaret’s balance will be below
$3000.
c) How many payments of $100 have been made?
Solutions for Activity 2:
b) 9, 27, 81 t1= 1, t n  3n 1 or
1. a) 14, 18, 22 t1 = 6, tn = 4n + 2
F
I
G
HJ
K
F
IJ
G
HK
F
I
G
HJ
K
n 1
n
n
1 1
1
1
c) 1, ,
t1  9, t n  9
or 27
3 9
3
3
1
2. a) t1  81, t n   t n 1 b) t1  1, tn  tn 1  n
3
3.
bg
1
3
3
c) t1  8, tn  tn 1  4
t1  6,000,000 tn  102
. tn 1  30,000
End of year 1 2
3
4
5
6,150,000
6,303,000 6,459,060 6,618,241 6,618,241
4. 2162; 2246; 28 months
5. The recursive definition works well on problems where new amounts are added on a
regular basis.
a)
rI
F
G
H nJ
KA  P
F1  0.04 IJA  P
A  100, A  G
H 12 K
A1  100, An  1 
1
n
n 1
n 1
b)
September October November December January February March 1
100
150.33 200.83
251.50
302.34 353.35
404.53
c) $465.03 after he added to the account on March 1.
 .08 
6. a) A1  50, 1 
b) 5 years 5 months
c)$21,589,86
 An 1  50
12 

7. a) A1  5000, 1.01 An 1  100
b) 35 months 34 payments c) 71 months with the last
payment of $65.53.
Advanced Math IUnit 7Sequences and Series
85
Activity 3: Applications of Series (GLEs: 4, 24, 26)
1. A sum of $5000 is invested at a compound interest rate of 6.5% per year. To the
nearest dollar, what will be the total value of the investment at the end of 5 years?
2. The chain letter reads:
Dear Friend,
Make six copies of this letter and send them to six of your friends. In twenty days
you will have good luck. If you break this chain you will have bad luck!
Assume that every person who receives the letter sends it on and does not break the
chain.
a) Fill in the table below to show the number of letters sent in each mailing:
a) 1st
b) 2nd
c) 3rd
d) 4th
mailing
mailing
mailing
mailing
g)
h)
i)
j)
b) After the 10th mailing, how many letters have been sent?
c) The population of the United States is approximately 294,400,000. Would the
11th mailing exceed this population? How do you know?
3. A company began doing business four years ago. Its profits for the last 4 years have
been $32 million, $38 million, $42 million and $48 million. If the pattern continues,
what is the expected total profit in the first ten years.
4. A production line is improving its efficiency through training and experience. If the
number of items produce in the first four days of a month are 13, 15, 17, and 19,
respectively, project the total number of items produced by the end of a 30 day month
if the pattern continues.
5. The Internal Revenue Service assumes that the value of an item which can wear out
decreases by a constant number of dollars each year. For instance a house depreciates by
1
of its value each year.
40
a) If your house is worth $125,000 originally, by how many dollars does it
depreciate each year?
b) What is your house worth after 1, 2, or 3 years?
c) Do these values form an arithmetic or a geometric sequence?
d) Calculate the value of your house at the end of 30 years.
e) According to this model is your house ever worth nothing? Explain.
Solutions: 1. $6850.43
2.a)
1st mailing 2nd mailing 3rd mailing 4th mailing … 10th mailing
6
62
63
64
610
b) 60,166,176 c) Yes, because the 11th mailing would be 362,797,056 letters
1. 590 million 4. 1,260 items 5. a) $3125 b)$121,875, $ 118,750, $115,625
Advanced Math IUnit 7Sequences and Series
86
e) …
k)
c) arithmetic sequence d) $34,375 e) Yes, since the model( V = 128125 – 3125n) does
not take into account the fact that usually property values usually appreciate over time.
Activity 4: Graphing Infinite Sequences and Determining their Convergence or
Divergence (GLEs: 4, 6)
Students are usually introduced to limits in the Sequences and Series Unit. This activity
is designed to first give students a visual picture of convergence and divergence of graphs
of sequences before learning to find the limits by working with the symbolic formulas.
Connect this activity to the end behavior of the functions previously studied. Ask the
students which functions they have studied that they would expect to be a) convergent
and b) divergent. Begin with the graphs shown below then follow with a series of
problems and ask the students to graph in sequence mode and decide on their
convergence or divergence.
Convergent
The values get closer and closer to
a fixed value. There is a horizontal
asymptote.
The values of this convergent
sequence oscillate back and forth
about one value. There is a
horizontal asymptote.
Divergent
The sequence diverges to  . the
values grow in size becoming infinitely
large.
The sequence is periodic. A set of
values is repeated at periodic intervals.
The sequence is both oscillatory and
divergent.
Advanced Math IUnit 7Sequences and Series
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Problems:
1) f(n) = 3n – 5
n
3
2) f (n)   
4
1
3) f (n) 
n 1
n 1
4) f (n)   2
5. f (n)   sin
nI
F
G
H2 J
K
cos n
n
n3
7. f (n)  2
n  n 1
n
9
8. f (n)  4 
10
6. f (n) 
F
I
G
H J
K
lim
Students are now ready to find the limits using the notation n   f(n). Use the problems
above that were convergent to show students how the table feature of their graphing
calculator will give the limit. Have them change the setting of the table to Indepnt: Ask
and Depend: Auto and then put in increasing large values for n. They will see the values
of f(n) converge towards a given value. Only after they understand what is happening
should they be shown the algebraic methods for working the problems.
Additional problems:
9.
lim
n
n 
e
2n  1
n  10
3n  12
11. lim
n 
n5
n 2  3n  4
lim
12. n
n4
3  4x
13. lim
n
x2  4
10.
lim
n
Solutions for Activity 4:
1) divergent an arithmetic sequence that is linear 2) Convergent; the values get closer
and closer to 0. 3) Convergent; the values get closer and closer to 0.
4) Divergent; the sequence is both oscillatory and divergent. 5) Divergent; the sequence
is periodic. 6) Convergent and oscillatory (It helps to go into FORMAT and turn off the
axes to see it clearly.) 7) Divergent; the values get larger and larger. 8) Convergent
and oscillatory 9) 0
10) 2
11) 3 12) divergent 13) 0
Activity 5: Using Summation Notation (GLE 26)
The final activity gives students a chance to find the sums of series using summation
notation. Problems include not only the familiar finite arithmetic and geometric series
Advanced Math IUnit 7Sequences and Series
88
but also infinite geometric series that have a sum as well as finite series that are neither
arithmetic nor geometric.
Part I: Problems:
Express the given series using summation notation then find the sum.
1. 5 + 9 + 13 + …+ 101
2. 48 + 24 + 12 …
3. 1 + 4 + 9 +…+ 144
1 1 1
1
4. 4. 1     ... 
2 4 8
512
Part II: Expand each of the following then find the sum:

1.
1
 

k 0  3 
8
k
4.
k
1
1
7
5.  5  2i
10
2.
3
 2i
i2
i 1
15
3.
6
j 1
Solutions:
Part I:
25
1.  4i  1  1325
i 1
F1 I
2.  48GJ  48
H2 K

i
i 1
12
3.  i 2  650
i 1
9
1I
171
F
G
H2 J
K  256
4.  
i 1
Part II
1 1
3
1) 1+ + + ...=
3 9
2
2) 2+ 4 +6 + ...+ 20 = 110
3) 6 +6 +6 + ...+6 = 90
4) 2+ 8 + 26 + ...+6560 = 9832
5) 1- 1- 3 - ...- 9 = -24
i 1
Sample Assessments
General assessments

The student will perform a writing assessment which covers activities of the unit
and which uses the glossary they have created throughout this unit. A possible
question could be: Explain the difference in a recursive and explicit (nth term
formula).The teacher will look for understanding in how the terms or concepts are
used, especially with verbs such as show, describe, justify, or compare and
contrast.
Advanced Math IUnit 7Sequences and Series
89


Spiral reviews should continue with this unit. The student will make connections
between an arithmetic sequence and a linear function and between a geometric
sequence and an exponential function. The teacher will mix questions about the
linear and exponential function with use of the explicit formulas for the arithmetic
and geometric sequences. Problems should also make a connection between the
end behavior of previously studied functions and the finding of limits of infinite
sequences. Continue to tie the spirals to the study guide for the written test or the
final exam.
The student will work with a group to model real-life problems using both
recursion and explicit formulas on problems involving sequences and series.
The scoring rubric should include:
 teacher observation of group interaction and work
 explanation of each group’s problem to class
 work handed in by each member of the group
Activity-Specific Assessments

Activity 1: The student will demonstrate proficiency in working with the finite
arithmetic and geometric sequences and series.

Activity 4: The student will demonstrate proficiency in determining convergence
or divergence and finding the limit of a convergent sequence.

Activity 5: The student will demonstrate proficiency in working with summation
notation.
Advanced Math IUnit 7Sequences and Series
90
Advanced Math I
Unit 8: Conic Sections and Parametric Equations
Time Frame: 2.5 weeks
Unit Description
This unit explores the characteristics of the conic sections, their graphs, and how they are
interrelated in the general form. The concept of eccentricity gives a common definition
to the conic sections and is used to identify the conic section and to write its equation.
Methods of graphing the conic sections both by hand and by using a graphing utility are
introduced. The polar form of a conic is also studied. The unit includes parametric
equations as well.
Student Understandings
Students recognize a conic as the intersection of a plane and a double-napped cone. They
are also able to identify the degenerate conics. Each of the conics is reviewed and
applications showing how they are used in real-life situations are presented. Students are
able to graph conics with and without the use of a graphing utility. Students are able to
define conic sections in terms of eccentricity and to classify a conic section by looking at
its equation. They are able to write and use equations of conics in polar form.
Guiding Questions
1. Can students recognize the conic as the intersection of a plane and a doublenapped cone?
2. Can students write and recognize equations of the conics in standard and general
form?
3. Can students use the properties of each of the conics to solve real-life problems?
4. Can students find the eccentricities of the parabola, ellipse, and hyperbola?
5. Can students use the eccentricity to write the equations of the parabola, ellipse,
and hyperbola?
6. Can students find the asymptotes of a hyperbola?
7. Can students use the asymptotes to write the equation of a hyperbola in standard
form?
8. Can students write equations of conics in polar form?
9. Can students evaluate sets of parametric equations for given values of the
parameter?
10. Can students graph curves that are represented by sets of parametric equations?
11. Can students use parametric equations to model motion?
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91
Unit 8 Grade-Level Expectations (GLEs)
GLE # GLE Text and Benchmarks
Algebra
4
Translate and show the relationships among non-linear graphs, related tables of
values, and algebraic symbolic representations (A-1-H)
9
Solve quadratic equations by factoring, completing the square, using the
quadratic formula, and graphing
10
Model and solve problems involving quadratic, polynomial, exponential,
logarithmic, step function, rational, and absolute value equations using
technology. (A-4-H)
Geometry
15
Identify conic sections, including the degenerate conics, and describe the
relationship of the plane and double-napped cone that forms each conic (G-1Ht
16
Represent translations, reflections, rotations, and dilations of plane figures
using sketches, coordinates, vectors, and matrices (G-3-H)
Patterns, Relations, and Functions
25
Apply the concept of a function and function notation to represent and evaluate
functions(P-1-H) (P-5-H)
Sample Activities
Ongoing activity: Glossary notebook
Terms to add to the vocabulary list: double napped cone, conic section, locus, radius,
center, standard form of the equation of a circle, parabola, standard form of equation of
a parabola, directrix, foci, vertex, tangent, ellipse, standard form of the equation of an
ellipse, vertices, major axis, minor axis, eccentricity, hyperbola, form of the equation of a
hyperbola, transverse axis, asymptotes, conjugate axis, parametric equations, plane
curve.
Activity 1: Working with Circles (GLEs 4, 9, 15, 16, 25)
Review the general second-degree equation Ax 2  Bxy  Cy 2  Dx  Ey  F  0 for the
conic sections. Students should know how to identify the conic section from this
equation. With each of the conic sections, students should know the standard form of the
equation of that conic section. This activity gives students additional practice in writing
equations of circles and working with semi-circles as functions.
Problems:
Part I: Find an equation of the circle. Write your answer in both standard and general
form.
1. With center at (2, -3) and radius equal to 4
Advanced Math IUnit 8Conic Sections and Parametric Equations
92
2. Circle passes through (1, 7) and has its center at (4, 2)
3. The endpoints of the diameter are (1, 5) and (-3, -7)
4. Center is at (-2, 5) and tangent to the line x = 7
Part II: Each of the problems below involve equations of semi-circles. For each one find
a) the center, b) the radius, c) the domain and range, and d) a sketch of the graph.
1. f ( x )  9  x 2
2. f ( x )   4  ( x  4) 2
3. f ( x )  4 x  3  x 2
4. f ( x )  16  6 x  x 2
Part III. Given the equation of the semi-circle y  1  x 2 .Write an equation of the semicircle if
a) the center is moved to (-3, 0)
b) the center is moved to (0, -2)
c) the center is moved to (3, -5)
d) the semi-circle is reflected over the x-axis
Solutions for Activity 1:
Part I
2
1. x  2  y  3  16
b g b g x  y  4x  6y  3  0
2. b
x  4g b
y  2g 34 x  y  8 x  4 y  14  0
3. b
x  1g b
y  1g 40 x  y  2 x  2 y  38  0
4. b
x  2g b
y  5g 81 x  y  4 x  10 y  52  0
2
2
2
2
2
2
2
Part II
1. a) center: (0,0)
2. a) center: (-4,0)
3. a) center: (2,0)
4. a) center: (3,0)
2
2
2
2
2
2
2
radius: 3 Domain: {x:-3 ≤ x ≤ 3} Range: {y: y ≥ 0}
radius: 2 Domain: {x:-4 ≤ x ≤ 0} Range: {y: y ≤ 0}
radius: 1 Domain: {x:1 ≤ x ≤ 3} Range: {y: y ≥ 0}
radius: 5 Domain: {x:-2 ≤ x ≤ 8} Range: {y: y ≥ 0}
Part III
a ) y  1  ( x  3) 2
b) y  1  x 2  2
c) y  1  ( x  3) 2  5 d) y = - 1- x 2
Advanced Math IUnit 8Conic Sections and Parametric Equations
93
Activity 2: Parabolas as a Conic Section (GLEs: 4, 9, 15, 16, 25)
Students have studied parabolas as a polynomial function. This activity looks at a
parabola as a conic section. Students will identify the vertex, focus, and directrix of the
parabola given its equation; write the equation given the identifying features, and look at
a series of functions that are “half parabolas”.
Part I: For each of the parabolas find the coordinates of the focus and vertex, an equation
of the directrix, and sketch the graph.
1. x 2  4 y
2. 2 y 2  9 x  0
3. y 2  8 x
4. x 2  y  0
Part II: Find the standard form of the equation of each parabola. Sketch its graph.
1. Focus: (-1,0); directrix, x = 1
2. Vertex (5, 2); Focus (3, 2)
39
41
; directrix is y = 3. Focus at 2,
8
8
F
IJ
G
H K
Part III: Give the domain and zeros then sketch the graph of each of the following.
Verify your answer with a graphing calculator.
1. x  y
2. y   x
3. y   x  3
4. x   y  3  2
5. y  x  2  1
Solutions for Activity 2:
Part I:
1. focus: (1, 0); vertex: (0,0); y = -1
9
9 
2. focus:  ,0  ; vertex: (0, 0); x  
8
8 
3. focus: (-2, 0); vertex: (0,0); x = 2
4. focus (0, -¼ ); vertex: (0,0); y = ¼
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Part II.
1
1. x   y 2
4
2. (y – 2)2 = -8(x – 5)
3. y + 5 = 2(x – 2)2
Part III.
1. domain: {x: x ≥ 0} zeros: x = 0
2. domain: {x: x ≥ 0} zeros: x = 0
3. domain: {x: x ≥ 3} zeros: x = 3
4. domain: {x: x ≤ 2} zeros: 2  3
5. domain: {x: x ≥ 2} zeros: x = -1
Activity 3: Eccentricity (GLEs: 4, 9, 15, 16)
Introduce eccentricity of a conic section. Students should know prior to this activity the
definition of eccentricity and how they can identify the type of conic section by its
eccentricity e. If e = 1 the conic is a parabola. If e > 1 the conic is a hyperbola and if e < 1
the conic is an ellipse. If e = 0 we have a point. In this activity students work with
ellipses and hyperbolas and then end with a set of problems that asks the students to
identify the type of conic and then find its general equation.
Part I: Ellipses
Find the eccentricity, center, foci, and directrices of the given ellipse and draw a sketch of
the graph:
1. 6 x 2  9 y 2  24 x  54 y  51  0
2. 4 x 2  4 y 2  20 x  32 y  89  0
Part II: Hyperbolas
Find the eccentricity, center, foci, directrices, and equations of the asymptotes of the
given hyperbolas and draw a sketch of the graph.
1. 9 x 2  18 y 2  54 x  36 y  79  0
2. 3 y 2  4 x 2  8 x  24 y  40  0
3. 4 y 2  9 x 2  16 y  18x  29
Part III: Identify the conic and find its equation having the given properties:
1. Focus at (2, 0); directrix x = -4; e = ½
2. Focus at (-3, 2); directrix: x = 1; e = 3
3. Center at the origin ; foci on the x-axis ; e = 2 ; containing the point (2, 3)
4. Center at (4, -2); one vertex at (9, -2) and one focus at (0, -2)
5. One focus at (3  3 13,1) , asymptotes intersecting at (-3, 1), and one
asymptote passing through the point (1, 7).
Advanced Math IUnit 8Conic Sections and Parametric Equations
95
Solutions for Activity 3:
Part I
1
3 ; center (2, 3); foci: (2  3 ,3) directrices: x  2  3 3
3
 5 
2. point-circle   ,4 
 2 
1. e 
Part II
2 
2

1. e  3 center : (3, 1); foci :  3, 1 
6  directrices : y  1 
6
3 
9

asymptotes : x  y 2  2  3  0 and  x  y 2  2  3  0
7
; center : (1, 4); foci : ( 1, 3), ( 1,11); directrices : y  0, y  8;
2
asymptotes : 2 x  y 3  2  4 3  0 and  2 x  y 3  2  4 3  0
2. e 



13
; center : (1, 2); foci : 1, 2  13 and 1, 2  13
3
9
9
directrices : y  2 
13 and y  2 
13
13
13
asymptotes : 3x  2 y  1  0 and 3x  2 y  7  0
3. e 

Part III
1. 3x 2  24 x  4 y 2  0
2. 8 x 2  24 x  y 2  4 y  4  0
3. 3x 2  y 2  3  0
( x  4) 2 ( y  2) 2

1
25
9
( x  3) 2 ( y  1) 2
5.

1
36
81
4.
Advanced Math IUnit 8Conic Sections and Parametric Equations
96
Activity 4: Polar Equations of Conics (GLEs: 16, 25)
Problems:
The equations below are those of conics having a focus at the pole. In each problem (a)
find the eccentricity; (b) identify the conic; and (c) write an equation of the directrix
which corresponds to the focus at the pole. Have students verify their answers by
graphing the polar conic and the directrix on the same screen.
2
1. r 
Solutions for Activity 4:
1  cos
1. a) 1 b) parabola c) rcosθ = -2
6
2. r 
2. a) 2/3 b) ellipse c) rcosθ = -3
3  2 cos
3. a) ½ b) ellipse c) rsinθ = 5
5
4. a) 6/5 b) hyperbola c) rsinθ = -5
3. r 
2  sin 
9
4. r 
5  6 sin 
Activity 5: Sketching a Plane Curve given by Parametric Equations (GLEs: 4, 10)
Parametric equations differ from rectangular equations in that they are useful in finding
both the time and positions of an object. By plotting the points in the order of increasing
values of t, one traces the curve in a specific direction. This is called the orientation of
the curve. Using this it is also possible to see if two curves merely intersect or if they
collide.
For each of the problems set up a table such as the one below:
t
x
y
Part I. Fill in the table and sketch the curve given by the following parametric equations.
Describe the orientation of the curve.
1. Given the parametric equations x  1  t and y  t for 0 ≤ t ≤ 10
a. Complete the table
b. Plot the points (x, y) from the table labeling each point with the
parameter value t.
c. Describe the orientation of the curve.
2. Given the parametric equations x  6  t 3 and y  3 ln t 0 < t ≤ 5
a. Complete the table
b. Plot the points (x, y) from the table labeling each point with the parameter
value t.
c. Describe the orientation of the curve.
Advanced Math IUnit 8Conic Sections and Parametric Equations
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Part II. Using technology to graph the curve and then eliminating the parameter
Students will note that eliminating the parameter and rewriting the equations in
rectangular form can change the ranges of x and y. In such cases x and y should be
restricted so that its graph matches the graph of the parametric equations.
For each of the following problems
a) Graph using a graphing utility
b) Eliminate the parameter and write the equation with rectangular coordinates.
c) Answer the following questions
i. For which curves is y a function of x?
ii. What if any restrictions are needed for the two graphs to match?
1. x = 4cost, y = 4sint for 0 ≤ t < 2π
2. x = cost, y = sin2t for 0 ≤ t < 2π
3. x  3 t , y  e t for 0 ≤ t ≤ 10
t
2
4. x  , y =
for 0 ≤ t ≤ 10
2
t -3
Part III: In the rectangular coordinate system the intersection of two curves can be found
either graphically or algebraically. Parametric equations make it possible to distinguish
between an intersection point (the values of t at that point are different for the two curves)
and a collision point (the values of t are the same).
1. Consider two objects in motion over the time interval 0 ≤ t ≤ 2π. The position of the
first object is described by the parametric equations x1  2 cos t and y1  3 sin t . The
position of the second object is described by the parametric equations
x2  1  sin t and cos t  3 . At what times do they collide?
2. Find all intersection points for the pair of curves
x1  t 3  2t 2  t ; y1  t and x2  5t ; y2  t 3 . Indicate which intersection points are true
collision points. Use the interval -5 ≤ t ≤ 5.
Solutions for Activity 5:
Part I:
1.
t 0 1 2
3
4 5
6
7
8
9 10
x 1 0 -1 -2 -3 -4
-5
-6
-7
-8 -9
y 0 1 1.4 1.7 2 2.24 2.25 2.65 2.83 3 3.16
The orientation is from right to left.
2.
t 1 2
3
4
5
x 5 -2
-21 -58 -119
y 0 2.08 3.30 4.16 4.83
The orientation is from right to left.
Advanced Math IUnit 8Conic Sections and Parametric Equations
98
Part II:
1. x2 + y2 = 16
not a function, no
restrictions needed
2. y 2  4 x 2 (1  x 2 )
not a function
no restrictions needed
3.
x2
9
y  e 0 ≤ x ≤ -9.49
1 ≤ y ≤ 22,026.47
4. y 
2
0 ≤ x < 1.5 or 1.5 < x ≤ 5
2x  3
2
2
y   or y >
3
7
Part III:
1. Set x1  x2 and y1  y 2 and solve the resulting system of equations. The object
3
collide when t =
. This occurs at the point (0, -3). Graphing the two parametric
2
equations shows another point of intersection but it is not a point of collision.
2. Point of collision is (0, 0). Other points of intersection: ≈ (-5.22, -1.14) and ≈ (6.89,
2.62)
Advanced Math IUnit 8Conic Sections and Parametric Equations
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Activity 6: Group Problem Solving: Modeling Motion using Parametric Equations
(GLE: 10)
Students have learned how to solve problems dealing with the height of a projectile at
time t in earlier units. Parametric equations describe both the horizontal and vertical path
of a projectile at time t. If a projectile is subject to no other force than gravity then
x(t) = horizontal position of the projectile at time t
y(t) = vertical position of the projectile at time t
The path of the projectile is described by the equations
x  tvo cos
1
y  tvo sin   gt 2  yo
2
where vo is the initial velocity, yo is the initial height, θ is the angle at launch, t is time
and g is the acceleration due to gravity.
Problems:
1. A rocket is launched from ground level with initial velocity of 1200 mph at an angle
of inclination θ = 32o. After 15 seconds how far has the rocket traveled horizontally and
vertically (The acceleration due to gravity is -32 ft/sec2.)
a) Write parametric equations to model the path of the project
b) Graph the path
c) Solve, writing your answers to the nearest hundredth.
2. A baseball player hits a fastball at 146.67 ft/sec (100 mph) from shoulder height (5
feet) at an angle of inclination 15o to the horizontal.
a) Write parametric equations to model the path of the project.
b) A fence 10 feet high is 400 feet away. Does the ball clear the fence?
c) To the nearest tenth of a second when does the ball hit the ground? Where does
it hit?
d) What angle of inclination should the ball be hit to land precisely at the base of
the fence?
e) What angle of inclination should the ball be hit to clear the fence?
3. A bullet is shot at a ten-foot square target 330 feet away. If the bullet is shot at the
height of 4 feet with the initial velocity of 200 ft/sec and an angle of inclination of 8o,
does the bullet reach the target? If so, when does it reach the target and what will be its
height when it hits?
4. A toy rocket is launched with a velocity of 90 ft/sec at an angle of 75o with the
horizontal.
a) Write the parametric equations that model the path of the toy rocket.
b) Find the horizontal and vertical distance of the rocket at t = 2 seconds and t = 3
seconds.
c) Approximately when does the rocket hit the ground? Give your answer to the
nearest tenth of a second.
Advanced Math IUnit 8Conic Sections and Parametric Equations
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Solutions:
1. a) x  1760t cos32o
y  1760t sin 32o -16t 2
(1200 mph is 1760 ft/sec )
b) no
c) x(15) = 22,388 ft.; y(15) = 10,390
2. a) x  146.67t cos15o
y  146.67t sin15o  16t 2  5
b) no
c) 2.5 seconds; 354.18 feet
d) 17.4o (Note: Have the students set the xmax to 400) Change the angle in
increments and trace to the point where the graph touches the x-axis.)
≈ 19.2o (Change the angle in increments until the graph intersects the point (400, 10.1))
3. 5 feet 10 inches at ≈1.67 seconds
4. a) x  90cos 75o t
y  90sin 75o t  16t 2
b) At t = 2 seconds x = 46.59 feet and y = 109.87 At t = 3 seconds x = 69.88 feet and
y = 116.8 feet
c) ≈ 5.4 seconds
Sample Assessments
General Assessments


The student will perform a writing assessment to explain how all of the conic
sections can be obtained by intersecting a double right cone and a plane
Spiral reviews continue with this unit. Type of problems to include are:
o writing the general equations and equations in standard form of each of the
conics
o identifying domain, range, and zeros of functionsFinally they have to once
again work with polar coordinates
o changing from rectangular to polar coordinates and vice versa
Advanced Math IUnit 8Conic Sections and Parametric Equations
101

Group competition: The teacher will make out a set of questions that covers all of
the material in the unit and then will divide that set so that each group gets one
portion. Each group will have a certain number of minutes to work on the
problems and then they must pass them on to the next group. At the end of the
time period, all of the group will hand in their answer sheet, showing work done
on the problems. The number of questions will depend on the number of groups
and the time available for the activity. The scoring rubric will be based on teacher
observation of group interaction and work as well as the number of correct
answers.
Activity-Specific Assessments

Activities 1 and 2: The student will demonstrate proficiency working with
functions that result from conic sections.

Activity 3: The student will demonstrate proficiency in identifying various
conic sections through the given eccentricity or through the general equation.

Activity 5: The student will demonstrate proficiency in sketching curves of
parametric equations.
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