CONTENTS
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Assessment Resources
All Answers Included
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Contents
Teacher’s Guide to Using the Chapter 7
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Lesson 7-5
Anticipation Guide (English) ..............0..ccceeeee 3
Anticipation Guide (Spanish).................ccccceee 4
Parametric Equations
Study Guide and Intervention................. 26
PRACTICE Fitacucd Sit a hes eheccaet Oech ae eeapaie. a 28
Word: Broblem Practices...
teste tates ene 29
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Spreadsheet Activity ......c2..4s1@ise8t--uebes 31
Lesson 7-1
Assessment
' Chapter Resources
Student-Built Glossary..............:cscccssessessssetes 1
Parabolas
Study Guide and Intervention...............ccccccce 5
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Lesson 7-3
Hyperbolas
Study Guide and Intervention...............
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Lesson 7-2
Ellipses and Circles
Study Guide and Intervention....................008
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49
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18
19
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Lesson 7-4
Rotations of Conic Sections
Study Guide and Intervention............... eee 21
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Word Probiem Practice Gaiva7Aeneik. sas 24
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Copyright
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Chapter 7
iii
Glencoe Precalculus
Teacher's Guide to Using the
Chapter 7 Resource Masters
The Chapter 7 Resource Masters includes the core materials needed for Chapter 7. These
materials include worksheets, extensions, and assessment options. The answers for these
pages appear at the back of this booklet.
Chapter Resources
Student-Built Glossary (pages 1—2) These
masters are a student study tool that
presents up to twenty of the key vocabulary
terms from the chapter. Students are to
record definitions and/or examples for each
term. You may suggest that students
highlight or star the terms with which they
are not familiar. Give this to students before
beginning Lesson 7-1. Encourage them to
add these pages to their mathematics study
notebooks. Remind them to complete the
appropriate words as they study each lesson.
Anticipation Guide (pages 3—4) This
master, presented in both English and
Spanish, is a survey used before beginning
the chapter to pinpoint what students may
or may not know about the concepts in the
chapter. Students will revisit this survey
after they complete the chapter to see if
their perceptions have changed.
Lesson Resources
Study Guide and Intervention These
masters provide vocabulary, key concepts,
additional worked-out examples and Guided
Practice exercises to use as a reteaching
activity. It can also be used in conjunction
with the Student Edition as an instructional
tool for students who have been absent.
Practice This master closely follows the
types of problems found in the Exercises
section of the Student Edition and includes
word problems. Use as an additional
practice option or as homework for
second-day teaching of the lesson.
Word Problem Practice This master
includes additional practice in solving word
problems that apply to the concepts of the
lesson. Use as an additional practice or as
homework for second-day teaching of
the lesson.
Enrichment These activities may extend
the concepts of the lesson, offer an historical
or multicultural look at the concepts, or
widen students’ perspectives on the
mathematics they are learning. They are
written for use with all levels of students.
Graphing Calculator, TI-Nspire, or
Spreadsheet Activities These activities
present ways in which technology can be
used with the concepts in some lessons of
this chapter. Use as an alternative approach
to some concepts or as an integral part of
your lesson presentation.
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Chapter 7
Glencoe Precalculus
Assessment Options
Leveled Chapter Tests
The assessment masters in the Chapter 7
Resource Masters offer a wide range of
assessment tools for formative (monitoring)
assessment and summative (final)
e Form 1 contains multiple-choice questions
and is intended for use with below grade
level students.
assessment.
Quizzes Four free-response quizzes offer
assessment at appropriate intervals in
the chapter.
Mid-Chapter Test This one-page test
provides an option to assess the first half of
the chapter. It parallels the timing of the
Mid-Chapter Quiz in the Student Edition
and includes both multiple-choice and
free-response questions.
Vocabulary Test This test is suitable for
all students. It includes a list of vocabulary
words and questions to assess students’
knowledge of those words. This can also be
used in conjunction with one of the leveled
chapter tests.
e Forms 2A and 2B contain multiple-choice
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
e Forms 2C and 2D contain free-response
questions aimed at on grade level
students. These tests are similar in
format to offer comparable testing
situations.
e Form 3 is a free-response test for use
with above grade level students.
All of the above mentioned tests include a
free-response Bonus question.
Extended-Response Test Performance
assessment tasks are suitable for all
students. Sample answers are included
for evaluation.
Standardized Test Practice These
three pages are cumulative in nature. It
includes two parts: multiple-choice questions
with bubble-in answer format and shortanswer free-response questions.
Answers
e The answers for the Anticipation Guide
and Lesson Resources are provided as
reduced pages.
e Full-size answer keys are provided for the
assessment masters.
Glencoe/McGraw-Hiil,
©
Copyright
Inc.
Companies,
McGraw-Hill
The
of
division
a
Chapter 7
Glencoe Precalculus
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Student-Built Glossary
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This is an alphabetical list of the key vocabulary terms you will learn in
_, Chapter 7. As you study the chapter, complete each term’s definition or
description. Remember to add the page number where you found the term. Add
these pages to your Precalculus Study Notebook to review vocabulary at the end
of the chapter.
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of
division
Companies,
Inc.
a
Chapter 7
(continued on the next page)
1
Glencoe Precalculus
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Anticipation Guide
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Conic Sections and Parametric Equations
4)
Before you begin Chapter7
ced)
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Write A or D in the first column OR if you are not sure whether you agree or
disagree, write NS (Not Sure).
1. The graph of y? = 4x is a parabola.
2. The eccentricity of an ellipse is greater than 1.
3. The general form of the equation of a conic section is
y=mx
+ b.
There are formulas to find the coordinates of a point on the
graph of a rotated conic section.
A conic section could be a triangle or a square.
The graph of a degenerate conic is a line or a point.
The transverse axis of a hyperbola is longer than the
conjugate axis.
A circle is a special type of ellipse.
" After you complete Chapter 7
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Reread each statement and complete the last column by entering an A or a D.
e
Did any of your opinions about the statements change from the first column?
e
For those statements that you mark with a D, use a piece of paper to write an
example of why you disagree.
Glencoe/McGraw-Hill,
©
Copyright
division
McGraw-Hill
The
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Inc.
Companies,
a
Chapter 7
3
Glencoe Precalculus
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Lee cada enunciado.
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Decide si estas de acuerdo (A) 0 en desacuerdo (D) con el enunciado.
e Escribe A o D en la primera columna O si no estas seguro(a), escribe NS (no estoy
seguro(a)).
1. La grafica de y? = 4x es una parabola.
he
La forma general de la ecuacién de una secci6n cénica es
y=mx
t+ b.
Existen formulas para calcular las coordenadas de un punto
en la grafica de una seccion cénica rotada.
Una seccion conica puede ser triangular o cuadrada.
La grafica de una conica degenerada es una recta o un punto.
El eje transversal de una hipérbola es mas largo que su eje
conjugado.
Un circulo es un tipo especial de elipse.
"Después de que termines el Capitulo 7
e
Relee cada enunciado y escribe A o D en la ultima columna.
e
Compara la ultima columna con la primera. {Cambiaste de opinion sobre alguno
de los enunciados?
e
En los casos en que hayas estado en desacuerdo con el enunciado, escribe en
una hoja aparte un ejemplo de por qué no estas de acuerdo.
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PERIOD
7 Study Guide and Intervention
Parabolas
Analyze and Graph Parabolas A parabola is the locus of all points in a plane
_ equidistant from a point called the focus and a line called the directrix. The standard
’ form of the equation of a parabola that opens vertically is (x — h)? = 4p(y— k). When p is
negative, the parabola opens downward. When p is positive, it opens upward. The standard
form of the equation of a parabola that opens horizontally is (y — k)? = 4p(x — h). When
Pp is negative, the parabola opens to the left. When p is positive, it opens to the right.
Example
| For 6!_3)? ==_ 12(y ++4), identity the vertex, focus, axis of symmetry,
aol directrix. Then graph the parabola.
The ce
is in standard form and the squared term is x, which means that the parabola
opens vertically. Because 4p = 12, p = 3 and the graph opens upward.
vr
N
S
oO
VW)
Vv)
The equation is in the form (x — h)? = 4p(y — k), soh = 3 andk = —4. Use the values
of h, k, and p to determine the characteristics of the parabola.
vertex:
(3, —4)
(h, k)
directrix:
y=-T7
y=k-p
focus:
(3, -1)
(h, k + p)
axis of symmetry:
x bed
Xe=uh
Graph the vertex, focus, axis, and directrix of the parabola. Then make a table of values to
graph the general shape of the curve.
Exercises
For each equation, identify the vertex, focus, axis of symmetry, and directrix.
Then graph the parabola.
1. (y + 1)? = 8(@ — 3)
2. (x + 2)? = 4(y — 1)
8. (y — 3)? = 2(x — 6)
4. +(x - 3) =(y + 2)
Copyright
Glencoe/McGraw-Hill,
©
Companies,
McGraw-Hill
The
of
division
Inc.
a
Chapter 7
5
Glencoe Precalculus
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‘Study Guide and Intervention
| (continued)
Parabolas
Equations of Parabolas Specific characteristics can be used to
determine the equation of a parabola.
_ Example Arc equation for and graph a parabola with
. focus (—4, —3) and vertex (1, —3).
Because the focus and vertex share the same y-coordinate, the graph is
horizontal. The focus is (h + p, k), so the value of p is —4 — 1 or —5. Because
p is negative, the graph opens to the left.
Write the equation for the parabola in standard form using the
values of h, p, and k.
ye
ly _ (—8)>|
4
4(—5)(x — 1)
(y + 3)? = —20(« — 1)
Standard form
p=-5,h=1,andk=-3
Simplify.
The standard form of the equation is (y + 3)? = —20(x — 1).
Graph the vertex, focus, and parabola.
Exercises
Write an equation for and graph a parabola with the given
characteristics.
1. focus (—1, 5) and vertex (2, 5)
2. focus (1, 4); opens down; contains (—3, 1)
3. directrix y = 6; opens down; vertex (5,3)
4. focus (1.5, 1); opens right; directrix x = 0.5
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NAME
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DATE
.
PERIOD
Identify the vertex, focus, axis of symmetry, and directrix for each
equation. Then graph the parabola.
s
1. (x — 1)? = 8(y — 2)
2. y2+ 6y+9=12—
12x
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Write an equation for and graph a parabola with the given
characteristics.
3. vertex (—2, 4), focus (—2, 3)
4. focus (2, 1); opens right; contains (8, —7)
5. Write x? + 8x = —4y — 8 in standard form. Identify the vertex, focus, axis
of symmetry, and directrix.
Copyright
©
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division
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6. SATELLITE DISH
Suppose the receiver in a parabolic dish antenna is
2 feet from the vertex and is located at the focus. Assume that the vertex
is at the origin and that the dish is pointed upward. Find an equation that
models a cross section of the dish.
Chapter 7
7
Glencoe Precalculus
DATE 2.
NAME
BE ERIOD
’ Word Problem Practice
Parabolas
1. REFLECTOR The figure shows a
parabolic reflecting mirror. A cross
section of the mirror can be modeled by
4. ARCHWAYS The entrance to a college
campus has a parabolic arch above two
columns as shown in the figure.
x” = 16y, where the values of x and y
are measured in inches. Find the
distance from the vertex to the focus
of this mirror.
—_ —
Thee
a. Write an equation that models
the parabola.
~
»
b. Graph the equation.
. T-SHIRTS The cheerleaders at the high
school basketball game launch T-shirts
into the stands after a victory. The
launching device propels the shirts into
the air at an initial velocity of 32 feet per
second. A shirt’s distance y in feet above
the ground after x seconds can be
modeled by y = —16x? + 32x + 5.
a. Write the equation in standard form.
. BRIDGES
b. What is the maximum height that a
T-shirt reaches?
. FLASHLIGHT A flashlight contains a
parabolic mirror with a bulb in the center
as a light source and focus. If the width
of the mirror is 4 inches at the top and
the height to the focus is 0.5 inch, find an
equation of the parabolic cross section.
The cable for a suspension
bridge is in the shape of a parabola.
The vertical supports are shown in
the figure.
dilert
|
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400
a. Write an equation for the
parabolic cable.
b. Find the length of a supporting wire
that is 100 feet from the center.
Chapter 7
Glencoe Precalculus
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Enrichment |
Tilted Parabolas
, ‘The diagram at the right shows a fixed point F'(1, 1)
and a line d that has an equation y = —x — 2. If P(x, y)
satisfies the condition that PD = PF, then P is ona
parabola. Our objective is to find an equation for the
tilted parabola, which is the locus of all points that are
the same distance from (1, 1) and the line y = —x — 2.
First find an equation for the line m through P(x, y)
and perpendicular to line d at D(a, b). Using this
equation and the equation for line d, find the coordinates
(a, b) of point D in terms of x and y. Then use
(PD)? = (PF)? to find an equation for the parabola.
Sy
Ss
c
oO
ma
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—l
Refer to the discussion above.
1. Find an equation for line m.
2. Use the equations for lines m and d to show that the coordinates
pO
of point D are D(a, pepe
hy ay
AS
2 et
3. Use the coordinates of F, P, and D, along with (PD)? = (PF)? to find an
equation of the parabola with focus F and directrix d.
4. a. Every parabola has an axis of symmetry. Find an equation for
the axis of symmetry of the parabola described above. Justify
your answer.
Copyright
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a
b. Use your answer from part a to find the coordinates of the vertex of
the parabola. Justify your answer.
Chapter 7
9
Glencoe Precalculus
NAME
DATE:
-PenioD
Study Guide and Intervention
Ellipses and Circles
Analyze and Graph Ellipses and Circles An ellipse is the locus of
points in a plane such that the sum of the distances from two fixed points,
called foci, is constant.
The standard form of the equation of an ellipse is
(x—a)?
, (y—)?
= 1 when the major axis is horizontal. In this
case, a? is in the denominator of the x term. The standard form is
(y—k)? | («—h)
a
2
2
= 1 when the major
axis is vertical. In this case,
a’ is in the denominator of the y term. In both cases, c? = a? — 6’.
Graph the ellipse given by the equation
Ca),
"d (x'+
oy
2)?
»
The equation is in standard form. Use the values of h, k, a, and 6 to
determine the vertices and axes of the ellipse. Since a? > b?, a? = 25
and b? = 9, ora = 5 and b = 3B. Since a? is the denominator of the
y-term, the major axis is parallel to the y-axis.
orientation:
vertical
center:
(—2, 1)
(h, k)
vertices:
(—2, 6), (—2, —4)
(h, k + a)
co-vertices:
(—5, 1), (4; 1)
(h + b, k)
major axis:
minor axis:
x=-—2
Vien
x=h
y=k
Exercises
Graph the ellipse given by each equation.
(x +5)? | (y +2)?
t
64
25
ih
2.
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Chapter 7
10
Glencoe Precalculus
NAME
BY5 Ssfop
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Study Guide and Intervention
os1)
(continued)
Ellipses and Circles
Determine Types of Conic Sections If you are given the equation
for a conic section, you can determine what type of conic is represented using
‘ the characteristics of the equation. The standard form of an equation for a
circle with center (h, k) and radius r is (x —h)? + (y —k)? =r’.
Write each equation in standard form. Identify the
ated conic.
a. 4x? + Dy? + 24x — 36y + 36 = 0
4x? + Oy? + 24x — 36y + 86 = 0
A(x? + 6x +?) + 9(y? Ae
Original equation
Complete the square.
st: ?) = -36+?+?
(8) =9,(-4)°=
A(x? + 6x + 9) + 9(y? — 4y + 4) = -36 + 36+ 36
Factor.
A(x + 3)? + Iv bee 2)? = 36
(x+ 3), (ve2ye
9
eee
Divide each side by 36.
(x adil 7 (y —k)? = 1, the graph is
Because the equation is of the form
an ellipse with center (—3, 2).
ay
Bs
Ss
c
2]
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cD)
b. x? — 16x — 8y + 80=0
x? — 16x — 8y + 80 =0
Original equation
(x? — 16x + ?) — 8y + 80 =0
Complete the square.
(x2 — 16x + 64) — 8y + 80 - 64 =0
(x =
8)? =
8(y =
2) =
a
(48)"
=64
(0)
Factor.
(x — 8)? = 8(y — 2)
Standard form
Because only one term is squared, the graph is a parabola with vertex (8, 2).
Exercises
Write each equation in standard form. Identify the related conic.
1. y? + 2y + 6x? — 24x = 5
2.y? + Qy + x? — 24x = 14
3. 4x —-8 + y?+ 4y = 0
4.x? + 4x + y? — 2y —-49 =0
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5. 4x? + 8x + 5y? — 30y — 11 =0
Chapter 7
6. 6x? + 24x + 2y — 10 = 0
11
Glencoe Precalculus
|
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Ellipses and Circles
Graph the ellipse given by each equation.
1. 4x? + 9y? — 8x — 36y +4 = 0
2. 25x? + Oy? — 50x — 90y + 25 = 0
Write an equation for the ellipse with each set of characteristics.
» }
3. vertices (—12, 6), (4, 6); foci (—10, 6), (2, 6)
4. foci (—2, 1), (—2, 7); length of major axis 10 units
Write each equation in standard form. Identify the related conic.
5. y? — 4y = 4x + 16
6. 4x? — 32x + 3y? — 18y = —55
7. x7 + y? — 8x — 24y = 9
8. x? + y? + 20x — 10y + 4=0
Determine the eccentricity of the ellipse given by each equation.
(x+1)? | Geely:
i
25
16a
10. yeas
=
64
rb
9
a, 1
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11. CONSTRUCTION A semi-elliptical arch is used to design a headboard
for a bed frame. The headboard will have a height of 2 feet at the center
and a width of 5 feet at the base. Where should the craftsman place the
foci in order to sketch the arch?
Chapter 7
12
Glencoe Precalculus
NAME
DATE
_.
PERIOD
Word Problem Practice
Ellipses and Circles
1. WHISPERING GALLERY A whispering
_
4. RETENTION POND A circular retention
gallery at a museum is in the shape of an
ellipse. The room is 84 feet long and
pond is getting larger by overflowing and
flooding the nearby land at a rate of
46 feet wide.
100 yards per day, as shown below.
a. Write an equation modeling the shape
of the room. Assume that it is centered
at the origin and that the major axis is
horizontal.
b. Find the location of the foci.
2. SIGNS A sign is in the shape of an
ellipse. The eccentricity is 0.60 and the
length is 48 inches.
a. Graph the circle that represents the
water, and find the distance from the
center of the pond to the house.
a. Write an equation for the ellipse if the
center of the sign is at the origin.
ty
Ss
a
oO
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%)
cd)
=al
b. What is the maximum height of
the sign?
3. TUNNEL
The entrance to a tunnel is in
the shape of half an ellipse as shown in
the figure.
b. If the pond continues to overflow at
the same rate, how many days will it
take for the water to reach the house?
a. Write an equation that models
c. Write an equation for the circle of
the ellipse.
water at the current time and an
equation for the circle when the water
reaches the house.
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b. Find the height of the tunnel 10 feet
from the center.
Chapter 7
13
Glencoe Precalculus
NAME
DATE
asco
ee
me
Superellipses
The circle and the ellipse are members of an interesting family
of curves that were first studied by the French physicist and
mathematician Gabriel Lamé (1795-1870). The general equation
for the family is |=" + || = 1, witha 40,640, andn>0.
For even values of n greater than 2, the curves are called
superellipses.
1. Consider two curves that are not superellipses.
Graph each equation on the grid at the right.
State the type of curve produced each time.
x [2
eat
a. |5| +|3[=1
2. In each of the following cases, you are given
values of a, b, and n to use in the general equation.
Write the resulting equation. Then graph.
Sketch each graph on the grid at the right.
adie
20 = ohn = 4
Dade
0 23, = 6:
CH SOS
or S38
3. What shape will the graph of ||"+Fal af
approximate for greater and greater even,
whole-number values of n?
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Chapter 7
14
Glencoe Precalculus
OR
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_) Graphing Calculator Activity
Pep
ht 2A
Translations of Ellipses
t
&
To graph an ellipse, such as
(x 18— 3? "6(yTp
+ 2)
ae 1l,ona
graphing calculator, you must first solve for y.
_
oy
3)2
2)?
ai os
i
|
Original equation
32(x — 3)? + 18(y + 2)? = 576
Multiply each side by 576.
18(y + 2)? = 576 — 32(x — 3)?
be
si
(y + 2)? = Ras
=
yor
athe
2
nae
ae
Subtract 32(x — 3)? from each side.
Divide each side by 18.
2
a)
—2
Take the square root of each side.
The result is two equations. To graph both equations in Y1, replace +
with {1, —1}. Be careful to include the proper parentheses or you will get
an error message.
Gad) (1 2 ©) Gee 0) Ge
6
S30 20 @ 18H B 2 eaea) Coon) «.
o
N
c
BS 82 eee
ie)
Vv)
o
ad
VW
Like other graphs, there are families of ellipses. Changing certain values
in the equation of an ellipse creates a new member of that family.
Exercises
Graph each equation on a graphing calculator.
Gea
0
2) |
Lhergeostcrgy =. b
3.
(Geass
pea lceee
kn
ele
1
(eet)?
Ses 2)
(pea)
TRS
aT ee
oie sige tte
4,
et Ca
1
5. Describe the effects of replacing x — 3 in Exercises 3 and 4 with (x + c) forc > 0.
Copyright
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6. Describe the effects of replacing y + 2 in Exercises 1 and 2 with (y + c) forc > 0.
Chapter 7
15
Glencoe Precalculus
NAME
DATE sence
PERIOD
Hyperbolas
Analyze and Graph Hyperbolas A hyperbola is the locus of all
points in a plane such that the difference of their distances from two foci is
constant. The standard form of the equation of a hyperbola is
=h)?
dae
j
;
Bisel)
— Wiehe
= 1 when the transverse axis is horizontal, and
a?
b?
—k)?
—h?
Y s ) _ a
Tes
= 1 when the transverse axis is vertical. In both
cases, a? + b? =’.
2
Graph the hyperbola given by the equation PRG = ent be
16
The equation is in standard form. Both h and & are 0, so the center is at the
origin. Because the x-term is subtracted, the transverse axis is vertical. Use,
the values of a, b, and c to determine the vertices and foci of the hyperbola.
Because a? = 16 and b? = 4,a = 4 and b = 2. Use the values ofa and 6b to
find the value of c.
ce=a’+
Ca
b?
Equation relating a, b, andc
Ae
a=4andb=2
c = V20 or about 4.47
Simplify.
Determine the characteristics of the hyperbola.
center:
(0, 0)
vertices: (0, 4), (0, —4)
(h, k)
foci:
(0, V20), (0, -V20)
ke)
(h, k + a)
asymptotes:
y = 2x,y = —2x
y—k=+2(—h)
Make a table of values to sketch the hyperbola.
—5.65, 5.65
aaa
Exercises
Graph the hyperbola given by each equation.
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2
lox Oa
ap
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Chapter 7
16
‘
Glencoe Precalculus
Date. oe
6RERIOD
Study Guide and Intervention — coninued
Hyperbolas
Identify Conic Sections You can determine the type of conic when the
equation for the conic is in general form, Ax? + Bxy + Cy? + Dx + Ey + F = 0.
‘The discriminant, or B? — 4AC, can be used to identify a conic when the
equation is in general form.
:
tion
" Use the discriminant to identify each conic section.
a. 2x? + Gy? — 8x + 12y
—-2=0
A is 2, B is 0, and C is 6. Find the discriminant.
B’ — 4AC = 0? — 4(2)(6) or —48
The discriminant is less than 0, so the conic must be either a
circle or an ellipse. Because A # C, the conic is an ellipse.
b. 5x? + 8xy — 2y? + 4x — 3y +10 =0
A is 5, B is 8, and C is —2. Find the discriminant.
B? — 4AC = 8? — 4(5)(—2) or 104.
The discriminant is greater than 0, so the conic is a hyperbola.
c. 12x? + 12xy + 3y? — 7x + 2y —6=0
A is 12, B is 12, and C is 3. Find the discriminant.
B? — 4AC = 12? — 4(12)(8) or 0
The discriminant is 0, so the conic is a parabola.
Exercises
Use the discriminant to identify each conic section.
1. 4x? + 4y? — 2x —-9y +1=0
2. 10x? + 6y? -x+ 8V+1=0
3. —2x? + 6xy + y? — 4x —- By + 2=0
4. x7 + 6xy + y? — 2x +1=0
Copyright
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a
5. 5x? + Qny + 4y? +44
2y+17=0
7. 25x? + 100x — 54y = —200
Chapter 7
6.x? + 2xny+y?+x+10=0
— 54y? = —100
8. 16x? + 100x
17
Glencoe Precalculus
NAME
0).| enemas
ree comowree
RAL DNASE
SANT
Practice
f=) (312)
AARNE
Hyperbolas
Graph the hyperbola given by each equation.
1. x? — 4y? — dx + 24y — 36 = 0
2
fi Bagh
soe S|
Write an equation for the hyperbola with the given characteristics.
3. vertices (—10, 6), (4, 6);
4. foci (0, 6), (0, —4); length of transverse
foci (—12, 6), (6, 6)
axis 8 units
5. Determine the eccentricity of the hyperbola given by the equation
7)?
(ital
Olle l
36
lea
os
6. ENVIRONMENTAL NOISE Two neighbors who live one mile apart hear
an explosion while they are talking on the telephone. One neighbor hears
the explosion two seconds before the other. If sound travels at 1100 feet
per second, determine the equation of the hyperbola on which the
explosion was located.
Use the discriminant to identify each conic section.
7. 5x? + xy + 2y? — 5x + BV +9=0
8. 16x? — 4y? — 8x — By +1=0
9. 4x? + 8xy + 4y? +x+1ly+10=0
10. 2x? + 4y? — 8x —-6v +2 =0
Chapter 7
18
Glencoe Precalculus
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NAME
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PERIOD
NESE:
‘Word Problem Practice
Hyperbolas
1. EARTHQUAKES The epicenter of an
earthquake lies on a branch of the .
hyperbola represented by
(x-—50)? — (y — 35)?
1600
as
2500.
2
1¢ where
3. PARKS A grassy play area is in the shape
of a hyperbola, as shown.
the
seismographs are located at the foci.
a. Graph the hyperbola.
a. Write an equation that models the
curved sides of the play area.
b. If each unit on the coordinate plane
represents 3 feet, what is the
narrowest width of the play area?
b. Find the locations of the seismographs.
2. SHADOWS A lamp projects light onto
a wall in the shape of a hyperbola.
The edge of the light can be modeled
We
hey
4, SHADOWS The path of the shadow
cost by the tip of a sundial is usually
a hyperbola.
.
ie
ise alate
a. Write two equations of the hyperbola
in standard form if the center is at the
origin, given that the path that
a. Graph the hyperbola.
contains a transverse axis of
24 millimeters with one focus
14 millimeters from the center.
b. Graph one hyperbola.
b. Write the equations of the asymptotes.
Copyright
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c. Find the eccentricity.
Chapter 7
19
Glencoe Precalculus
7-3
Less
NAME
DATE
SSE
NNER ESE
LAG
LT SANTA
Enrichment
PERIOD
AEE
Moving Foci
Recall that the equation of a hyperbola with center
at the origin and horizontal transverse axis has the
2
2
equation = - = = 1. The foci are at (—c, 0) and
(c, 0), where c? = a? + Db’, the vertices are at (—a, 0)
and (a, 0), and the asymptotes have equations
Siete
Lex. Such a hyperbola is shown at the right.
What happens to the shape of the graph as c grows
very large or very small?
Refer to the hyperbola described above.
1. Write a convincing argument to show that as c approaches 0,
the foci, the vertices, and the center of the hyperbola become
the same point.
2. Use a graphing calculator or computer to graph x? — y? = 1,
x” — y?=0.1, and x? — y? = 0.01. (Such hyperbolas correspond
to smaller and smaller values of c.) Describe the changes in the graphs. What shape do the graphs approach as c approaches 0?
3. Suppose a is held fixed and c approaches infinity. How does the
graph of the hyperbola change?
4, Suppose b is held fixed and c approaches infinity. How does the
graph of the hyperbola change?
Chapter 7
20
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NAME
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Study Guide and Intervention
LGD
LOSE LAIDBHii
Rotations of Conic Sections
Rotation of Conic Sections An equation Ax? + Bxy + Cy? + Dx + Ey +F =0
_im the xy-plane can be rewritten as A(x’)? + C(y’)? + Dx’ + Ey’ + F = 0 in the x’y’-plane
“ by rotating the coordinate axes through an angle 0. The equation in the x’y’-plane can be
found using x = x’ cos 6 — y’sin
@and y = x’ sin 0 + y’ cos 6.
ote
ton
Use 0 = 45° to write the standard form of
x? — 2xy — Ay? + $ = 0 in the x’y’-plane. Then identify the conic.
The conic is a hyperbola because B?— 4AC > 0. Find the equations for x and y.
x =x’ cos 0—y’sin 0
2a
Peas
Bee»
oun)
Rotation equations forxand y
See
y =x’ sin 0+ y¥’ cos 6
eae
2h ety BgOhoene he
sin 7 =—5- and'€os =
ae
Substitute into the original equation.
xe
(V2
5
— Qxy
— 4y?
+5 =)
A/D4’ i 2 V 2x! = V2y' \V2x' + V2y' |= | V2x' + V2y’
5
9
2
|+3 = 0
Replace x and y.
Kael)? = x'y’ + 309)? — (2'P + (9)? — 2(e'? — 4x'y’ — 20)? + 4 = 0 expand the
binomials.
em 5
INO
5 (x’)?
— 5x'yVN,
x
9 (y’)?\2 + thy
9 = 0
.
5
Simplify.
The equation of the hyperbola after the 45° rotation is 5(x’)? + 10x’y’+ (y’)? = 1.
Exercises
a
a
Write the standard form of each equation in the x’y’-plane for the
given value of 0. Then identify the conic.
1.x? - 4x +y?=2,0=7
\e)
a
2. 8x2 — By? = 40, 6 = 30°
al
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Copyright
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a
Chapter 7
21
Glencoe Precalculus
NAN
2)
or
8
ee
ee
Study Guide and Intervention
ee
(coninuea)
Rotations of Conic Sections
Graph Rotated Conics
When the equations of rotated conics are given
for the x’y’-plane, they can be graphed by finding points on the graph of the
conic and then converting these points to the xy-plane.
Graph
ph = +HOES)
= 1 if it has been rotated 30° from
its position in the epee
The equation represents an ellipse, and it is in standard form. Use the
center (0, 0); the vertices (0, 8), (0, —8); and the co-vertices (6, 0), (—6, 0)
to determine the center and vertices for the ellipse in the xy-plane.
Find the equations for x and y for 0 = 30°.
x =x’ cos 6— y’ sin 0
V3,
erred: ~sy'
a
Rotation equations forx and y
:
{
y=
V3
x’ sin 0+ yy’ cos 0
EY,
sin 30° = = and cos 30° = —>-
V3
way? LEGS
/
Use the equations to convert the x’y’-coordinates of the center mo?
xy-coordinates.
x =
3
¥3.(0) - +(0)
=.)
x’=Oandy’=0
y =
Multiply.
V3
(0) arte OY,
=)
Likewise, convert the coordinates of the vertices and co-vertices
into xy-coordinates.
(Ops Wwas( 449/38) 4(0,.8) =>. (4,543) 2(670)s—> (8:35.83)
(26, 0) > (—3'V3, —3)
The new center, vertices, and co-vertices can be used to sketch the
graph of the ellipse in the xy-plane.
Exercises
Graph each equation at the indicated angle.
we _ OF 4 60:
N\2
1. (x’ + 6)? = 12(y’ + 2); a
2.
N2
9
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Chapter 7
22
Glencoe Precalculus
Oi ree A.
eee
ere
DATE... _. PERIOD
| Practice ent
Rotations of Conic Sections
Write the standard form of each equation in the x’y’-plane for the
given value of 6. Then identify the conic.
1. xy =1,0=7
2. 5x? + 6y? = 30; 6 = 30°
Write an equation for each conic in the xy-plane for the given
equation in x’y’ form and the given value of 0.
8. (x’)? = 16(y’); 6 = 45°
Ne A4 Eg 7St
7 Nee
25
(se)
Determine the angle 0 through which the conic with each equation
should be rotated. Then write the equation in standard form.
5.2 + xy +y?=2
6. 18x? — 8xy + Ty? — 45 = 0
7. 16x? — 24xy + Dy? — 30x — 40y = 0
8. 18x? + 12xy + 13y? — 198 = 0
9. COMMUNICATIONS Suppose the orientation of a satellite
dish that monitors radio waves is modeled by the equation
Ax? + Ixy + 4y? + V2x — V2y = 0. What is the angle of
rotation of the satellite dish about the origin?
10. Graph (x’ + 1)? = —16(y’ + 3) if it has been rotated 45° from its position
7-4
Less
in the xy-plane.
ly
Copyright
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division
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Chapter 7
23
Glencoe Precalculus
NAMB: ey
ee
DA
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‘Word Problem Practice
Rotations of Conic Sections
1. COMMUNICATIONS A satellite dish is
3. ART Mimi is drawing a picture with
rotated parabolas. She wants to graph
(x’ — 3)? = 12(y’ — 4) if it has been
rotated 45° from the xy-plane.
modeled by the equation y = ax
when it is directly overhead. Later in the
day, the dish has rotated 60°.
a. Find the vertex in the xy-plane.
a. Write an equation that models the
new orientation of the satellite dish.
b. Find the equation for the axis of
symmetry in the xy-plane.
.
b. Draw the graph.
s
:
c. Draw the graph in the xy-plane.
2. GEARS Suppose the equation of
an elliptical gear rotated 60° in the
as,
pur
(&.)4
‘3
4, LOGIC A hyperbola has been rotated
40° clockwise. Through what angle
sg SMR: F
164120 anak
must it be rotated to return it to its
a. Write an equation for the ellipse
formed by the gear in the xy-plane.
original position?
b. Draw the graph.
5. SHAPES The shape of a reflecting
mirror in a telescope can be modeled by
25x? + 13xy + 2y? = 100. Determine
whether the reflector is elliptical,
parabolic, or hyperbolic.
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Chapter 7
24
Glencoe Precalculus
DATE: sax
Enrichment
PERIOD
Graphing with Addition of y-Coordinates
.
Equations of parabolas, ellipses, and hyperbolas
that are “tipped” with respect to the x- and y-axes
are more difficult to graph than the equations you
have been studying.
y
|
[14
i"
| | |B
Often, however, you can use the graphs of two
simpler equations to graph a more complicated
equation. For example, the graph of the ellipse in
the diagram at the right is obtained by adding the
y-coordinate of each point on the circle and the
y-coordinate of the corresponding point of the line.
¢
+
;
eA
++
ne
7
Were
Graph each equation. State the type of curve for each graph.
ly=6-x+4+V4-27
2
=
a
HELA
iy
7-4
Less
Copyright
Glencoe/McGraw-Hill,
©
division
Companies,
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The
of
Inc.
a
Use a separate sheet of graph paper to graph these equations.
State the type of curve for each graph.
Bi y= 2% tW.7 + 6x — x”
Chapter 7
4,.y=—2x+
25
V-2x
Glencoe Precalculus
NAME. a
re
DA
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Study Guide and Intervention
Parametric Equations
Graph Parametric Equations
Parametric equations are used to
describe the horizontal and vertical components of an equation. Parameters
are arbitrary values, usually time or angle measurement.
» Example 1 ' Sketch the curve given by the parametric equations
x=-3+ 4tand y=? + 3 over the interval —4 <t <4.
Make a table of values for —4 <¢t < 4.
Plot the (x, y) coordinates for each t-value and
connect the points to form a smooth curve.
y
T Write x = 4t — 2and y = # + 1 in rectangular form.
Parametric equation for x
Ss; Z =A
—
Solve for t.
x+2
| ri
:
|—
__ x7+4x4+4
ic
2
Substitute X42 for t in the equation for y.
x42
Square “7.
2
BHM ar Mors
IG
4
4
impli
Simplify.
miss
xe
The rectangular equation is y = eta
pas
Foal
a
Exercises
Sketch the curve given by each pair of parametric equations over
the given interval.
l.x=#+4andy=4-3,-4<¢<4
2.x=Landy
at
oo
=Vi+2;0<t<8
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26
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NAME
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-_ Study Gu
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Interventio
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4 37 )P)
(continued)
Parametric Equations
Projectile Motion
.
Parametric equations are often used to simulate
projectile motion. For an object launched at an angle @ with the horizontal
at an initial velocity v,, where g is the gravitational constant, t is time, and
h, is the initial height, the horizontal distance x can be found by x = tu, cos 8
;
anh
:
1
and the vertical position y by y = tu, sin 6 — 3 yen
i
>
Tigi isAraceae sa soccer ball: He kicks the ball with
an aatial velocity of 35 feet per second at an angle of 48° with the
horizontal. The ball is 2 feet above the ground when he kicks it. How
far will the ball travel horizontally before it hits the ground?
Step 1 Make a diagram of the situation.
Q”
aa
~
Step 2 Write a parametric equation for the vertical position of the ball.
y=
tv, sin 6 — Sgt? = h,
= t(35) sin (48) — $(82)¢ +2
Step 3
Parametric equation for vertical position
v, = 35, 6 = 48°, g = 32, andh, = 2
Graph the equation for the vertical position and the line y = 0. Use
5: INTERSECT on the CALC menu of a calculator to find the point of
intersection of the curve with y = 0. The value is about 1.7 seconds.
You could also use 2: ZERO and not graph y = 0.
Step 4 Determine the horizontal position of the ball at 1.7 seconds.
x=
tv, cos
8
Parametric equation for horizontal position
= 1.7(85) cos 48
~—-v, = 85, 0 = 48°, andt = 1.7
=~ 39.8
Use a calculator.
The ball will travel about 39.8 feet before it hits the ground.
Exercises
1. Julie is throwing a ball at an initial velocity of 28 feet per second and an
angle of 56° with the horizontal from a height of 4 feet. How far away will
the ball land?
Copyright
©
Glencoe/McGraw-Hill,
McGraw-Hill
The
of
division
Companies,
Inc.
a
2.
Jerome hits a tennis ball at an initial velocity of 38 feet per second and an
7-5
Less
angle of 42° with the horizontal from a height of 1.5 feet. How far away
will the ball land if it is not hit by his opponent?
Chapter 7
27
Glencoe Precalculus
UAE
ca
a
LALLA
’ Practice
ciety AT
rs eee
RLS
Parametric Equations
Sketch the curve given by each pair of parametric equations over
the given interval.
l.x=f+landy=5—6,-Sistz5
2.0=2+6andy=—4;-5<t<5
Write each pair of parametric equations in rectangular form.
SB.
=i 2b-+-38) y= t 4
4.x=t+5,y
= —3??
oo
= 3 sill o, y = 2 cos'@
6. y = 4sin 0,x = 5 cos 0
eu
Use each parameter to write the parametric equations that can
represent each equation. Then graph the equation, indicating the
speed and orientation.
7.t =2=*
fory
3 — x
=2=*
9. MODEL ROCKETRY
8.¢=4«¢ —1lfory =x?+2
Manuel launches a toy rocket from ground level
with an initial velocity of 80 feet per second at an angle of 80° with
the horizontal.
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a. Write parametric equations to represent the path of the rocket.
b. How long will it take the rocket to travel 10 feet horizontally from its
starting point? What will be its vertical distance at that point?
Chapter 7
28
Glencoe Precalculus
Dna
ee
Ta
LC
DATE.. ——
~PPERIOD
Word Problem Practice
Parametric Equations
1. PHYSICS A rock is thrown at an initial
5. GOLF Julio hit a golf ball with an
velocity of 5 meters per second at an
initial velocity of 100 feet per second at
angle of 8° with the ground. After |
an angle of 39° with the horizontal.
0.4 second, how far has the rock traveled
horizontally?
a. Write parametric equations for the
flight of the ball.
2. PLAYING CATCH Tom and Sarah are
b. Find the maximum height the ball
playing catch. Tom tosses a ball to Sarah
rearhon,
at an initial velocity of 38 feet per second
at an angle of 28° from a height of 4 feet.
Sarah is 40 feet away from Tom.
a. How high above the ground will the
6. BASEBALL
ball be when it gets to Sarah?
Micah hit a baseball at an
initial velocity of 120 feet per second
from a height of 3 feet at an angle of 34°.
b. What is the maximum height
of the ball?
3. TENNIS Melinda hits a tennis ball with
an initial velocity of 42 feet per second at
an angle of 16° with the horizontal from
a height of 2 feet. She is 20 feet from the
net and the net is 3 feet high. Will the
a. How far will the ball travel
ball go over the net?
horizontally before it hits the
ground?
b. What is the maximum height the
ball will reach?
4. BASKETBALL Mandy throws a
basketball with an initial velocity of
28 feet per second at an angle of 60° with
the horizontal. If Mandy releases the ball
from a height of 5 feet, write a pair of
equations to determine the vertical and
horizontal positions of the ball.
c. If the fence is 8 feet tall and 400 feet
from home plate, will the ball clear
the fence to be a home run? Explain.
fh
N
=
Copyright
Glencoe/McGraw-Hill,
©
Companies,
McGraw-Hill
The
of
division
Inc.
a
2)
Ww
vi
rf
a |
Chapter 7
29
Glencoe Precalculus
NAME
DATE 2a
‘Enrichment
ene
AAAS
Coordinate Equations of Projectiles
The path of a projectile after it is launched is a parabola when graphed
on a coordinate plane.
The path assumes that gravity is the only force acting on the
projectile.
The equation of the path of a projectile on the coordinate plane is
given by
y
2 + (tan
Bs? &GO }x
iy) |
eee
t a)x,
where g is the acceleration due to gravity, 9.8 m/s? or 32 ft/s”,
U, 1s the initial velocity, and a is the angle at which the
projectile is fired.
Write the eaetan neea prjccuie fired at an angle of
10° to ce horizontal with an initial velocity of 120 m/s.
Pore
9.8
eZ”
oo
tan
eT
10°
y = —0.00035x? + 0.18x
Find the equation of the path of each projectile.
1. a projectile fired at 80° to the
2. a projectile fired at 40° to the
horizontal with an initial velocity
of 200 ft/s
Chapter 7
horizontal with an initial velocity
of 150 m/s
30
Glencoe Precalculus
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NAME
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DILLER
DATE
—
PERIOD
TS
Parametric Equations
: You have learned that the motion of orbiting planets can be modeled
using parametric equations. A spreadsheet can be used to evaluate
parametric equations and to provide a graph of a planet’s orbit.
Use the parametric equations for Saturn’s
position in its orbit in a spreadsheet. To
calculate the x position, enter the following
formula in B2: = 9.5*COS(PI()15*A2). To
Ds | 2sds661 |o.0te0i6
Ta |-o9ecce
|9.426000
calculate the y position, enter the
formula = 9.48*SIN(PI()15*A2) in C2. Use
the fill handle to paste these formulas into
the remaining cells in columns B and C.
In the spreadsheet, the values in column A
represent t, and the values in columns B and C
represent the calculated x and y values of Saturn’s
position at time ¢.
Exercises
1. Complete the spreadsheet to find the position of Saturn every
2 years from t = 2 to t = 32. Use the spreadsheet to make an
X-Y graph of Saturn’s position.
2. Find the coordinates representing Saturn’s position after
23 years.
3. Given the parametric equations x = 6 sin 3t and y = —4 cos f¢, write the
formulas to put into your spreadsheet if time is found in cell A2.
Copyright
Glencoe/McGraw-Hill,
©
division
McGraw-Hill
The
of
Companies,
Inc.
a
7-5
Less
Chapter 7
31
Glencoe Precalculus
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NAME
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6)
—
PERIOD
SCORE
(Lessons 7-1 and 7-2)
1. Write 16x? + 4y? — 96x + 8y + 84 = 0 in standard form.
Identify the related conic.
; 2. Graph the ellipse given by ee
+ Es Ls ail
64
100
1.
2
‘
:
~
3. Identify the vertex, focus, axis of symmetry, and directrix
for x? — 4x + 8y + 12 =0.
.)
eVv
Ab
%)
cd)
mA
%)
4. Write an equation for the ellipse with vertices
(—8, 5), (4, 5) and foci (—7, 5), (3, 5).
5. MULTIPLE CHOICE
Write an equation for the parabola with
vertex (0, 0) and directrix x = —8.
Asy2=.—32x,.
-Buy? = —16x..
.C. y? =1S8x
we
w we
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ee
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ee
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Pero
SCORE
G
S
5
(Lesson 7-3)
Soe
1. Graph the hyperbola given by pee. _ oe
2
= I
1.
cs
&
=
CG)
F
2
q
=
G
2
:
:
&
8
2. Write an equation for the hyperbola with vertices
(—4, —1), (2, —1) and foci (—5, —1), (3, —1).
3. Find the ee
(x + 3)?
ogee
V8)
aot
of the ellipse given by
ls
3.
4, Use the discriminant to identify the conic given by
24x? — 4y? — 150x — 16y = —109.
5. MULTIPLE CHOICE
2.
4.
Write an equation for the hyperbola with
vertices (0, —6), (0, 6) and foci (0, —9), (0, 9).
y
Men
a
a6
46
Cane
B
Ne pa
D
iy?
ree
Chapter 7
tees
a5
Via eae
36
rule
33
5.
Glencoe Precalculus
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i 55 |)
rn Oe moenemmone BM: 7 OMe
PERIOD
48 7.12
hapter 7 Quiz 3
ELLE
TNO
EE
ONES
SCORE
(Lesson 7-4)
1. Identify the conic x? + xy — y? = 6. Write the standard
form of the equation in the x’y’-plane for the given value
of d=.
4
2. Identify the conic 9x? + 4y? = 36. Write the standard
form of the equation in the x’y’-plane for the given value
of 6. =:30:.
3. Determine the angle @ through which the conic given by
8x? + 8xy — Ty? = 15 should be rotated.
4, Write an equation in the xy-plane for the conic given by
N\2
a = we = 1 in «2 form and for @= 60;
5. MULTIPLE CHOICE
A
Identify the conic xy = —8.
circle
C ellipse
B_ parabola
ee
ee
ee
ee
ew ee
ee
ee
ee
ee
D
ew
ee
ee
ee
ee
ew ee
ee
eB ee
ew ee
ee
hyperbola
Be em ee
ee
ee
Be ee
ee
ee
ee
ee
ee
wee
ee
ee
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ee
em em em em ee
em ee
ee
ee
ee
ee
eK
PERIOD
Chapter 7Quiz 4
SCORE
(Lesson 7-5)
1. Sketch the curve given by x = —3t? and y = 2t, -2 <t < 2.
2. Write
x = 5 cos 0 and y = 7 sin @ in rectangular form.
3. MULTIPLE CHOICE
rectangular form.
A y=(x+5)?
oo
Rewrite y = #? + 4 and x = 2t — 5in
C y= 4x? — 20x + 29
2
D
aca)
2
While positioned 25 yards directly in front of the goalposts,
Bill kicks the football with an initial velocity of 65 feet per
second at an angle of 35° with the ground.
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4, Write the position of the football as a pair of parametric
equations. If the crossbar is 10 feet above the ground,
does Bill’s team score?
5. What is the elapsed time from the moment the football is
kicked to the time the ball hits the ground?
Chapter 7
34
Glencoe Precalculus
Chap
7Mitd-eChaprter Test
NAME
DATE
PERIOD
ow
(Lessons 7-1 through 7-3)
| Part I| Write the letter for the correct answer in the blank at the right of each question.
, 1. Which is the graph of (y — 6)? = 4(% — 2)?
A
B
y
i
8
4
O
4 | 8 | 12x
‘4
=
2. Which is the equation for an ellipse with vertices (—3, 4), (11, 4)
and foci (—i, 4), (9, 4)?
F
(x —
Fi
4)?
Ce
oes
AR
(
—
ALE
4)’
aes
1
H
Cee eet
<
sqit ey =e
toe
y?
(e- 4)?
a.
25may
n7)
oY
vn
= 4)’
Say ancmpareay Tsse
ines
2
3. Which are the equations of the asymptotes of “+ — a = beg
4jpo6.5
Arye
CyAsh
sous
Byaree
D
86:
95 &
425
¥>= eae
3.
4, Use the discriminant to identify the conic given by
Ax? ++ 10x + 2v+8=0.
F
ellipse
G
hyperbola
H_ parabola
J circle
5. Write an equation for the hyperbola with vertices
(2, 5), (2, —3) and conjugate axis of length 10.
eS
6. The entrance to a zoo is in the shape of a parabola. Write
~ an equation for the parabola if the point of origin is the
center of the figure.
6.
4.
Copyright
Glencoe/McGraw-Hill,
©
division
Companies,
McGraw-Hill
The
of
Inc.
a
= 6)
1)?
7. Graph the hyperbola given by sean ~e
Chapter 7
35
= bP
ce
Glencoe Precalculus
‘Cha7pVtocaebulrary Test
NAME) ae
ach
axis of symmetry
conic section
conjugate axis
co-vertices
degenerate conic
directrix
eccentricity
ellipse
focus (foci)
hyperbola
locus
major axis
minor axis
parabola
eo
parameter
parametric equation
parametric form
rectangular form
transverse axis
vertex (vertices)
State whether each statement is true or false. If false, replace the
underlined term or expression to make a true sentence.
1. A parabola is the set of all points in a plane equidistant from
a point and a line in the same plane.
1.
2. In an ellipse, the relationship between the values a, b, and c
is given by the equation a? + b? = c’?.
=
Z,
3. The length of the major axis of an ellipse is greater than
the length of the minor axis.
3.
4, The eccentricity of a hyperbola is less than one.
4,
5. The foci of a hyperbola are on the conjugate axis.
5.
6. An equation for a graph written in rectangular form uses
the variable t or the angle 0.
6.
7. The intersection of a double-napped right cone and a plane
is a conic section.
7.
8. The axis of symmetry of a parabola is a line through the
focus, parallel to the directrix.
8.
9. The equation of a line tangent to a parabola can be found
using an isosceles triangle.
3:
10. ae slopes of the asymptotes of a vertical hyperbola are
Ing
10.
Define each term in your own words.
11. hyperbola
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Chapter 7
36
Glencoe Precalculus
NAME
DATE
:
PERIOD
LLOYD RASPY SORORITIES
Chapter 7 Test, Form 1
SCORE
Write the letter for the correct answer in the blank at the right of each question.
For Questions 1-3, refer to the ellipse represented
2
by ————
Cra Ay a
= 1,
“1. Find the coordinates of the center.
As (1,.2)
B
(1, —2)
C
(-1, 2)
D
(-2, 1)
Loess
2. Find the coordinates of the foci.
Be (tet, 7, —2)
H (5, —2), (-3, —2)
Ga (4,
oP ly Ayes)
2/7)
2.
3. Find the coordinates of the vertices and co-vertices.
5
A
(1, 2), (1, -6), (4, -2), (—2, —2)
C
(5, —2), (—3, —2), (1, 1), (1, —5)
B
(4, 2), (—2, 2), (1, 1), (1, -5)
D
(5, —2), (—3, —2), (1, 2), (1, -6)
So
4, Determine the angle of rotation 0 through which the conic given by
4x? — Axy + y? = 4 should be rotated.
Ke.
G
63°
Ha 7
J
153°
Ay ee
5. Write the pair of parametric equations x = 5 cos 0 and y = —sin 0
in rectangular form.
ke
ee
LeNorcnd
gene
B=ame
teak
C
L+y'=1
D Lae
=
by a lb
On
J circle
Ghee
6. Use the discriminant to identify the conic section
Oy? + 4x? — 108y + 24x = —144.
F parabola
G
hyperbola
H
ellipse
7. Write the standard form of the equation y? — x? = 5 in the x’y’-plane
after a rotation of 45”.
A x'y’ = —2.5
C) iy’)
B xy’=—-5
D
Oa) a2
(x)? = 2.5y’
yfd geestapeuees
8. Marcy hit a golf ball with initial velocity of 130 feet per second at an
angle of 28° with the ground. Write parametric equations to represent
this situation.
Bevel 30r cos 265
y = 130¢ sin 28° — 16¢?
H x = 28 cos 130°
y = 28t sin 130° — 16¢?
G x = 130t cos 28° — 16¢?
J x = 28 cos 130¢°
y = 130¢ cos 28°
y = 28 cos 1308”
8, ues Be
Glencoe/McGraw-Hill,
©
Copyright
Companies,
McGraw-Hill
The
of
division
Inc.
a
9. A cross section of the reflector shown is in the shape
of a parabola. Which of the following is an equation
for the cross section?
A y?= 4x
B y? = 8x
Chapter 7
C
x2 =4y
D
x7 = 8y
37
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tia:
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2 in.
beppdaceca
Glencoe Precalculus
2
7%
an
NAME
DATE ee Oe
l Chapter | 7 Test, Form
1
(continued)
2
For Questions 10 and 11, refer to the hyperbola represented by = ~
10. Write the equations of the asymptotes.
Poy
3 + 2x
1
Gy=+5%
—s.)
ks
y=4+v2x
dhad tSendCoy
mer
105,77" >
C (+Vv2, 0)
D (+V6, 0)
ihease!
H
11. Find the coordinates of the foci.
A (0, +V2)
B (0, + V6)
12. Write the standard form of the equation of the hyperbola for which
the transverse axis is 4 units long and vertical and the conjugate
axis is 3 units long.
Cee
ee
Vt
G@So
a a ee
~2—=1
Ges) a1
- Se
H CsA
= enn ee
Leoa 1
OS
J G1Soe a1
ig
12.
;
13. Darius serves a volleyball with an initial velocity of 34 feet per second 4 feet
above the ground at an angle of 35°. What is the maximum height, reached
after about 0.61 second?
A
2.14 ft
B
9.94 ft
C
5.94 ft
D
6.14 ft
13a
14. Write the standard form of the equation of the parabola with
directrix at x = —1 and with focus at (5, —2).
F (y +2)?
= 12(x + 2)
H x+2=4(y
+2)?
G y—2=12(x +2)?
J x-2=4
(y+ 2)?
iyBee
15. Identify the graph of the equation 16x? — 24xy + 9y? — 30x — 40y = 0.
A hyperbola
B
ellipse
C
parabola
D
circle
15.
D
parabola
ue
16. Which graph represents a curve with parametric equations
x=t-—4andy =?’ over the interval -3 < t < 3?
F
7G
x
H
O
17. Identify the conic that may have an eccentricity of 2.
A
circle
B
ellipse
C
hyperbola
18. Write the standard form of the equation of the circle with center at (2, —7)
and radius 5.
F (x —2)?+(y +7)’ = 25
H («—2)’+(y+7)’=16
G («@-2)? + (y+7)?=5
Bonus’
J (x +2)? + (y —7)? = 25
18. —__—
Write the equation of the line tangent to y = x”
through (2, 4).
Chapter 7
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38
Glencoe Precalculus
NAME
Chapter 7 Test, Form 2A
.
DATE
LLL
DEE ERR
PERIOD
so
MEE LLEE
Write the letter for the correct answer in the blank at the right of each question.
.
_-‘For Questions 1-3, refer to the ellipse represented by the equation
(x — 3)?
ae
oe
+(y — 2)? =1.
1. Find the coordinates of the center.
A
(2, 3)
B
(3, 2)
C
(-38, —2)
D
(-2, —3)
|Pl,
ee
2. Find the coordinates of the foci.
F @,2+2V6)
“G@*(—2) 2)p(8)2)°
Hh (ae ever2)?
a" (@ Paves)
~
—
cD)
2
&
Vv)
Vv
3. Find the coordinates of the vertices and co-vertices.
eS;
225
2) as 3), (3, 1)
VY
C
(4,2), (2, 2)s(3.5),.co5L)
D
(4, 2), (2, 2), (3, NG (3, —3)
Vi
7)
<x
B
(8, 2), (—2, 2), (3, a); (3 ? —3)
SIGS
es
4. Determine the angle of rotation @ through which the conic
given by equation 2x? + 3xy + y? = 1 should be rotated.
F
-9°
G
36°
H
—36°
J 324°
5. Write the pair of parametric equations x = 3 cos 8 and y = sin 0
in rectangular form.
Re) yal
Bie9 erat
CO aka
1
3
4, ses
Dy Ree
3
ees
J circle
6.
6. Use the discriminant to identify the conic section
By? — ox
F
12y + 18x ="42:
parabola
G
hyperbola
H
ellipse
7. Write the standard form of the equation y? — x? = 2 in the x’y’ plane
after a rotation of 45°.
A x'y’=-1
B x’y’=—-2
Cy)
er
2 Dr (a) aay
rp
8. Randall hit a golf ball with initial velocity of 120 feet per second at
an angle of 38° with the ground. Write parametric equations
to represent this situation.
F
x = 120¢ cos 38° — 16??
4 = 120% sin’38"
G
H
x = 38¢ cos 120°
y = 38t sin 120° — 16¢?
J x = 38 cos 120¢
x = 120¢ cos 38°
y = 120¢ sin 38° — 16??
8.
y = 38 cos 120¢°
Copyright
Glencoe/McGraw-Hill,
©division
Companies,
McGraw-Hill
The
of
Inc.
a
9. A cross section of the reflector shown is in the
shape of a parabola. Write an equation for the
cross section.
A y?=4%x
Cx" = 4y
By? = 8x
D x’ = 8y
Chapter 7
.
oo
2 om
3. ——
39
Glencoe Precalculus
NAME
DA TE tect
AALS
ee
PR RIOD
AANA
: ‘Chapter 7 Test, Form 2A
(continued)
2
For Questions 10 and 11, refer to the hyperbola represented by oa
2
—x=1.
10. Write the equations of the asymptotes.
F y-1=
Gavi
+6(« — 2)
H y+2=
+6 — 1)
6x
J
yt+2= +6x
LOSces5
11. Find the coordinates of the foci.
Ap len) 3702 | eBoy 97? a OR
Gee OT?
eke)
37
ee
12. Write the standard form of the equation of the hyperbola for which
a = 2, the transverse axis is vertical, and the equations of the
asymptotes are y = + 2x.
Ne
F pe
ae
Oe
G ee Bie
o
H ery? ae
y*
J rey
Sel
PA
ee 2ome
13. Aaron kicked a soccer ball with an initial velocity of 39 feet per second
at an angle of 44° with the horizontal. After 0.9 second, how far has the
ball traveled horizontally?
A
24.4 ft
B
12.3 ft
C
11.4 ft
D
25.2 ft
|Rs yoraed Ca
14. Write the standard form of the equation of the parabola with
directrix at y = —4 and focus at (2, 2).
Fi (y=2)?
= 124.2)
G y+1=12(x — 2)’
H (& + 2)°=
12(y — 2)
J (x — 2)? = 12(y + 1)
oa
15. Identify the graph of the equation 4x? — 5xy + 16y? — 32 = 0.
A
circle
B
ellipse
C
parabola
D
hyperbola
15.
16. Which graph represents a curve given by x = 4¢ + 5 and y = 2?” over the
interval —3 <t < 3?
17 Identify the conic that has an eccentricity of =
A
circle
B
ellipse
C
hyperbola
D
parabola
ioe
18. Write the standard form of the equation of the circle with center
at (—3, 5) that is tangent to the y-axis.
F (x + 3)?+(y—5)?=9
G (x +3)? +(y —5)? = 25
Bonus’
1Sqicec
Write the equation of the line tangent to
(y — 2)” = 8x through (8, —6).
Chapter 7
&
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H (x +3)?+(y—5)?=8
J (x—3)?+(y+5)*=9
B:
40
Glencoe Precalculus
NAME
DATE
ae
PERIOD
Chapter 7 Test, Form 2B
SCORE
Write the letter for the correct answer in the blank at the right of each question.
_
.
1/2
.'For Questions 1-3, refer to the ellipse represented by = + wea srk,
1. Find the coordinates of the center.
A
(-1, 0)
B
(0, —-1)
G10)
D
(0,1)
| oe
2. Find the coordinates of the foci.
F (0,1+ V5)
H (V5, 3), (—V5, 3)
G (V5, 1), (= V5, 1)
Tolle Ba lee)
2.
3. Find the coordinates of the vertices and co-vertices.
A
(2, 1), (—2, 1), (0, 4), (0, -2)
B, (3-1), (—2,. 1),.(0, 8)eeOr-=0)
C
Ww
®
(8, 1), (—3, 1), (0, 4), (0, —2)
D> (2/2) (2); (0, 8)(0) 21)
3.
4. Determine the angle of rotation 6 through which the conic given by
2x?
+ xy + 2y? = 1 should be rotated.
Foi”
G
150°
H
45°
J —30°
4s5 se
5. Write the pair of parametric equations x = — cos 6 and y = sin 0
in rectangular form.
Ay? =x? =1
B x’+y’=-1
Cig
y=.)
D
Aye
pee ee
6. Use the discriminant to identify the conic section
x? — y? + 12y + 18x = 42.
F
parabola
G
hyperbola
H
ellipse
J circle
i Sena
7. Write the standard form of the equation x? + y? = 16 in the x’y’-plane
after a rotation of 45°.
Ai (x')? 4.~)? = 16
C(x’)? — 2x’y’ + (y’)? = 16
B
D
(x)? — (y’)? = 16
(x)? + 2x'y’ + (y’)? = 16
i Re
8. Rusty hit a baseball with initial velocity of 125 feet per second at an angle
of 32° with the ground from a height of 3 feet. Write parametric equations
to represent this situation.
F x = 125t cos 32° — 16¢?
H
y = 125t sin 32° + 3
G
x = 32 cos 125¢° + 3
Nee
x = 32h cosi25:
y = 82t sin 125° — 162?
ei
25? C08
ce
y = 125t sin 32° — 16¢? + 3
COS 1 2Dt
ie Pee
Copyright
Glencoe/McGraw-Hill,
©
of
division
McGraw-Hill
The
Companies,
Inc.
a
9. A cross section of the reflector shown is in
acon
the shape of a parabola. Write an equation
for the cross section.
A y= 4x
B y’? = &
Chapter 7
i
C x? =4y
Dix = oy
(ee eee
41
=
Glencoe Precalculus
DAT Bige
Chapter
7 Test, Form
2B
a
e
-PERIOD
(continued)
— 3/2
For Questions 10 and 11, refer to the hyperbola represented by LP ai
16
2
lsre > =f)
10. Write the equations of the asymptotes.
F y-8=45x
G y-8=43x
H y = 4+4(@ - 3) J y=+3(¢-3)
10huvt1
C
filha s
11. Find the coordinates of the foci.
A
(5, 0), (=9; 0)
B
(0, 5), (0, =3)
(3, 5), (3, =)
D
(8, 0), (2) 0)
12. Write the standard form of the equation of the hyperbola for
which a = 5, b = 6, the transverse axis is vertical, and the
center is at the origin.
Veptlh &
25
36. =
ae
36.0)
:
25 Fl
i
pe
Meo
=
Rg
Jee
te
Se
ynKesid
a
13. Jana hit a golf ball with an initial velocity of 102 feet per second
at an angle of 67° with the horizontal. After 2 seconds, how far has
the ball traveled horizontally?
A
27.9 ft
B
123.8 ft
C
79.7 ft
DOT
it
To.
14. Write the standard form of the equation of the parabola with
directrix at x = —2 and focus at (2, 0).
F (y — 2)? = 8(x« — 2)
Hoge
= 8x
G (y — 2)? = 4(x — 2)
oy x = Sy.
145s
15. Identify the graph of 6x? — 8xy — 2y?+6=0.
A
circle
B
ellipse
C hyperbola
D
parabola
16
16. Which graph represents a curve given by
x=t—2andy =? — 2 over theinterval —3 < t < 3?
PG oe
17. Identify the conic with an eccentricity of 1.
A
circle
B
ellipse
C
hyperbola
D
parabola
Lig
18. Write the standard form of the equation of the circle with center at (—4, 8)
that is tangent to the x-axis.
F x?+y? = 64
G @w+4) + vy -— 8) =16
Bonus
H (x —4)?+ (vy +8)? = 16
J (x + 4)? + (y — 8)? = 64
18.
Write the equation of the line tangent to
(x — 2)? = 20y through (8, 5).
Chapter 7
YBUAdOD
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‘I[IH-MeIDO\W/a
B|IIH-MeINOW|
UOISIAIP
©
JO
OUL
‘Ou
B:
42
Glencoe Precalculus
NAME
‘Chapt
7eTerst, Form 2C
DATE
PERIOD
SCORE
For Questions 1-3, use the ellipse represented
by 3x? + 2y? + 24x — 4y + 26 = 0.
1. Find the center.
2. Find the foci.
3. Find the vertices and co-vertices.
~
|
ce)
4. Determine the angle of rotation @ through which
the conic given by 4x? + 12xy — 5y? = —3 should be
rotated, to the nearest degree.
=
VW
vw)
co)
Wi
Va)
=
5. Write the pair of parametric equations x = 8 cos 0
and y = —2 sin @ in rectangular form.
6. Use the parameter t = 2x — 3 to determine the
parametric equations that can represent y = x? + 1.
7. Write the standard form of x? — xy + y? = a
in the x’y-plane after a rotation of 45°
about the origin.
8. Tomas hit a
100 feet per
ground from
equations to
baseball with an initial velocity of
second at an angle of 35° with the
a height of 4 feet. Write parametric
represent this situation.
9. The figure shows a parabolic archway. Write an
equation for the parabola.
For Questions 10 and 11, use the hyperbola given
©
Copyright
Glencoe/McGraw-Hill,
Companies,
McGraw-Hill
The
of
division
Inc.
a
(y — 6)?
(~¥-—9)? _
ry eas ey
10. Find the foci.
10.
11. Find the equations of the asymptotes.
11.
Chapter 7
43
Glencoe Precalculus
NE
en
en
ee
PERIOD
ee
Chapter
7Test, Form 2C
NSS
(continued)
12. Write the standard form equation of the hyperbola
with foci at (—2, —3 + 2/3) and conjugate axis of
length 6 units.
12.
13. Find the coordinates of the vertex and the equation
of the axis of symmetry for the parabola represented
by x = y? — 2y — 5.
13.
14. Write the standard form equation of the parabola
with a vertex at (—2, 1), and focus at (—2, —4).
14.
15. Use the discriminant to identify the conic given
by xy + 3y + 4x = 0.
15.
16. Graph the curve given by the parametric
equations x = 2¢ + 3 andy = ¢? — 1 over the
interval —3 <¢t < 3.
For Questions 17 and 18, Lisset throws a softball
from a height of 4 feet with an initial velocity of
60 feet per second at an angle of 45° with respect
to the horizontal.
17. When will the ball be a horizontal distance of
30 feet from Lisset?
17.
18. What is the maximum height the ball will reach?
18.
19. Identify the conic section with eccentricity 2.
19.
20. Write the equation of a circle with center (2, 7) and
diameter 30.
20.
Bonus’
BWYyGuAdoo,
‘SAIURGWOD
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JO
OUL
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Write a rectangular equation for the
curve given by x = 3 cos (40) + 2 and
y = 2 sin (46) — 5.
Chapter 7
44
Glencoe Precalculus
NAME
Chapter 7Test, Form 2D
DATE
PERIOD
SCORE
For Questions 1-3, use the ellipse represented by 4x? + 9y? — 24x + 18y + 9 = 0.
‘LL
Find the center.
1:
2. Find the foci.
2.
. Find the vertices and co-vertices.
~
. Determine the angle of rotation 6 through which the
conic given by 3x? — 8xy — 3y? = 3 should be rotated, to
the nearest degree.
Cc
)
&
wv
7)
cd)
rv,
Vv)
. Write the parametric equations x = 10 cos 8 and y = 3 sin 0
in rectangular form.
~
. Use the parameter t = 2x + 4 to determine the parametric
equations that can represent y = x? + 2.
. Write the standard form of 5x2 + 3V3xy + 2y? = 13
in the x’y-plane after a rotation of 30° about the origin.
. Lucinda hit a softball with an initial velocity of 110 feet
per second at an angle of 32° with the ground from a
height of 3 feet. Write parametric equations to represent
this situation.
. The figure shows a parabolic archway. Write an equation
for the parabola.
For Questions 10 and 11, use the hyperbola given
by 4x? — y? + 24x + 4y + 28 = 0.
©
Copyright
Glencoe/McGraw-Hill,
Companies,
McGraw-Hill
The
of
division
inc.
a
10. Find the foci.
10.
11. Find the equations of the asymptotes.
11.
Chapter 7
45
Glencoe Precalculus
PERIOD
DATE
NAME
Chapter 7 Test, Form 2D
SLA ARAN
(continued)
12. Write the standard form equation of the hyperbola
with foci at (—1, -2 + 2\/5) and conjugate axis of
length 8 units.
12.
13. Find the coordinates of the vertex and the equation of
the axis of symmetry for the parabola represented
13.
by —2x +y?—- 2y+5=0.
14. Write the equations in standard form of the parabola
with vertex (2, —5), directrix y = —1, and opens to the left.
14.
15. Use the discriminant to identify the conic given
by 4x? + 8xy — 2y? = 17.
15.
de
16. Graph the curve given by the parametric equations
x= 2t+5andy =? — 2 over the interval —3 <t < 3.
For Questions 17 and 18, Alfie throws a softball from a
height of 5 feet with an initial velocity of 65 feet per
second at an angle of 42° with respect to the horizontal.
17. When will the ball be a horizontal distance of 30 feet
from Alfie?
17.
18. What is the maximum height the ball will reach?
18.
19. Identify the conic section with eccentricity 2
19.
20. Write the equation of a circle with center (—6, —9)
and diameter 22.
20.
Bonus
JYSuAdoo
‘IIIH-MeIDI\\/
‘SAIUBGWOD
©JO
&|IIH-MEINOY|
UOISIAIP
SUL
“Qu|
Write a rectangular equation for the curve given
by x = 5 cos (30) — 4 andy = 2 sin (36) + 1.
Chapter 7
46
Glencoe Precalculus
NAME
DATE
PERIOD
at
est, Form
3
SCORE
For Questions 1-3, use the ellipse represented by 9x? + 4y? + 54x — 16y + 61 = 0.
¢
e
1. Find the center.
1
2. Find the foci.
2.
3. Find the vertices and co-vertices.
3.
4. Identify the graph of 14x? — 7xy — 10y? = 5. Find 6,
the angle of rotation about the origin, to the nearest degree.
7)
5
=
4.
vi
vd)
5. Write the parametric equations x = 2 cos 30 and y = —3
sin 36 in rectangular form.
v
<
5.
6. Use the parameter ¢t = 3x — 2 to determine the parametric
equations that can represent y = —x? + 4.
7. Write the standard form of 5x? — V/3xy + 4y? = 20 in
the x’y-plane after a rotation of 60° about the origin.
6.
7.
8. Jim threw a ball with an initial velocity of 38 feet per second
at an angle of 29° from a height of 4 feet to Jennifer, who
was standing 42 feet away. Jim’s dog Rex was between them,
4 feet in front of Jennifer. If Rex can jump 5 feet into the air,
can Rex intercept the ball? Write the parametric equations
for the flight of the ball and explain your answer.
8.
9. The figure shows the entrance to a tunnel in the shape of
a parabola. Write an equation for the parabola.
9.
Copyright
Glencoe/McGraw-Hill,
©
division
McGraw-Hill
The
of
Companies,
Inc.
a
For Questions 10 and 11, use the hyperbola represented
by y? + 4y — 4x? —- 84 + 16 = 0.
10. Find the foci.
10.
11. Find the equations of the asymptotes.
iA
Chapter 7
47
Glencoe Precalculus
Chapt7eTerst, Form 3 onnui
PERIOD
DATE
ERA
12. Write the equation in standard form of the hyperbola
with a horizontal transverse axis of length 10, center
at (—2, 4), and asymptotes y — 4 = +5(x + 2).
12.
13. Find the coordinates of the vertex and the equation of
the axis of symmetry for the parabola represented by
y? + 4y — 4x + 12 = 0.
13.
14. Write the equation in standard form equation of the
parabola that passes through the point (—8, 15), with
14.
a vertex at (2, —5), and opens to the left.
15. Use the discriminant to identify the conic given by xy = 36.
15.
i
16. Graph the curve given by parametric equations x = —=??
andy = yeas 3 over the interval —4 <t < 4.
2
16.
For Questions 17 and 18, Shannon kicks a soccer ball
with an initial velocity of 45 feet per second at an angle
of 12° with respect to the horizontal.
17. After 0.5 second, what is the height of the ball?
17.
18. The goal is 4 feet high and 20 feet in front of Shannon.
Will the ball go into the goal before it hits the ground,
assuming she kicks it straight ahead of her and the
goalkeeper does not stop it? Explain.
18.
19. Find the eccentricity of the conic represented by
Cee nC
4
8
a
19.
ch
20. Write the equation of a circle with a diameter whose
endpoints are (—4, 6) and (30, 18).
20.
BJYybuAdog
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Bonus _ Find the equation of the ellipse with a center at
(—1, 3), a vertical minor axis of length 40 units,
and eccentricity of e
Chapter 7
48
Glencoe Precalculus
DA ieeee et ert
is
ee
AD
DATE
Extended-Response Test _
PERIOD
—
Demonstrate your knowledge by giving a clear, concise
_ solution to each problem. Be sure to include all relevant
drawings and justify your answers. You may show your
solution in more than one way or investigate beyond the
requirements of the problem.
1. Consider the equation of a conic section written in the form
Ax?+ By?+Cx+Dy+E=0.
~
=
co)
a. Explain how you can tell if the equation is that of a circle.
Write an equation of a circle with a center not at the origin.
Graph the equation.
=Vv)
WV
co)
Va)
VW
=g
b. Explain how you can tell if the equation is that of an ellipse.
Write an equation of an ellipse with a center not at the origin.
_ Graph the equation.
c. Explain how you can tell if the equation is that of a parabola.
Write an equation of a parabola with its vertex at (—1, 2).
Graph the equation.
d. Explain how you can tell if the equation is that of a hyperbola.
Write an equation of a hyperbola with a vertical transverse axis.
e. Identify the graph of 3x? — xy + 2y? — 3 = 0. Then find the angle
of rotation 6 to the nearest degree.
2. a. Describe the graph of x? — 4y? = 0.
b. Graph the relation to verify your conjecture.
ce. What conic section does this graph represent?
Copyright
Glencoe/McGraw-Hill,
©
Companies,
McGraw-Hill
The
of
division
Inc.
a
3. Give a real-world example of a conic section. Discuss how
you know the object is a conic section and analyze the conic
section if possible.
Chapter 7
49
Glencoe Precalculus
Smamacanvaiteaeals
NAME
DAW
Ee
Semen
(Chapters 1-7)
Part 1: Multiple Choice
Instructions:
Fill in the appropriate circle for the best answer.
1. Find the remainder when 5x? + 8x — 2 is divided by x — 3.
A
-161
B
19
C
67
D’ 157
1.@®@OoO
2. Find the period and phase shift of the graph of f(x) = —4 cos 2(x + 7).
F
2,4 right
G
1, 27 left
H
7, 7 left
J
27, 7 right
2,.00@0O
\
*
3. Solve 2 cos? x + 5 cos x — 3 = 0 where 0 <x < 27.
4. Find cos In
ee
F eV 2.
gti
2
H V2 -V6
4
J V2 +V6
4
4.®©@O
4
5. Write the expression for the numerator of x for this system of equations
using Cramer’s Rule.
2x —-5By+z=6
—38x + 4y — 2z = -9
x+2y+4z=5
ZnO
A|-3
4-9|
TS
2-5
1
B|-3
4 -2
12.
4
6-5
1
C{l-9
4-2)"
ppm
pee!
PA celSi
_D-—3--9 —2
145
4
5.®®©®
6. Find sin oe
V3
Le sere
G5
1
1
V3
H =>
Or
V24
Camonget
D
6.©®©@O
7. Find sin (arcos -3).
2
1
25
Baa
2/6
7
7 FOR ORCMO)
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8. Find f-\(x) if f(x) = 2x — 6.
a aee
H f(x) =51,
G f-\(x)
= 2x +6
J f(x) =jx +6
cay |
Chapter 7
me
x
aie 6
=i]
50
6X
il
8.©©@O
Glencoe Precalculus
NAME
DATE
PERIOD
Standardized Test Practice
(Chapters 1-7)
LBB
oninuea
9. Find the equation of the horizontal asymptote of g(x) = ae.
A y=5
B y=-3
C y=2
Dy=->
9.©@®@O©O
10. The graph of g(x) = (x — 2)? + 3 can be obtained from the graph
of f(x) = x? by doing which transformation?
F move right 2 and up 3
G move right 2 and down 3
H move left 2 and up 3
J move left 2 and down 3
11. If f(x) = —2x°, where is f(x) decreasing?
10.00@90O
y
A (—00, 00)
C (0, 00)
B_ (—oco, 0)
D (—10, 10)
O
x
11..0®0®
Ass
12. Find the eccentricity of the conic represented by
te See
100
5
144
Teri
6
Nil
23
13. Find the value of }—1
A
—49
V1l1
12.0
©@®
O
4 —2J.
5
0-3
B
—63
C
—67
D
-79
13.@ ®O®@
14. Change 105° to radians.
bn
Beco:
OG;Tn
15sif cosix = -— and 1 <x
ie3m
dies5m
14.0 ©@O
= find sin 2x.
336
336
ae
p _527
625
625
25
625
15.@
© ©©
16. Find the domain of f(x) = ae
F {x |x € R}
H {x |x #-2,x
€ R}
G
J {x|x#-2,3,xeR}
{e(x4pxeR}
16.0
© ®@O
17. If f(x) = 2x + 7 and g(x) = x? — 3, find fig(«)).
A
2x? + 7x?=— 6x — 21
C 4x? + 28x + 46
B
x7+2x+4
D
Copyright
Glencoe/McGraw-Hill,
©
division
McGraw-Hill
The
of
Inc.
Companies,
a
;
17.© ®OO
2x7+1
:
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a. Write parametric equations to represent the flight
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Quiz 3 (Lesson 7-4)
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