West Essex Regional School District
Algebra 2 (Honors)
Summer Assignm ent 2011
The topics for Algebra 2 (Honors) include
. LinearRepresentations
o Functions
. Systems of Linear Equations and lnequalities
. Matrices
. Quadratic Functions
. Polynomial Functions
. Rational Functions
. Powers, Roots, and Radicals
. Conic Sections
. Exponential and Logarithmic Functions
. Sequences and Series
. Counting Principles and Probabílity
. Trigonometry
The first maior theme of the curriculum, linear representations, is a review
of
concepts learned in Algebra I and Geometry and will be compieted over the
summer. Upon completing the summer assignment, each student should be able
to:
'
'
.
.
.
Represent a real-world linear relationship ín a table, graph or equation.
ldentify linear equations and linear relationships between variables in a
Graph linear equations.
Find the slope of a line.
Write an equation in slope-intercept form for a line:
* given the graph of the line
* given the slope and
ond poínt on the line
* given two poínts on the
'
line
that contains gíven point and is parallel to the given line
*" that contains aa given poínt
and is perpendicular to the given line
'
Find the constant of variation and write an equation of direct variation.
'
Determine whether values in a table represent a direct variation.
table.
Assignments
toward homework grades. No credit will be given for assignments turned in
late.
assessments will be announced prior to the quizzes.
Complete all work on a separate piece of paper showing all work for each problem
and using Pencil, credit will not be given othen¡¡ise. lt is also recommended that
students use a three-ríng binder for all notes, assignments, and worksheets. Complete
alf problems in the exercises unless othenrise indicated.
Show allwork!
l.
ll.
lll.
D
D
Suggested Due Date: 811111
Read: pp.4-7
Complete pp.7-101 # 1, 4,7-29 (odd), 31-34, 49, 51-54
tr
J
Suggested Due Date: 81811'l
Read. pp. 12-16
complete: pp. 17-201#1 ,3, 19-25 (odd), 3o-39, 41-45(odd), s3-59(odd), 6s-69
fl
D
Suggested Due Date: 8115111
Read: pp.21-24
complete pp. 25-281 # 3,7,11-15(odd), 23-27(odd), s2-34, 35, 37 ,43,4s,62-64
lV.
Su gested Due
Date:
8l,22111
D Read: pp.29-32
D Complete: pp. 33-36t # 2'-9, 5,7, g-13, 1S-21(odd), 33, 35, !3-57(odd), 66-71
Have a great summer....
Tables and Graphs
of Linear Equations
A salesperson in a music store usually eørns cotnmßsrcn'
M"q
Investigating Gommission
APPLICATION
You
will need: graph
store. Yoú earn
Suppose that you work Part-time in a music
plus a 10% commission on all of the sales you make.
1- Discuss what
PROBLEM SOLVING
p¿iPer and a straightedge
it
means to earn commission
this relationship between weekly s{e9 1nd weekly
2-- You can rePresent a table' Copy and complete the table below'
by mat ing
;;;.t
WeeHy sales, s (in dollars)
100
200
300
w (in dollars)
Weekly
40+0.10(100)=50
?
?
400
500
?
entries in the weekly
s. what observations can you make about successive
sales column? in the weekly wages column?
ordered pair (s, w)'
¿. you can represent each row in the table as an
last points with a
Plot each ordered pair, and connect the first and
straightedge.Does.eachpointthatyouplottedapPeartobecontained
in this line segment?
cHEcKPoTNT
4
c¡¡¡PreR I
y' s.Writeanequationtorepresenttherelationshipbetweensandw'
In the Activity on page
4, a linear relationship that can u.
-oa.t.a uy
linear equation is described. Example I provides another instance of a "linear
relationship
APPTICATION
An attorney,charges a fixed fee of 9250 for an initial meeting and gl50 per
hour for all hours worked after that.
a. Make a table of the total charge for I,2,3, and 4 hours worked.
b. Graph the points represented by your table and connect them.
'Write
c.
a linear equation to model this situation.
d. Find the charge for 25 hours of work.
SOLUTION
a.
Hours worked
Total charge
I
2
3
4
$4oo
$ss0
$70o
$8so
c
b.
t(
)
ø
c. Tianslate the verbal description into
an equation involving c and h.
total variable
+
.c=150h+250
charge= charge
A lawyer, or øttorney, in
court
Thus,
fixed
ct
fee
o
F
c=150h+250.
h
d. Use the equation. Substitute 25 for h
the equation.
c
in
Hours worked
= I50h+250
150(25) +250
c:
c: 4000
CHECK
GRAPHICS
CATCUTATOR
From the graph of y = l50x + 250,
you can see that when x - 25,y: 4000.
Thus, the attorney charges $4000 for
Guide, page 80
TRY THIS
25 hours of work.
A water tank already contains 55 gallons of water when Darius begins to fi.ll it.
Water flows into the tank at a rate of 9 gallons per minute.
a. Make¿ table for the volume of water in the tank after 1,2.,3,and
4 minutes.
!
b. Graph the points represented by your table and connect them.
c. Write a linear equation to model this situation.
d. Find the volume of water in the tank 20 minutes after Darius begins filling
the tank.
The equations in Example I and the Try This exercise above have a
characteristic in common. Each has the form shown below.
total amount = variable amount + fixed amount
In general, if a relationship between x and y can be written ãs / = mx + b,
where m and b are real numbers, then x and y are linearly related. The
equation y = lnx + b is called a linear equation. The graph of a linear equation
is a straight line.
LESSoN
r.t
TABLES AND GRApHS oF LTNEAB
EouATtoNS E
)
cHECKpOtNT
7
what are the values
of.
m
and'b in the equation c =
l50h+ 250 from
Example l?
Graphy--þ-t.
SOLUTION
y--2.3xgraph i, u ståigh
ãrrty *o ordered
b'where
Because
d by
*=trandb=-f
its
wo pointi' so you need to plot
- I and draw the line through them.
v
3
1C
0
v
,=!tÐ-l=-l
r--!tzl -t
I
(3
Plot (0, -l) and (3, 1), and draw
a
line through
GRAPHICS
CATCUlATOR
x
t
them.
.TECHNOLO
l)
(o
-1
)
CHECK
=Z{- I on a graphics calculator'
(3' 1)
and veriff íhat the points (0, -i) and
Graph y
are on the line.
Guide, page 80
TRY TH¡S
Graph
,--2nx+3.
there is
related and
when variables represented in a table of values are linearþ
is also a constant difference in the
a constant difference in the x-values, there
F.r example, consider the linear equation 7 = -2x t 5'
;-;;ñ;;.
have a constant difference'
Make a.table of values by choosing x-values that
I
such as l,2,3,4,andso on'
+1
+1
+1
x
I
2
3
4
v
3
1
-1
-3
x-values results
In a linear relationship, a constant difference in consecutive
in a constant difference in consecutive 7-values'
cHEcKpotNr
,l
J
I
the
y' Suppose thatyou are making a table ofvalues to determine whether
exist
must
;;;
lirr.arþ related]Describe the relationship. what
"*åù1.,
between the x-values that you choose?
Does the table of values at right represent a
linear relationship between x and' y? Explain.
If the relationship is linear, write the next
ordered pair that would appear in the table.
x
7
T2
v
11
8
t7 22 27
5
{
32
) -1 -4
SOLUTION
Find differences in consecutive x-values and consecutive 7-values.
+5 +5 +5 +5
1C
v
1
t2
t7
22
27
32
11
8
5
2
-l
-4
-3
PROBLEM SOLVING
+5
-3
-3
-3
-3
Look for a pattern. Because there is a constant difference in the x-values and a
constant difference in the 7-values, the relationship between x and 7 is linear.
The next table entry for x is 32 + 5, or 7.
The next table entry for 7 is -4 + (-3), or
TRY THIS
(
CRITICAL THINKING
-7.
Does the table of values at right represent a
linear relationship between x and 7? Explain. If
the relationship is linear, write the next ordered
pair that would appeár in the table.
Does the table at right represent a linear
relationship between x and 7? Explain.
x -2 2
v 1 )
6
10
14
t8
4
8
16
32
x
9
6
3
0
_J
-6
v
5
5
5
5
5
5
ftâommunrcate
the relationships among the
equation, and graph shown here.
table,
i
/
x
x
v
m
2.
-4
-7
-2
0
2
4
-4
-1
z
5
y=zft- |
TAX Suppose that a state sales tax ís 7o/o. Make a table showing the
amount of tax on items with prices of $6' $8' $10' and $12' How can you
find whether the price and amount of sales tax are linearly related?
SALES
e. Explain how to verifr that the points
the same line.
LESSON
(-I,7),
1.1
(0, 4), and (2,
-2)
arc all on
TABLES AND GRAPHS OF LINEAR EOUATIONS
7
Ç Quìded Skills Practlce
NCOME SupPose that You work
part-time at a dePartment store,
earning a base salarY of $50 Per
week plus a l5o/o commission on
Weekly sales, x
Weeklyincome'Y
100
s0+(0.1s)(100)=65
?
200
?
300
all sales that you make'
(EXAMPLE 1)
?
400
a. Copy and comPlete the table.
?
x
b. Graph the Points rePresented
in the table and connect them.
the weekly
c- Write a linear equation to represent the relationship between
sales, r, and the weeklY income 7'
u. Find the weekly income,7, for weekly sales of $1200'
5- Graph
f = 3x - 2. GxarøPLE
2)
and 7? If the
6- Does the table below represent a linear relationship between x
in the
appear
relationship is linear, *iit" th" next ordered pair that would
table. lexaMPLE 3)
El inte¡netconnect
x -4
Activities
0nline
Go
I3
v
To: go.hrw.com
1
6
il
I6 2l
19
25
31,
5/
43
Keyword:
MBI Paycheck
ÇPractice and APPIV
State whether each equation is a linear equation'
=
8-
-3x
10.Y=5-4x
11.
@r=I
14-
Homework
Help Online
=þ-
=
*2x
=-x2+I
-
1s,.y=17-1.2x
2
Graph each linear equation'
20 i=4x+3
=3x-6
l:3 - 5x
y=þ+a
/=5-2x
@
2€.Y:-x-2
Go To: go.hrw.com
Keyword:
MBI Homework HelP
for Exercises 19-30
12
e.y=-þ
z
y _-4
x
:5.5x
1e,.y=2+5x2
El inte¡netconnect
y:-x
@r t 3=x*6
30-y+4=x-3
For Exercises 31-38, determine whether each table represents a linear
next
relationship between x and y. lf the relationship is linear, write the
ordered pair that would appear in the table'
8
cHRptR I
@
@
x
v
-5
3
3
-1
4
-5
I
34
6
J
5
6
46
9
x
v
0
1C
v
10
0
I
)',
2
3
6
ll
@
x
v
-2 I
-3
.,
4
-5
8
35.
36.
x
v
8
28
12
6
22
9
v
-3
-8
4
I6
6
2
l0
3
,t
37
x
v
6
38.
,c
v
115
-6
58
9
100
-9
44
-13
T2
85
-r2
32
-18
15
75
-15
20
For each graph, make a table of values to represent the points. Does
the table represent a linear relationship? Explain.
39_
v
40.
I
x
Ã
41. Make a table of values for the equation / = 4x - 1, and graph the line.Is
the point (2,6) on this line? Explain how to answer this question by using
the table, the graph, and the equation.
Use a graphics calculator to graph each equation. Then sketch the
graph on graph paper.
WÅy:-3x+1.5 Wy:-x-2.5
Wly: -o.Sx ffiy:ir-t
ffi
m
ffir =12-2.5x
ffiy:-Z{*T
.¿s. What can you determine about the graph of y = lnx + b when ¡ : 0?
lVhat can you determine about the graph of y : mx + b when 7 = gz
NCOME A
video rental store charges a $6
membership fee and $3 for each
video rented. In the graph at
o
right, the x-axis represerrts the
É
o
number of videos rented by a
o
ú
customer and the y-ans
x
represents the store's revenue
from that customer.
Numberof
videotapes rented
a. Make a table for the data
points on the graph. b. If 15 videos are rented, what is the revenue?
c. If a new member paid the store a total of $27, how
many videos werJrented?
d. Explain how to find answgrs to parts b and c by
using an extended table and an extended graph.
LESSoN
1.1
TABLES AND GRAPHS oF LINEAR EoUATIoNS
9
50. DEMoGRAPHICS City Community College plans to increase its enrollment
capacityto keep upwith an increasing number of student applicants. The
.o'll.g. .rrr.ttÎy hìs an enroilment capacity of 2200 students and plans to
increãse its capacity by 70 students each year.
a. Let x represent ihe'number of years from now, and let 7 represent the enrollment capacity.Make a ta6le of values for x and y with x-values of
0, 1,2,3, and 4.
b. What will the enrollment capacity be 3 years from now?
c. Write a linear equation that could be used to find the enrollment
capacity, /, aftet x Year s.
51.
An airport parking lot charges
a basic fee of $2 plus $l per half-hou¡
USINESS
Half-hours
Total charge ($)
0
2+(1)(0)=2
parked.
a. Copy and comPlete the table.
b. Graph the points represented in the
table. Label each axis, and indicate
commuruty
I
?
2
?
3
?
your scale.
?
12
c- Write an equation for the total charge,
c, in terms of the number of halfhours parked, h'
d. How many half-hours is 72 hours? What is the total charge for parking
in the lot for 72 hours?
52-
is.the
ivil!ng to
.prlce.
At 6:00 4.M., the tempera ture was 67"F. As a cold front
passed, the temperature began to droP at a steady rate of4"F Per hour.
a. Write a linear equation relating the tempera ture in degrees Fahrenheit,
t, to the number of hours, h, after the initial temPerature reading.
b. Estimate, to the nearest 15 minutes, how long it would take to reach
freezing (32"F) if the drop in temperature continued at the same rate.
ROLOGY
MIGS This table gives the
Price ($)
SINESS Casey has a small business
making dessert baskets. She estimates
that
rent,
electricity,
her fixed weekly costs for
and salaries is $200' The ingredients for
one dessert basket cost $2.50.
a. If Casey makes 40 dessert baskets in a
given week, what will her total weekly
costs be?
b. Casey's total costs for last week were
$500. How many dessert baskets did
she make?
1O
cHaPren I
Supply
Demand
price, the supply, andthe demand
s00
150
20
for a video game.
400
250
30
a. Graph the points rePresenting
and
the
price and supplY
200
450
50
points representing Price and
demand on the same coordinate plane.
b. Estimate the price at which the supply of video games meets the
price'
'.demand. Estimate the supply and demand at this
the price of the
when
c. What happens to the s"tply and to the {emand
video game is higher thin the price in part b? lower than the price in
part b?
ffitook
Back
The followingRules of DivßibilirT are useful in finding factors of numbers.
If a number
'
'
'
'
is
divisible by 3, the sum of the digits of the number is diviiible by
divisible by 6, the number is divisible by 3 and is even..
l.
divisible by 9, the sum of the digits of rhe number is divisible by 9.
divisible by 4, the number formed by the last rwo digits is divisible by 4.
Find numbers a and þ that meet the following
a*b=13
and ø-tb=43
sg.ab:l2Banda*b=24
61. øb =48 and a-fb:19
55.ab:36
57. ab:82
"onU¡tìon"t
ø*b=20
se. ab=72anda+b:22
€¡0.ab=56and a.I-b:15
az. øb:52anda*b=IT
and
sø. ab =
5l
and
Evaluate each expression. Write your answer in simptest form.
i"i
63.
67.
Portfolio
Extension
Go
To: go.hrw.com
Keyword:
MBl Linear
,*l
xt
o+.1xft
øs. z
oe. s +-f
as.2I
+{
oa.
ftx
z
2o.10-f
(ffi took BeVond
71. Graph the equations ./ - 2x, y = 2x + 3, and y : 2tc 4 on the same
coordinate plane. How are the graphs alike? How are they different?
t.
Describe three real-world situations in which a distance changes at
a fatùy constant rate over time. For example, the distance driven
on an interstate highway or the distance walked in a walkathon changes
at a fairly constant rate.
z. Choose one of your three real-world situations from Step l, and
determine a suitable way to collect some time lnd distance data. collect
and record a minimum of seven data values. This data will become your
portfolio data
set.
3- organize the data from your portfolio data set in a table of values.
4. [ße graph paper to graph your þortfolio data set. Label the x-axis with
units of time, and label the 7-axis with units of distance. The point that
represents the first distance measure that you took should have an
x-coordinate of 0.
WORKING ON THE CHAPTER PROJECT
You should now be able to complete Activity
LESSoN
r.1
I of the chapter project.
TABLES AND GRApHS oF LTNEAR
EouATtoNS .1 I
Slopes and lntercepts
Trends
such as
in
the real world,
the increase in cellular
phone use, can often he
modeled by a linear equation,
in which the sloPe indicates a
rate of change'
states become cellular phone
Every year more and more people in the.united
phone
;br:r'ib.tr. The table bel, w reþresents the recent trend in celiular
number
The third columï in the table gives the change in the
ffi;ilt"ns.
of subsiribers from one year to the next'
Year
Number of subscribers
(in millions, rounded to
the nearest 100,000)
1990
5.3
I99I
7.6
t992
11.0
1993
16.0
1994
24.0
r995
33-8
i990-199r
1997-1992 3.4
1992-1993 s.g
1993-1994 &0
1994=1995 9:8
oo
-ôo
üo
ao
of new subscribers from
Notice in the highlighted row above that the n¡rmber
ttt' also be represented by the
1990 to i991 is 7.6 - 53, ot Z.3-This differettb
ratio shown below.
Change in subscribers _ 7.6-5.3 -_2.3 =2.3
= l99l - 1990 I
Uhange ln Years
E¿
All entries in the third column
U.S. Cellular
Subscribers
ga
can be found by using a ratio' Each of the
ratios gives the rate of change from one year to the next'
øQ
qk
:E
oH
-ôo
you can also find the averøge røte of change from i990 to 1995. on average,
per
¡ear'.T^his
from 1990 to 1995, there wãre about s.z áillion new subscribers
at left'
graph
the
on
line
dashed
red
the
by
inàicated
ir
u.,r.rug. rate of .tr""f"
in subscribers 33.8 - 5-3
Total
= tJ
EÉ
Ao
012345
Years after 1990
rn years
cHEcKporN
12
2.3
cnnPreR I
r y'
Estimate the average rate of change
from
1995
-
1990
to
1990
=";t
1993 and
from
1993
to 1995'
In a graph, the slope of a line is the change in vertical units divided by the
corresponding change in horizontal units.
Slope of a Line
If points
(xt,yì
v
(x2,y) lie on a line,
then the slope, m,,of the line is gîven by
the ratio below.
and
(ry
fz- /t
(xr, y¡)
_ fz-lt
Xz - lct
m=
y2)
xz- xt
x
You-can find the slope of a line if you know the coordinates of two points
on the line. This is shown in Example 1.
Find the slope of the line containing the points (0, 4) and (3, 1).
SOLUTION
Use a formula. Let
of slope.
PROBLEM SOLVING
(n,y) :
(0,
m= lz-lt
xz- xt
4) and (x2,y) = (3,1). Apply the definition
r-4
-3-0- -33 -
r
The line containing (0, 4) and (3, 1) has a slope of
-1.
Find the slope of the line containing the poinrs (-5, 3) and (3,
-4).
TRY THIS
You
t.
GRAPHICS
CALCUI.ATOR
will
need: a graphics calculator
: lrrx,where m is the slope. Graph
eachpair of equations. Describe how the slopes for each paiiof linei are
alike and howìhey are different.
\
Each equation below has the form y
a.y=þ*d
Guide, page 80
y=-L5*
b.y:.þand,y:-Zt
c.y=xandy=-¡
d.y=þand,y=-þ
e.y=5xand/=-5x
z. Make
a conjecture
about the slopes of y = mxwhen m <0 and when
m > 0. Explain how the graphs of these lines are related.
CHECKPOTNT
y'
e. Verift your conjecture from Step 2 by writing and graphing another
pair of equations with the relationship that you described in Step 2.
LEssoN 1.2 sLopESANDtNTERcEprs
13
linear,equation crosses the
The /-coordinate of the point where the graph of a
below, the /-intercept
graph
the y-inteìcept of the line. In the
u-uÁris called
of titt. r is 3 and the 7-intercept of line s is -2'
v
Í
((
/
I
(
r)
s
for the line'
To find the 7-intercept of a line, substitute 0 fot x in an equation
/
Line s
Line r
/=2x-2
,: -þ+t
y=2(0)-2
y=-lliÐ +z
/:-2
/=3
Slope-lntercePt Forrn
The slope-intercept form of a line is y =
and b is the 1-intercePt.
mx*
b' where rn is the slope
of the line' The
The slope of a line tells you about the steepness and direction
the line crosses the,/-axis'
7-intelc-ept tells you where
Varying theT intercePt
Varying the sloPe
y=2x+b
Y=mx+l
/
¡n=2
v
=l
m=O.5
)
t
b
*
m=-2
14
cunPrER I
I
I
/b =[
4- J- t
/
I
/
,/
/
m=-0-5
m=-l
/
I
t
/
/,
/
Í.
.E.,tx
y=
to graph the equatio¡-Zri
ØUse the slope andT-intercept
'A
-3
*
SOLUTION
v
r. Write the equation in slope-intercePt
form,Y-- mx+b'
I
2x*f__3
v :2x-3
/
)
I
is
The slope is 2 and the 7-intercept
I
I
r
-3"
/
the slope
2. Plot the point (0, -3), and use
to find a second Point'
,r:@
change in Y
-?_i
the line'
3. Connect the two points to graph
TRY THIS
Use the slope and y-intercept
to graph the equation2x *
f
= 3'
is graPhed'
write an equatron for a line that
Example 3 shows you how to
v
form'
Write the equation, in slope-intercept
for the line graPhed'
(0 2)
s0LuTl0N
on the grlph' the
Because the point (0, 2) is
point
u-lrrt....pt ii z. Use another convenient
sloPe'
the
frnd
to
such as (3,0)'
\l
L
å;ù;ii;",
(3' o)'
Let (xr, /r) = (0, 2) and (x2'Yù =
lz-ft O-2--2
rn=l;l=T4
(\z ,¿ =
(0, 21.
TRY THIS
3
form is y = -þ + 2'
The equation in slope-intercePt
Writetheequation,inslope-interceptform,forthelinethatpassesthrough
(1, 4) and hãs a 7-intercePt of 3'
Standard Form
B'
equation is Á¡ + By = C'where '4'
The standard form of a linear
not both o'
;;ã ä;;ãd ,rr,*bt's and A ut'¿ ¡ are
x- and 7-intercepts to sketch
of a graph is the x-coordinate
use the
e
"tt"pt
x-axis'
LESSON
1.2
SLOPES nrr¡o
tnrEnCeprs 15
Use intercepts to graph the equation2x
-
3y = 6'
SOLUTION
To find the x-intercePt, let
2x-3Y=$
2x-3(0)=6
v
/ = 0.
,(
X:3
t,9)1,
x
To find the 1-intercePt, let x = 0
2x-3Y=$
2(0)-3Y=6
/: -2
two pointsGraph the points (3, 0) and (0, -2)- Draw the line through these
TRY THIS
Use intercepts to graph the equation 5x +
3y:
15
v
For the horizontal line at right, the formula
for slope gi""t
t, which is o'
-3, r)
ffi:
\2 3)
A horizontal line is a line that has a slope of 0'
For the vertical line at right, lhe formula for
which is undefined'
slope gives
f,,
A vertical line is a line that has an undefined
+ffi=
slope.
cHEcKpotNr
y' Which type of line has a 7-intercept but no x-intercept?
Which type of line has
an r-intercePt but no 7-intercePt?
Graph each equation.
a'/ = -2
b.
x: -3
s0[uTl0N
v
b
v
a.
r=4
Ì
)
r
/
x
ttl
I
\t
CRITICAL THINKING
16
cHnPren I
veriff that every line of the form
the form Ax= C is vertical-
By = C is horizontal and that every line
of
Ç âouuunlcate
how to arrange the linear equations given below in ascending
order of steepness, that is, from least steep to most steep.
a.y=3x-5
d.y=5x+5
b.y:Lr**t
e.y=5
c.y=þ*t
l.y=3x+5
z. Describe how to sketch the graph of the line 3x
t
2y = 4.
how to find the 7-intercept for the graph of 2x - 5y = L0.
+. Describe how to write the equation x - y = 2 in slope-intercept form.
@xlhin
Q
Qtrlded
AkÌlls Pracilce
s. Find the slope of the line containing the points (-2,4) and (8,
-3).
(EXAMPLE 1)
6- Use the siope and 7-intercept to graph the equatio
(EXATVTPLE 2)
"*
*
/ = -4.
v
z. Write the equation in slope-intercept
form for the line graphed at right.
(EXAMPLE 3)
8. Use intercepts to graph -2x
(EXAMPLE 4)
-
4y
:
x
t-
8.
Graph each equation. GxaMpLE s)
s.x:+
rc.
y:+
QPractlce and Apply
write the equation in slope-intercept form for the line that has the
indicated slope, m, and y-intercept, b.
11. rn
=2,b:0.75
13.m=0,b:-3
15. rn = -3, b :7
17- rn = Lob =
üE
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for Exercises 1$-26
-]
12.m=¡5rb:0
14. m=-f,,b =z
16.m=-l,b=-l
18.m=0.08, b=-2.9I
Find the slope of the line containing the indicated points.
19. 0, 0),and (3, 30)
1
20. (I, -3) and (3, -5)
3, -2) and (4,5)
22. (-10, -4) and (-3, -3)
-6, -6) and (-3, 1)
24-
(å,*) ""a (:, -j)
(-2,8) and (-2,
26. (-4,8) and
-l)
(-3,-6)
LESSoN
1.2
sLopES AND
tNlEncgprs 17
line' Then graph
ldentify the slope, rn, and the y'intercept' b' for each
z7.y+2x:0
28-Y=2
2s'-þtY=-7
30..x+Y=6
31.y-x
32.-2x=8+4y
35. x= -3
34.2x*
I Y: -4
33. -0.6x
Y
=|
line.
Write an equation in slope-intercept form for each
v
v
I
)(.
I
-2
I
x
v
39.
v
\
\
t
\I
x
It
t
\
x
to graph each equation.
r3y=12
44.7x -l3Y =
/
47.-x-7Y=3
50.5t-By=16
-3xt Y: -9
oe,-þ* 3y:7
4G-
42.2x-/=8
@-.l-ly=-6
48-x-Y:-1
51. x+þ =-Z
Find the slope of each line. Then graph'
sz.
x=5
@r =-S
58.y=-g
il. x=-+
'
@x=-2
5G.
x=-l
@r=,
ez-Y=|
54.y=$
x=9
6O-,C=
as. Y
=!
I
)
Find a'
64. The points (-2, 4), (0, 2), and (3, a - I) are on one line'
he declares
Tristan buys a computer for $3600. For tax Purposes'
'v;i'iä;ãä;;"úon (loss^of value) of $600 per yeàr.Letybe the declared
value of the comPuter aftet x Years'
a. what is the slãpe of the line that models this depreciation?
b. Find the y-intercept of the líne'
' c. Write a linear .qoàtion in slope-intercept form to model the value
of the comPuter over time'
d. Find the value of TÏistart's computer aftet 4'5 years'
,6?)to*tt
18
cnRrren I
m
The graph at right
66.
shows data for the distance
d
of
an object from a motion
detector.
a- Describe how the motion of
the object changes with time
in the graph.
b. Model the distance of the
object from the detector by
using three linear equations.
c. For what times is each
a
J
É
c)
|J
ßl
ø
â
_'J_,i
rl
tì
equation valid?
d. During which time interval
is the object moving the
t
Time (s)
fastest?
e. Is the slope of the segment representing the fastest movement positive
or negative? Explain what the sign of the slope indicates about the
motion of the object.
t
METE0R0LOGY The fhermometer at left shows temperatures in degrees
Fahrenheit, F, and in degrees Celsius, C. Room temperature is 20"C, or
68"F.
a. choose two other Fahrenheit temperatures, and use the thermometer at
left to estimate the Celsius equivalents.
b. Let the temperature equivalents from part a represent points of the
form (F, C). Graph the three points and draw a line through the points.
c. Find the slope and 7-intercept of the line.
d. Write the equation of the line in slope-intercept form.
C0NSTRUCTI0N The slope of a roof is called the pitch and is defined as
@ follows:
pitch =
- rise of roof
f
x
span
ofroof
if the rise is 12 feet and the span is 30 feet.
b. Find the pitch of a roof if the rise is lB feet and the span is 60 feet.
c. Find the pitch of a roof if the rise is 4 feet and the span is 50 feet.
d. If the pitch is constant, is the relationship between the rise and span
a. Find the pitch of a roof
linear? Explain.
LESSON
1.2
SLOPESANDINTERCEPTS
19
I
LTURE
The table àt right shows the
Pounds of
milk
Year
per daY
averagemilk production of dairy cows in the
1996'
to
1993
from
years
the
for
Uniteã States
42.7
t993
a- Let rc = 0 represent 1990' Make a line
44.3
r994
graph with years on the horizontal axis
45.2
r995
átrá pounds of milk on the vertical axis'
45.2
r996
b. In what year did the average milk
Dept. of
lSource:
p-roduction increase tþe most? What is
the slope of the graPh for that Year?
In whaì year did"thã averagemilk production increase the least? What is
". the slope of the graPh for that Year?
cow, one type of dairY
cow.
ffiLook Back
the equatiortv = Iwhwhenl:2,w =3,andh= 5'
principal'
71. Use the formula ¡: prtto find the interest,l, in dollars when the
2
years.
is
t,
time,
the
and
8%;
p, is $1000; the annual interest Íate,r,is
T2.IJsethe formula P -- 4sto find the perimeter of a square' P, in feet when
the length of a side, s' is 16 feet.
zo. Find the value of
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Extension
To: go.hrw.com
Keyword:
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MBl Rate
vin
73. Does the table of values at right represent
a linear relationship between x and y?'
Explain. If the relationship is linear, write
thá next ordered pair that would appear in
the table. r¿EssoN 1-1)
x -8 -5
.|
I
4
7
5
3
I -1
v
9
line' Is
z+' Make a table of values for the equation / = -3x * 7 ' anð graph the
,fr" point (4, -4) on this line? Eiplain how to answer this question by
using the table, the graph, and the equation' {rassoN 1'1)
Look Beyond
'*
É,i7Éà
Graph the equatio ns y - 2'L2x - 3'7 and / = x + 5'4 on the same screen'
Fittã the coordinates of any points of intersection'
Refer to your
portfolio data set from the Poitfolio Activity on page ll.
1.
oints that
points
others.
graP!'
2. Use a straightedge to draw a line through the pot-"l: on your
This line *iU U"þ"t [near model for yo.ur portfolio data set. Find the
equation of your linear model.
3. What rate of change is indicated by your linear model? Include the
appropriate units of measurement.
WOR KING ON THE CHAPTER PBOJECT
You should now be able to complete Activity 2 of the Chapter Project.
2O
cuePrrR I
Linear Equations
in T\ryo Variables
Tim\eaves his house and drives at a constant speed to go camping. on his way
to the campgrounds, he stops to buy gasoline. Three hours after buying gas,
Tim has traveled 220 miles from home, and 5 hours after buying gas he has
traveled 350 miles from home. How far from home was Tim when he bought
gas? To answer this question, you can use the information given to write a
linear equation in two variables. You will solve this problem in Example 3.
Example
I
shows you how to write a linear equation given two points on
the line.
write an equation in slope-intercept form for the line containing the points
(4,-3) and (2, t).
SOLUTION
t. Fiid the slope of the line.
\
--1 --)
*-r-(-t)
2-4-t-þ
z. Find the r/-intercept of the line.
Substitute -2 lor m and the coordinates of either point into y = mx
/:
mx+
*
b.
b
t=-2(2)+b
L =-4+b
s. Write an equation.
5=b
/ =mx+h
/=-2x+5
LESSoN
r.3
LTNEAB EouATtoNS tN
Two vnnnslËs 21
Youcanusethepoint-slopeformtowriteanequationofalineifyouaregiven
line'
th" ,iop. and the coordiÃates of any point on the
Point-SloPe Form
(xr, yr), then
If a line has a slope of m anð,contains the point
p.itri-tfãp" form of its equation is y - yt= rn(x - x) '
write
the
line that has a slope of
an equation in slope-intercept form for the
and contains the Point (-8' 3)'
]
SOLUTION
f - h=
ffi(lc
xr)
-
y-3=|t'-t-slt
Begin with Point-sloPe form'
Substitute 1, f or
n,
3 for yr, and
-8 for xr
^t
y-3=-/t*4
y=
I
)x+7
Write the equation in slope-intercept form'
constant speed can be
The distance traveled by a motorist driving at a
modeled bY a linear equation'
beginning of the lesson'
Refer to the travel problem described at the
/no*far from home was Tim when he bought gas?
v
SOLUTION
.
Ë
l
Wria. a linear equation to model Tim's distance' 1'
in terms of iime, x. Three hours after buying
gas, Tim has traveled 220 miles, and 5 hours
ãfter buying gas, he has traveled 350 miles'
The line contains
rL¿ trr. slope.
(3,22ü and
m
1z 2
(5, 350)'
=?Þff = 65
\
x
Write an equation. Begin with point-slope form'
f-ft=m(x-xr)
y-220=65(x-3)
IJse either point. The point (3, 220) is used here'
y -220 = 65x- 195
Write the equation in slope-intercept form'
/ = 65x +25
with respect to
Thus,7 = 65x * 25 models Tim's distance from home
bought
traveled afterhe
time. Since x represents the number of hours he
bought gas when he was 25 miles
gas, he bought gas when x 0. Thus, he
from home.
\
GHECKPOINT
22
cne"real
y'
What does the slope of the line
/=
65x + 25 inExample 3 represent?
Parallel and Perpendicular Lines
The graph at right shows Iine hparallel to
f3 perpendicular to (t atpoint P.
(
11 and
(2
(r
In the Activity below, you can explore how
parallel lines and perpendicular lines are related.
.
ffi"æ";ry
Exploring Parallel and Perpendicular Lines
TEGHNOTOGY
GRAPHICS
CATCUI.ATOR
Keystroke Guide,
You
need: a graphics calculator
1. Graph y = 2x * 1 On the same screen, graph y : 2x, y : 2x - 2.5, and
f = 3x + 1. Which equations have graphs that appear to be parallel to
that of y :2xi l? What do these equations have in common?
80
GHECKPOTNT
will
y'
z. Write an equation in slope-intercept form for a line whose graph you
think will be parallel to that of y :2x i L.Verifr by graphing.
3. Graph y = 2x + 1. On the same screen, graph y = -þ + Z,
/ = þ -f 2, and, y = -,!r+ 3. Which equations have graphs that appear to
be perpendicular to that of y = 2x + I? What do these equations have in
common?
CHECKPOTNT
y'
¿.
write an equation in slope-intercept form whose graph you think will
be perpendicular to that of y = 2x + I.Verifr by graphing.
The relationships between the slopes of parallel lines are stated below.
Parallel Lines
If'two lines have the same slope, they are parallel.
If two lines are parallel, they have the same slope.
All vertical lines have an undefired slope and are parallel to one another.
All horizontal lines have a slope of,0 and are parallel to one another.
v
The graphs of three parallel lines are
shown at right.
/=2x+3
/=b-l
/=2x-4
/
r
'=fi¡¡:J..
/
x
/
Notice that the lines do not intersect.
Because they are parallel, the lines
wlll never intersect.
LtsùsuN
t.r
LrNÈAr'i
EuuAÍiONS ¡N
tw_-O
Vnn¡nSLES
23
Write an equation in slope-intercept form for the line that contains the
point (-1,3) and is parallel to the graph of y = -2x t 4.
o
GRAPHICS
CATCUI.ATOR
Guide, page
soluilON
Because the line is parallel to the graph
= -2x * 4, the slope is also -2.
of
CHECK
/
/-lt=m(x-x)
y-3=-z[x-(-t\]
y
8l
-3 =-2x-2
/=-2x+I
TRY TH¡S
Write an equation in slope-intercept form for the line that contains the point
(-3, -4) and is parallel to the graph of y -- -4x - 2.
The relationships between the slopes of perpendicular lines are stated below.
.
.,Perpendicular Lines
If a nonvertical line is perpendicular to another line, the slopes of.the
lines are negative reciprocals of one another.
All vertical lines are perpendicular to all horizontal lines.
All horizontal lines are perpendicular to all vertical lines.
v
The graphs of two perpendicular lines
are shown at right.
| :2xf I and y:
-Lr*
-
2
24
3
-4
-6
Write än equation in slope-interceþt form fol the line that contains the
point (4, -3) and is perpendicular to the graph of y - 4x t 5.
SOLUTION
Because the line is perpendicular
GRAPHICS
CATGUTATOR
graph of y = 4x + s,the slope is
/-ft:ffi()c-xt)
Keystroke Guide, page 8l
,___Lnx_2
24
SHAPTERI
-+.
y-(-3):-l{r-+)
y¡3=-LnxIl
to the
CHECK
x
TßY THIS
Write an equation in slope-intercept form for the line that contains the point
(-1, 5) and is perpendicular to the graph of y =
-4x - 2.
The graph of 3x - / = 4 is perpendicular to the graph of Ax -t Zy =
value of A. Find A.
CRITICAL THINKING
Q
B
for some
âommunlcate
t. Describe how to write an equation in slope-intercept form for the line
containing two given points, such as (t,
j)
and (4, -2).
z. Explain how to use the different forms of
a linear equation to write the equation of
the line graphed at right.
v
(àDescribe how to determine whether the
lines 5x * 6y = 12 and 6x - 5y = 15 are
parallel, perpendicular, or neither.
+. Explain how to write the equation for the
line that contains the point (3, -l) and is
perpendicular to the line x t 2y = 4.
(ft
m
QuÌded
9kÌlls
x
Practtce
s-
write an equation in slope-intercept form for the line containing the
points (3,3) and (-5, -1). GXAMqLE 1)
o.
write an equation in slope-intercept form for the line that
3 and iontains the point (4,7). (EXAMqLE 2)
has a slope
of
Tina leaves home and
drives at a constant speed to
college. On her way to the campus,
shé Stops at a restaurant to have
lunch. Two hours after leaving the
restaurant, Tina has traveled 130
miles, and 4 hours after leaving
the restaurant, she has traveled
240 miles. How far from home
was Tina when she had lunch?
(EXAMPLE 3)
g. Write an equation in slope-intercept form for the line that
contains the point (6, -5) and is parallel to the ßne 2x - 5y =
(EXAMPLE 4)
-3.
g. write an equation in slope-intercept form for the line that contains the
point (-7, 3) and is perpendicular to the line 2x + 5y = 3. (ExArøpLE s)
LEssoN 1.3
LTNEAR
EouATtoNs tN TWo vARIABLES
Zs
ÇPractÌce and ApPly
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Homework
Help 0nline
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Keyword:
MBI Homework Help
for Exercises 10-21
Go
Write an equation for the line containing the indicated points.
1, -3) and (3, -5)
1o. (0,0) and (3,30)
( -10, -4) and (-3, -3)
12. (-4, -4) and (-3, -3)
( -2,8) and (-2, -l)
14. (-6, -6) and (-3, 1)
17. (8, -3) and (-8,3)
1e.. (4, -B) and (3, -6)
r
å)
zo. (-s,1) and (-+, t)
'". (+,-:)
ra. (-z!, 2) ana (-*
and
(r,-å)
zt. (-5,4) and (-t'-å)
write an equation in slope-intercept form for the line that
has the
indicated slope, m, and contains the given point.
_],
t)
24. m: -4, (5,-3)
@ *=-tr,(6, -5)
26' rn = 0, (2,3)
Ø
2a.m=4,(9,-3)
so. rn= _], {s, _z)
29.m=3,(-4,9)
22. m=
{a,
@ m = 5, (-1, -3)
ffi = 0, (-7,8)
31. m
=,_!, {s, _+)
Write a linear equation to model each table of values' For each
on, state what the slope represents.
32-
Hours
Miles
Items
Cost ($)
)
t35
4
14.00
5
225
7
21.50
34.
Parking
Hours
fee ($)
6.s0
J
t2.50
Write an equation in slope-intercept form for the line that contains the
given point and is parallel to the given line.
(-2,3),y=-3xr2
(0,-4), y=lr--l
39. (-1, -3),2x't 5y =
t
41. (3, 0), -x
*
2y
36. (5,
-3),y=4x+2
38. (-6,2),/=-þ-l
15
: l'/
4o. (4,
-3),3x't
4y = g
42.(4,a3),-4x+l=-7
Write an equation in stope-intercept form for the line tþat contains the
given point and is perpendicular to the given line'
foâ)1-2,5),f
\â(
:-2x+4
@(8,5),y=-x+2
47. (2,5),6x*2y =)4
4s-(-2,4),x-6y=15
++-(r,-4),y=3x-2
46- (0,
4a.
-5),Y=1c-5
(3,-l),l2xr
50- (5,
4y=$
-2),2x-5y=15
Sr. Write an equation for the line that is perpendicular to the line 2x * 5y =
lJ
at the y-intercePt.
sz. Write an equation for the line that is perpendicular to the line x
the x-intercePt.
26
cnRptn I
-
3y = 9 at
C00RDINATE GEOMETRY
For Exercises 53-58, refer to the lines graphed on
the coordinate plane below.
53. Use slopes to determine whether
is parallel to 12.
v
11
(,1
I
54. Use slopes to determine whether f3
is parallel to la.
55. Use slopes to determine whether
is perpendicular to (y
{r
-')
x
t-
57. Use slopes to determine whether
is perpendiailar to (,a.
d2
58. Use slopes to determine whether
is perpendicalar to (,a.
fr
60.
/¡
1)
/
(2
56. Use slopes to determine whether fz
is perpendicular to {3.
ss.
t
I
-3 4)
l
-t
?l
/
I
(,
eo
v
C00RDINATE cEOMETRY
Opposite
sides of a parallelogram are parallel.
Use slopes to determine whether
the quadrilateral graphed in the
coordinate plane at right is a
parallelogram.
CO0RDINATE GE0NIETRY
f"
1'(
tc
A rectangle
has opposite sides that are parallel
and four right angles. Use sloper to
determine whether the quadrilateral
graphed in the coordinate'plane at
right is a rectangle.
(0, a)
(a, a)
(0,0)
(a, o)
61. Use the diagram ai right to proye that the
diagonals-of any square are pert'endicular
\-
m
'l
bikes at nonconstant rate of speed from home through
@ town. HeMac
then begins his training ride at constant speed of 25 miles per
TBAVEL
a
a
hour. After 3 hours of biking at a constant speed, his odometer shows that
he has traveled 83 miles since he left home.
a-'write a [near equation in slope-intercepq form for the distance , d, in
miles that Mac has traveled in terms of tlie time, f; in hours since he
began his training ride.
b. When Mac began his training ride, how far from home'was he?
LESSoN
1,3
LINEAR EoUATIoNS IN TWo VARIABLES
27
@@'oo,M|csAprofessorgive¡atest,andthescoresrangefrom40to80.
vTh. professor decides to scale the test in order to make the scores range
from 60 to 90. Let r represent an original scole, andlety represent a
converted score.
a. Use the ordered pairs (40, 60) and (S0,90) to write the equation that
the professor will use to scale the test scores.
b. 'vVhat will an original score of 45 become?
c. If a converted score is B4, what was the original score?
is a salesperson who earns a wee6y salary and a
that is 7o/o of hisweekly sales. In one week Trevor's sales were
$952.00 and his weekly income was $466.64. In another week his sales
were $2515.00 and hisweekly income was $576.05
a. Write a linear equation in slope-intercept form for Trevor's weekly
Trevor
Æì'ucotrE
v.orn-ission
income,7, in terms of his weekly sales, x.
b. What is Tievor's weeklY salary?
e
Look Back
copy and complete the table. write the fractions in simplest form.
Decimal
Percent
65.
0.33
33!o¡,
66.
0-875
Fraction
67.
2o/o
I
68.
20
Izjvo
69.
2
70.
3
1
71.
6
0.0Io/o
72.
73-
0.80
2
74.
5
75.
0.45
5
76.
6
77. Use the formula d = rt to find the distance, d, in meters when the rate, r, is
50 meters per second and the time, f, is 4 seconds.
28. Use the formula C: nd to find the circumference, C, in centimeters when
the diameter, d,is B centimeters. Use 3.I4for n.
@Look Beyond
79.Lety = {Y.
f
a.-=
'
b.lfv=3,thenx=
lr. I
1C
8o.Let
28
crnpren I
f=mx.Ify=4andx=2,then m= ?
-
?
Direct Variation
and Proportion
Each day |ohnathon rides his bicycle for exercise. When traveling at a constant
rate, he rides 4 miles in about 20 minutes. At this rate, how longiould it take
Johnathon to travel 7 miles? To answer this question, you can use a directvariation equøtion or a proportion. you wiil sorve this probtem in Example 2.
Recall that distance, d, tate, r, and elapsed time, ú, are related by the equation
d: rt- You can say that d varies directþ as f because as time increases, ihe
distance traveled increases proportionally.
Direct Variation
The variable 7 varies directþ as x if there is a nonzero constant k such
tltat y = kx. The equation 7 : k¡ is called a direct-variation equation and
the nurnber k is called the constant of variation.
Find th'e constant of variation, k, and the direçt-variation equation
7 varies directþ as x and, y = -24wl¡ren x = 4.
if
SOLUTION
/: kx
-24:k.4
-24 _ k
4
Use
th
e di re ct-va ri ation eq u ation.
Substítute 24 for y and 4 for x.
Solve for k.
-6=k
The direct-variation equation is y = -6*.
TRY THIS
Find the constant of variation, k, and the direct-variation equation tf y varies
directly as x and f = Ilwhen r = 3.
LEssoN 1.4 DtBEcr vARtATtoN AND pnoponTloru
29
Refer to the problem described at the beginning of the lesson.
APP
t I C AT I O N
At the constant rate that Johnathon bikes, how long would it take him to
travel T miles?
SOTUT¡ON
r. Write a direct-variation equation,
as it varies with time.
4mi
20
min-
d = rt, that models Johnathon's distance
ffi
7mi?min-
Find the constant of variation, r.
r: #ft
=
f
mile per minute
Write the direct-variation equation.
distance ín miles
\
d-
time ín minutes
+Í
2- Use the direct-variation equation to solve the problem.
it:Lt
5
7
Lt
5
35=t
TECH NOTOGY
GRAPHICS
CATCUI.ATOR
ide, p age
Substitute 7 for
Solve for
d.
t.
CHECK
Graph the equatio n y = ]''-,and check to see that the point (35, 7) is on the
line.
the
81
variation,
of the
y
Thus, àt the rate given it will take ]ohnathon
TRY THIS
Xx.
minutes to travel 7 miles.
Suppose that when lohnathon is riding, he travels 5 miles iq about 30 minutes.
At this rate, how long would it take Johnathon to travel 12 miles?
The Proportion Property given below applies to all direct-variation
relationships.
Proportion Property of Direct Variation
Forq *0andx2*0:
If (n, y.)
'1,1'
1i
t:
T'?;
_i
tÍ
1,r,
30
CHAPTEB
1
and, (x2,
y2) satisfy y = Icx,thenf" =
O
=';.
In the Activity below, you can see a connection between the concepts of
geometric similariry proportion, and direct variation.
#æ"ry
Exploring Similarity and Direct Variation
You
will need: a calculator
Recall from geometry that similar
figures have the same shape. This means
that the corresponding angles of similar
polygons are congruent, and their
corresponding sides are proportional.
B,
B
A
C,
A
C
t.
Copy and complete the table below to compare the lengths of the sides
in LA'B'C' with the corresponding lengths in AABC.
AB=16
in LA'B'C'
A,B,:24
BC=20
B'C':30
-Eî=!
AC =24
A'C'=36
A'C'
Length in AÁBC
cHEcKPoINT
y'
Length
Ratio of LA'B'C'
A,B,
AB
AC
to LABC
?
=l
2.Do your calculations in the third column indicate a direct-variation
relationship between the lengths of the sides of AA'B'C' and those of
AABC? Explain your response.
It is said that If y varies directly
as
¿
then y is proportional to x.
A proportion is a statement that two ratios are equal. A ratio is the
comparison of two quantities by division. A proportion of the form ctc
bd
can be rearranged as follows:
a
_
c
6-ã
+.bd=!.bd
bd
ad=bc
The rêiult is called the Cross-Produbt Property pf Proportions.
Cross-Product Property of PropoÍtions
'
-
Forb*Oand d+0:
,c4
tl --- -c
b à,then ad: bc.
In a proportion of the formf,=â, o and d are the extremes andb and c are the
means. By the Cross-Product Property, the product of the extremes equals the
product of the means.
LESSoN
1.4
DrREcr vARtAT¡oN AND pnopoRTtoru
3tr
PttcATl0
compare the weight
Using Newton's law of universal gravitation, ratios that
on Earth with its weight on another planet can be calculated'
"i"î"Ui..t
For Mars and Earth, the ratio is shown below'
N
Hzle",i"ly:ï,=w=ffi
vehicle) that was sent
Sojourneris the name of the first rover (robotic roving
the size of a
to Mars. Sojourner weighs 2l.3pounds on Earth and is about
small wagon.
child's
.. pin¿ the weilht of Sojourner on Mars to the nearest tenth of a pound'
¡. Write a direCt-variatiôn equation that gives the weight of an object on
Mars, W¡r,t,in terms of its weight on Earth, I4l¿'
SOLUTION
a. Solve the proportion for the weight of Sojourner on Mars'
Wu-38
24.3 100
(24.3)(35) Use the Cross-Product
a,r
Q4.3)(38) <- weight on Mars
vv M = ---Tol<- weight on Earth
(W¡z)(100) =
Property'
Wm= 9'2
On Mars, Sojourner would weigh about 9'2 pounds'
The Sojourner
Wu_38
wn
131t, = weight on Mars
W¡¿=ff
*- wtigntonEarth
b.
W7¿= 0.3BWE
j TECHN0L0
CHECK
GY
GraphT = 0.38x, and confirm that a weight
of l+S portnds on Earth, r' corresponds to a
weight of about 9.2 pounds on Mars,7'
GBAPHICS
CAICULATOR
G
ide, page
81
Ç
SU*
-t
=
f. checkyour
answer
s0ruil0N
3x-I 5
(3x
-
x
2
1)(2) = (5)
CHECK
(x)
l[se the Cross-Product Properiy'
6x-2=5x
x-
CRIT¡CAL TH¡NKING
solve
Z-- 0
Yl
cunPren I
f. Checkyour
True
answer'
f
Let a>0. How many solutions does
your answer,
32
=
52 =t
2
5 -2
1=1
3(2)-r t
x=2
TRY THIS
3x- |
=
Lhave? Find the solutions.
|ustiff
ÇâorønunÌcate
t. Suppose thaly varies directly
as x and thaty = l8 when ¡ = 9. Describe
how you would find an equation of direct variation that relates these'
two variables.
p)wh-""
v
are linear equations not dirèctvariations? How do their graphs
differ from those of direct variations?
@D"sc.ibe
two methods for solving the following
: B when x
If y variesdirectly u.
"rrd.lz
what is the value of x"when
y = lZ?
problem:
ì
= -2,
Determine whether each equation describes a direct variation. Explain
your reasoning.
4-l=x+5
6'Y:5x
@v:*-t
Ø y:t
Q Qutded akÌlls practÌce
m
8- Find the constant of variation, k, and the direct-variation equation if
7
varies directly as x and J¿ = 1000 when x :200. GXAMzLE 1)
PHYslcs The speed of sound in air is about 335 feet per second. At this
rate, how far would sound travel in 25 seconds? GxarøpLE 2)
@lrtrcorvrr workers at a particular store earn hourþ wages. A person who
worked 18 hours earned $114.30. GXAMqLE s)
a. How many hours must this person work to earn$727?
u. write a direct-variation equation that gives the income of this person
in terms of the hours worked. what does the constant of variation
represent?
Solve each equation for x. Check your answers. (EXAMzLE 4)
4x-I _x
21 6
@+:
3x
36
QPractÌce and ApplV
ln Exercises 14-29, y varies directly as x. Find the constant of variation,
and write an equation of direct variation that relates the two variables.
14. y - 2l when x-- 7
y=2whenx=l
16'y=-16whenx=2
y:
1a.y-f,whenx=|
y=-+whenx=-#
2O.'y=-2whenx=9
22.y= l.$when x=30
24.y=24whenx=8
Iwhenx=T
.12=5whenx=-0.1
23. y = 0.4 when.x= -l
2r..y=tZwhenx'=ï
LESSoN
1.4
DtREcr vARtATtoN AND pnoponTlolr¡
33
27'y=4when x--0'2
29-y=-1'2whenx=4
x=-l
28.y=0.6when x=-3
2f'.y=-f,when
write an equation that describes each d¡rect variation.
3O. p varies directly as
q.
31. ø is directly proportionalto b'
For Exercises 32-36, a varies directly as b'
s2. Íf ø is 2.8 when b is 7 , frnd ø when b is -4'
6.3when b is 70, find b when a is 5.4'
34. If ø is -5 when b is 2-5, find b when ø is 6'
ø is -å, frnd awhen b is ].
b is
ø is
@rf
]when
@rr
36. If b ß -+when ø i,
-å,find
ø
when b is -N'
solve each proportion for the variable. check youf answers.
Homework
Help Online
Go
To: go.hrw.com
Keyword:
MBI Homework HelP
for Exercises 37-51
37
w=D
tt'2r--i
40.
9=9
l0r
41';=
43
x-4
412
*
t''å=rfu
_'7
or.?
ro
=
;
4g..+=x-6
/J
-5
ot.+= x-3
ou.+=ä
u'.å=+
n'-+=#
Determine whether the values in each table represent a direct variation.
lf so, write an equation for the variation' lf not, explain'
52.
54.
56.
s8. Show that if ¡ varies directþ
as 7,
then 7 varies directþ
59. If a varies directly as c and b varies directly
as c, show thaf ø + b varies directþ as c.
as
(t
x'
v
\
\
ffi
34
cHnprER t
60.
Which of the lines
shown in the graph at right represents a
direct variation? Explain your reasoning.
I
COORDINATE GE0METRY
)c
\r
/r
. {r
OULTURAI c0NNEcTlON: ASIA The
Harappan civilization flourished in an area
near present-day Pakistan around 2500 B.c.E. They used balancing stones in
their system of weights and measures. The vedic civilization, which followed
the Harappan civilization, used gunja seeds to weigh precious metals. The
smallest Harappan stone has the same mass as B gunja seeds.
6r.
The mass of a Harappan stone, m,
varies directly as the number of gunja
seeds, g. Find the constant of variation
and the direct-variation equation for
this relationship.
1.76 grams
os. The largest Harappan stone is
equivalent to 320 gunja seeds. What is
the mass of this stone?
with
ô
0.88 gram
€.
62. How many gunja seeds are equivalent
to a Harappan stone whose mass is
3.52 grams?
The scale is balønced
Smallest
Hørappan stone
e,
3.52 grams
320 gunjø seeds
Lørgest
Harøppøn stone
16 gunjø seeds
on the left and the second smallest
Hørøppan stone on the right.
64. GE0METRY In the figure at right,
the height of each object is direcrly
proportional to the length of its
shadow. The person is sj feet tall
and casts an 8-foot shadow, while
the tree casts a 33-foot shadow.
How tall is the tree?
65. GE0METRY In an aerial photograph,
a triangular plot of land has the
dimensions given in the figure at right.
If the actual length of the longest side of
the plot is 50 kilometers, find the actual
lengths of the two shorter sides.
7cm
6cm
l0 cm
mpuvsícbInanelectriccircuit,ohrnslawstateqthatthevoItage,V,measured
in volts varies directþ as the electric current, 1, measured in amperes according
to the equation V: IR. The constant of variation is the eleqrical resistance of
the circuit, R, measured in ohms.
iron is plugged into a 1lO-volt electrical outlet, creating a currenr
of 5.5 amperes in the iron. Find the electrical resistance of the iron.
P")4"
I
heater is plugged into a 1lO-volt outlet. If the resistance of the heater
@)n
v is 1l ohms,
find the current in the heater.
I,lllnïï,ï::'j;iffi
.
äî:','*îï,o;ïi;l$ffi;îfi äl;ïì."*T.Tll'""
the current, to the nearest hundredth of an ampere, in a lamp that
has a resistance of 385 ohms and is plugged into a I l0-volt outlet.
@)fina
-
LEssoN 1.4 DTREcTvAHtATtoNAND pRoPoRT¡ot¡
35
pressure vanes
As a scuba diver descends, the increase in water
However, the
surface.
directly as the increase in dePth below the water's
in salt water. For
constan t of variation is smaller in fresh water than
lake, the
example, at 80 feet below the surface of a typical freshwater
the
than
greater
Pressure at the
pressure is 34.64 pounds Per square inch
the Pressure at 80 feet
surface. In a typical ocean, where the water is salty,
at the surface.
is 35.6 pounds Per square inch greater than the pressure
variation for
direct
of
uation
eq
the
a. Find the constant of variation and
the increase in pressure in a typical freshwater lake'
a lake'
u. Find the incrçaie in pressure tOO feet below the surface of
"i
variation for
c. Find the constant ofìariation and the equation of direct
the increase in pressure in a typical ocean'
an ocean'
u. Find the increase in pressure at 100 feet below the surface of
of
The distance a spnng stretches varies directþ as the amount
spring
the
stretches
pounds
32
of
weight that is hanging on it. A weight
6 inches, and a weight of 48 Po unds stretches it 9 inches.
variation for
a. Find the constant of variation and the equa tion of direct
represent?
the stretch of the sPring. What does the constant of variation
is
stretched
it
when
spring
the
on
b. How heavy is the weight hanging
3 inches?
hanging
c. Find the stretch of the spring when a weight of 40 pounds is
on it.
@took Back
F
internetconnect
Activities
0nline
To: go.hrw.com
Keyword:
Go
MBl Metrics
Write the prime factorization for each number'
72.261 73. 860 74-315 75.180
76-154
Evaluate.
.,
l1
7A.
5
79
13
11
6
15
26
t2
ffitook
77'490
80.
3
4
2L
81.
5
28
25
BeVond
zero, is
An equation of the lorm xY k, where k is a constant greater than
k,
for and
called an inv erse-v ariation equøtion. Choose a positive value
graph the equation. Describe the graph
on page 11.
Refer to your portfolio data set from the Portfolio Activity
as the other
1. In your portfolio data set, does one variable vary directþ
variable? ExPlain.
of the
2. using data values from your portfolio data set, write a proportion
formy]=y].ßthe proportion true for some values? Is the proportion
true for all values? ExPlain.
3. using points on your linear model from the Portfolio Activity on
true
page 2¡,write a proportion of the fomy]=y].3 the proportion
for some values? Is the proportion true for all values? Explain.
36
csePren-l
I
1
'|
J
¡
I
i
¡l-t
t