Solutions Key 13 - Schilling Farms Middle School

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3EQUENCES AND &UNCTIONS
!2% 9/5 2%!$9 0!'% LINEAR EQUATION
SLOPE YINTERCEPT
POINTSLOPE FORM
4HE PATTERN IS TO ADD TO THE NUMERATOR AND
SUBTRACT FROM THE DENOMINATOR
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4HE NEXT THREE NUMBERS IN THE PATTERN ARE ???
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4HE PATTERN IS TO INCREASE EACH NUMBER BY THE NEXT
LARGER ODD INTEGER
4HE NEXT THREE NUMBERS IN THE PATTERN ARE 4HE PATTERN IS TO ADD TO EACH INTEGER
4HE NEXT THREE NUMBERS IN THE PATTERN ARE 4HE PATTERN IS TO DECREASE EACH NUMBER BY ??
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4HE NEXT THREE NUMBERS IN THE PATTERN ARE ??
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A S B DC FOR A B C S D S D
S D
S D
Y ??X SLOPE ?? YINTERCEPT 4HE LINE CROSSES THE YAXIS AT THE POINT S D AND
MOVES UNITS UP FOR EVERY UNITS IT MOVES TO THE
RIGHT
Y
X
/
Y ??X SLOPE ?? YINTERCEPT 4HE LINE CROSSES THE YAXIS AT THE POINT S D
AND MOVES UNIT DOWN FOR EVERY UNITS IT MOVES
TO THE RIGHT
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ALREADY IN SIMPLEST FORM
Y X Y
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Y ??X SLOPE ?? YINTERCEPT 4HE LINE CROSSES THE YAXIS AT THE POINT S D
AND MOVES UNITS UP FOR EVERY UNITS IT MOVES
TO THE RIGHT
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0!'%3 n
Y
4HINK AND $ISCUSS
0OSSIBLE ANSWER &IND THE DIFFERENCES BETWEEN
THE SUCCESSIVE TERMS TO SEE IF THERE IS A COMMON
DIFFERENCE
X
/
0OSSIBLE ANSWER &IND THE FIRST TEN TERMS BY
REPEATEDLY ADDING THE COMMON DIFFERENCE 4HE
TENTH TERM IS S D
!LSO BY USING THE FORMULA THE TENTH TERM IS S D S D Y X %XERCISES
Y X Y
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4HE SEQUENCE IS NOT ARITHMETIC
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4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
DIFFERENCE OF ??
X Y 4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
Y X DIFFERENCE OF Y
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4HE SEQUENCE IS NOT ARITHMETIC
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4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
COLORDIFFERENCE OF X
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#OPYRIGHT © BY (OLT -C$OUGAL
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4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
DIFFERENCE OF ??
4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
DIFFERENCE OF N
N
Y
N
N
N
&IND THE TH TERM A N A S N DD
A S D
A S D
A A &IND THE TH TERM A N A S N DD
A S D
A S D
A A A D A N A N A N D
N N N
N
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4HE PACKAGE WEIGHS OUNCES
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
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4HE SEQUENCE IS NOT ARITHMETIC
N
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Y N
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&IND THE ST TERM A N A S N DD
A S D S D
A S D S D
A S D
A &IND THE TH TERM A D A N A S N DD
A S D
A S D
A A ??
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&IND THE TH TERM AN A SN DD
A S D S D
A S D S D
A S D
A &IND THE RD TERM A N A S N DD
A S D
A S D
A A (OLT -C$OUGAL -ATHEMATICS
&IND THE TH TERM A D A N A S N DD
A S D
A S D
A A A D A A A 4HE FIRST FOUR TERMS OF THE SEQUENCE ARE &IND THE TH TERM A D A N A S N DD
A S D S D
A S D S D
A S D
A A A A A A A B A N A S N DD
A S D
A S D
A A C A N A S N DD
S N D
N N
N
N
LAPS
)DENTIFY THE ARITHMETIC SEQUENCE A D
A N A N A S N DD
S N D
N N
N
N
/SCAR RACED FOR HOURS
A D A A A A 4HE FIRST FIVE TERMS ARE A D A A A A 4HE FIRST FIVE TERMS ARE A D A A A A 4HE FIRST FIVE TERMS ARE )DENTIFY THE ARITHMETIC SEQUENCE A D
A N A N A S N DD
S N D
N N
N
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IS THE TH TERM
A D A A A A A 4HE FIRST FOUR TERMS OF THE SEQUENCE ARE #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
0OSSIBLE ANSWER 7HAT ARE THE FIRST TWO TERMS OF THE
ARITHMETIC SEQUENCE WITH A AND D !NSWER AND 0OSSIBLE ANSWER 3UBTRACT EACH TERM FROM THE NEXT
TERM TO FIND THE COMMON DIFFERENCE )F THE COMMON
DIFFERENCE IS POSITIVE THE TERMS OF THE SEQUENCE ARE
INCREASING )F IT IS NEGATIVE THEY ARE DECREASING
4HE CONSECUTIVE TERMS ARE A N AND A N A N S A N D A N A N A N SO A N 3OLVE FOR N
A N S N D
S N D
N N
N
N N 4HE CONSECUTIVE TERMS ARE AND AND THEY ARE
THE TH AND TH TERMS
# )DENTIFY THE SEQUENCE A D A N A S N DD
S N D
N N
N
N
A D A N A S N DD
A S D
A S D
A A (OLT -C$OUGAL -ATHEMATICS
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X ? K K &IND THE TH TERM R ?? A N A R N
A S D &IND THE ST TERM ?? ?? ?? ?? ?? ?
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A N A R N
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"IASED POSSIBLE ANSWER 4HE QUESTION LEADS PEOPLE
TO AGREE WITH hEXPERTSv
.OT BIASED POSSIBLE ANSWER 4HE QUESTION DOES NOT
LEAD PEOPLE TO GIVE A PARTICULAR RATING
0!'%3 n
4HINK AND $ISCUSS
0OSSIBLE ANSWER )N AN ARITHMETIC SEQUENCE A
CERTAIN NUMBER THE COMMON DIFFERENCE IS ADDED
TO EACH TERM TO FIND THE NEXT TERM )N A GEOMETRIC
SEQUENCE A CERTAIN NUMBER THE COMMON RATIO IS
MULTIPLIED BY EACH TERM TO GET THE NEXT TERM
0OSSIBLE ANSWER $IVIDE EACH TERM BY THE TERM
BEFORE IT
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UNDEFINED 4HE SEQUENCE IS NOT GEOMETRIC
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ?? ?? ?? ?? ?? x
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF (EATHERS STARTING SALARY IS A AND IF SHE GETS A
RAISE EVERY TIME SHE IS ELIGIBLE SHE WILL RECEIVE RAISES 3INCE EACH RAISE IS AFTER EACH RAISE SHE
WILL MAKE OR TIMES HER PREVIOUS SALARY
SO THE COMMON RATIO IS A R N A N A R N
A S D y )F (EATHER GETS A RAISE EVERY TIME SHE IS ELIGIBLE
AFTER YEARS SHE WILL MAKE PER HOUR
%XERCISES
A ?? S D ??
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
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4HE SEQUENCE IS NOT GEOMETRIC
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4HE SEQUENCE IS NOT GEOMETRIC
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ???
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
?? 4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
A R 4HE FIRST FIVE TERMS ARE &IND THE TH TERM ?? R ?? ??
A N A R N
A ?? S D A R S D S D S D S D 4HE FIRST FIVE TERMS ARE &IND THE TH TERM R ????? ????
A N A R N
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&IND THE TH TERM ?? ??
R ???
A N A R N
A ?? ???
A R 4HE FIRST FIVE TERMS ARE S D
A R ??
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&IND THE TH TERM R ?? A N A R N
A S D ???
?? ?? &IND THE ST TERM ???
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A N A R N
S D ??
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A R 4HE FIRST FIVE TERMS ARE &IND THE TH TERM R ???
A N A R N
A S D 4HE STARTING POINTS ARE 3INCE ONE FIFTH OF
THE TOTAL POINTS ARE ADDED AT THE END OF EACH LEVEL
AND ONEFIFTH IS THE NUMBER OF POINTS WILL BE TIMES HIGHER AT THE END OF EACH LEVEL
A R N A N A R N
A S D 4HERE ARE POINTS AT THE END OF THE FIFTH LEVEL
A COMMON RATIO ??
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? ?? 4HE NEXT THREE TERMS ARE A COMMON RATIO 4HE NEXT THREE TERMS ARE #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
A R S D S D S D S D 4HE FIRST FIVE TERMS ARE R A ???
A N A R N
A S D ???
A S D
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A R A N A R N
A S D A S D
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A ?? (OLT -C$OUGAL -ATHEMATICS
A A A ???
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A A N A R N
A ??
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A N A R N
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A A A ??
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A N A R N
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A 4HERE ARE TWO TEAMS
LEFT IN THE SIXTH ROUND
S D
S D
S D
S D
S D
S
D
4OWN ! A R N A N A R N
A S D A y 4OWN " A R N A N A R N
A S D A y #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
0OSSIBLE ANSWER 4HE STUDENT DIVIDED BY THE
COMMON RATIO INSTEAD OF MULTIPLYING BY IT 4HE
CORRECT ANSWER IS 0OSSIBLE ANSWER %VEN THOUGH BOTH SEQUENCES HAVE
AS THEIR SECOND TERMS THE SEQUENCE WITH THE
GREATER COMMON RATIO GROWS MUCH FASTER THAN THE
SEQUENCE WITH THE LESSER COMMON RATIO
A A 4HERE ARE DAYS IN THREE WEEKS 4HE CELLS TRIPLE
EVERY DAYS SO YOU WANT TO KNOW HOW MANY
CELLS THERE ARE AFTER THE CELLS HAVE TRIPLED TIMES
4HEREFORE N A R N A N A R N
A S D A S D
A 4HERE ARE CELLS AFTER WEEKS
3INCE YOU WANT TO KNOW HOW MUCH THE CAR IS WORTH
AFTER YEARS N A R N A N A R N
A S D A y 4HE CAR WILL BE WORTH APPROXIMATELY A #HECK TO SEE IF THERE IS A COMMON RATIO FOR THE
VALUES IN THE HEIGHT ROW
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4HE SEQUENCE COULD BE A GEOMETRIC SEQUENCE
WITH A COMMON RATIO OF ??
B A R ?? N A N A R N
A ??
A ??????
A y 4HE BALL WILL BOUNCE APPROXIMATELY FEET AFTER
THE TH BOUNCE
S D
)F THE RATE CONTINUES 4OWN " WILL HAVE A GREATER
POPULATION 4OWN ! WILL HAVE AN ESTIMATED
POPULATION OF AND 4OWN " WILL HAVE AN
ESTIMATED POPULATION OF A N A R N
A A R A A R A ????
A R ??
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A A R R R
A ???????
SO R ??
A 'UESS AND CHECK TO FIND R
SO R &IND A A N A R N
A S D A S D
A
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&IND A A N A R N
S D A ????
A # A R N A N A R N
A S D A #HECK FOR A COMMON RATIO
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
M ???
M S D S D
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M X
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X
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X
(OLT -C$OUGAL -ATHEMATICS
X ???
X S D S D
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IS FOLLOWED BY ONE MORE THAN ITS OPPOSITE 4HE NEXT
THREE TERMS ARE S P D P
P P
P ! POSSIBLE RULE IS THAT THE SEQUENCE REPEATS THE
PATTERN 4HE NEXT THREE TERMS ARE X X S X D
X X X S Y D Y Y
! POSSIBLE RULE FOR THE SEQUENCE IS THAT IT IS THE LIST
OF NEGATIVE PERFECT SQUARES
S D S D AND SO ON 4HE NEXT
THREE TERMS ARE 0!'%3 n
4HINK AND $ISCUSS
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A N ?????
N
A ?????
A ?????
A ?????
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FIRST DIFFERENCES SECOND
DIFFERENCES %XERCISES
3EQUENCE ST
$IFFERENCE
ND
$IFFERENCE
4HE FIRST FIVE TERMS ARE ?? ?? ?? ???
A N S N D S N D
A S D S D S D S D A S D S D S D S D A S D S D S D S D A S D S D S D S D A S D S D S D S D 4HE FIRST FIVE TERMS ARE 4HE NEXT THREE TERMS ARE 3EQUENCE ST
$IFFERENCE
ND
$IFFERENCE
N
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A N N
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A ?????
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A 4HE NEXT THREE TERMS ARE 3EQUENCE
ST
$IFFERENCE
ND
$IFFERENCE
4HE NEXT THREE TERMS ARE ST
$IFFERENCE
ND
$IFFERENCE
4HE PRODUCT AC IS EITHER ONE MORE OR ONE LESS
THAN B 4HE NEXT THREE TERMS ARE ??? x
?? ?? ?? ?? ??
3EQUENCE ! POSSIBLE RULE IS TO ADD TO THE NUMERATOR AND
DENOMINATOR OF THE PREVIOUS TERM 4HE NEXT THREE
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TERMS ARE ???
ST
$IFFERENCE
! POSSIBLE RULE IS THAT EACH POSITIVE TERM IS FOLLOWED
BY ONE LESS THAN ITS OPPOSITE %ACH NEGATIVE TERM
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
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4HE FIRST FIVE TERMS ARE ?? ?? ??
3EQUENCE
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ND
$IFFERENCE
4HE NEXT THREE TERMS ARE (OLT -C$OUGAL -ATHEMATICS
3EQUENCE ST
$IFFERENCE
ND
$IFFERENCE
4HE NEXT THREE TERMS ARE 3EQUENCE
ST
$IFFERENCE
ND
$IFFERENCE
4HE NEXT THREE TERMS ARE A N N S N D N
A S D S D S D A S D S D S D A S D S D S D A S D S D S D A S D S D S D 4HE FIRST FIVE TERMS ARE N
A N ?????
N
S D
A ?????
S D
A ?????
S D
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S D
A ?????
S D
A ?????
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4HE FIRST FIVE TERMS ARE ?? ?? ???
3EQUENCE ST
$IFFERENCE
ND
$IFFERENCE
4HE EVEN NUMBERS IN THE &IBONACCI SEQUENCE ARE
AND 4HESE ARE THE RD TH TH TH
TERMS 4HE NEXT FOUR EVEN NUMBERS WILL OCCUR IN THE
TH TH ST AND TH TERMS
! POSSIBLE RULE FOR THE SEQUENCE IS A PATTERN OF
A NUMBER AND ITS OPPOSITE THEN THE NEXT TERM IS
ONE MORE THAN THE ORIGINAL NUMBER ALONG WITH ITS
OPPOSITE AND SO ON 4HE NEXT THREE TERMS ARE
! POSSIBLE RULE FOR THE SEQUENCE IS TO FIRST ADD THREE
TO THE PREVIOUS TERM TO FIND THE NEXT AND THEN
SUBTRACT ONE FROM THE PREVIOUS TERM AND CONTINUE
THIS PATTERN 4HE NEXT THREE TERMS ARE 4HE MULTIPLES OF ARE AND 4HESE ARE THE
TH TH TH TERMS 4HE NEXT FOUR MULTIPLES OF WILL
OCCUR IN THE TH TH TH AND TH TERMS
3EQUENCE
ST
$IFFERENCE
! POSSIBLE PATTERN IS THAT A ZERO IS ADDED BETWEEN
THE DECIMAL POINT AND THE IN THE DECIMAL PART OF
THE NUMBER 4HE NEXT THREE TERMS ARE ND
$IFFERENCE
4HE NEXT THREE TERMS ARE ???
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! POSSIBLE RULE FOR THE SEQUENCE IS TO TAKE THE
RECIPROCAL OF THE CUBE OF A NUMBER THAT INCREASES BY
ONE EACH TERM 4HE NEXT THREE TERMS ARE
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C IS EITHER ONE MORE OR ONE LESS THAN AE AND BD
4HE NEXT THREE TERMS ARE ?????
A N N
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A ??
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4HE SEQUENCE OF THE FREQUENCIES ! ! ! ! IS ????
IT IS A GEOMETRIC
??? ????
"ECAUSE SEQUENCE 3O A AND R S
D
A N A R N
S N D
A N 4HE SEQUENCE OF HARMONICS IS "ECAUSE IT IS AN ARITHMETIC SEQUENCE WITH
A COMMON DIFFERENCE OF A A 4HE FREQUENCY OF % IS (Z
4HE FIRST FIVE TERMS ARE ?? ?? ?? ??
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
4HE SEQUENCE OF HARMONICS IS "ECAUSE IT IS AN ARITHMETIC SEQUENCE 3O A AND D A N A S N DD
A N S N D
A N N A N N
2%!$9 4/ '/ /. 0!'% 4HE SEQUENCE IS NOT ARITHMETIC
4HE SEQUENCE COULD BE ARITHMETIC WITH A COMMON
DIFFERENCE OF 4HE SEQUENCE IS ?????
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"ECAUSE ????
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IT IS A GEOMETRIC
?????
SEQUENCE 3O A AND R A N A R N
S
D
A N N
4HE NOTE & IS THE TH NOTE IN THE SEQUENCE SO
FIND A S
D
A A A y 4HE NOTE & WOULD HAVE A FREQUENCY OF ABOUT
(Z
4HE SEQUENCE IS NOT ARITHMETIC
&IND THE TH TERM A D A N A S N DD
A S D
A S D
A &IND THE TH TERM A D A N A S N DD
A S D S D
A S D S D
A &IND THE TH TERM ?? ?? A D ??
A N A S N DD
A S D??
A S D??
A ??
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# A N N
N
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A ?????
A N N
N
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A ??
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4HE SEQUENCE IS NOT GEOMETRIC
&IND THE TH TERM A D A N A S N DD
A S D
A S D
A ?? ???
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ?? ???
????
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4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF A D A N A N A S N DD
S N D
N N
N
N
#ARMEN WILL MAKE BRACELETS IN THE ST WEEK
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF ??
4HE SEQUENCE IS NOT GEOMETRIC
#OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
(OLT -C$OUGAL -ATHEMATICS
3EQUENCE
4HE SEQUENCE COULD BE GEOMETRIC WITH A COMMON
RATIO OF &IND THE TH TERM A R ???
A N A R N
A A A ST
$IFFERENCE
ND
$IFFERENCE
ND
$IFFERENCE
A N N A S D A S D A S D A S D A S D 4HE FIRST FIVE TERMS ARE 0OSSIBLE ANSWER 4HE RULE FOR THIS SEQUENCE COULD
BE THE TH POWERS 4HE NEXT THREE TERMS ARE 4HE NEXT THREE TERMS ARE #OPYRIGHT © BY (OLT -C$OUGAL
!LL RIGHTS RESERVED
?? ?? ?? ???
0OSSIBLE ANSWER !DD TO THE PREVIOUS NUMERATOR
AND DENOMINATOR
???
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4HE NEXT THREE TERMS ARE ???
ST
$IFFERENCE
4HE NEXT THREE TERMS ARE 3INCE THE MACHINE DEPRECIATES BY EACH YEAR
ITS VALUE IS OR TIMES THE PREVIOUS YEARS
VALUE 3O R 4HE END OF THE SIXTH YEAR IS THE
BEGINNING OF THE SEVENTH YEAR SO N A S D 4HE VALUE OF THE MACHINE AFTER YEARS IS ABOUT
3EQUENCE S D
S D
TH TERM A R ?????? ???
A N A R N
A ???
A ???
A 4HE NEXT THREE TERMS ARE S D
S D
ND
$IFFERENCE
&IND THE TH TERM ??
A R ???
N
A N A R
A ??
A ??
y A ????
&IND THE TH TERM ?? ?
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A R N
A N A R
A ?? S D A ?? S D A 3EQUENCE
ST
$IFFERENCE
A N N
A A A A A 4HE FIRST FIVE TERMS ARE A N S D N N
A S D S D A S D S D A S D S D A S D S D A S D S D 4HE FIRST FIVE TERMS ARE A N S N D A S D A S D A S D A S D A S D 4HE FIRST FIVE TERMS ARE (OLT -C$OUGAL -ATHEMATICS
0!'%3 n
7RITE THE FUNCTION IN SLOPEINTERCEPT FORM
F S X D X F S X D X 4HE FUNCTION IS LINEAR BECAUSE IT CAN BE WRITTEN IN
THE FORM F S X D MX B 4HE SLOPE M IS n AND THE
YINTERCEPT B IS 4HINK AND $ISCUSS
0OSSIBLE ANSWER 3UBSTITUTE THE YINTERCEPT AND THE
COORDINATES OF A POINT ON THE GRAPH INTO THE EQUATION
Y MX B AND SOLVE FOR M 7RITE THE EQUATION
USING THE VALUES FOR M AND B THAT YOU FOUND
7RITE THE FUNCTION IN SLOPEINTERCEPT FORM
F S X D X X
F S X D X
4HE FUNCTION IS LINEAR BECAUSE IT CAN BE WRITTEN IN
THE FORM F S X D MX B 4HE SLOPE M IS AND THE
YINTERCEPT B IS ?
4HE FUNCTION F S X D ?
X CANNOT BE WRITTEN IN SLOPE
INTERCEPT FORM SO IT IS NOT LINEAR
%XERCISES
7RITE THE FUNCTION IN SLOPEINTERCEPT FORM
F S X D X X F SX D X 4HE FUNCTION IS LINEAR BECAUSE IT CAN BE WRITTEN IN
THE FORM F S X D MX B 4HE SLOPE M IS AND THE
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