pre-apalgebra ii- chapter 8day#3 ii -- cha… · 2 we finally want to beableto relate...
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Pre-AP Algebra II - Chapter 8 Day #3Topic: Multiplying Binomials that have Radical ExpressionsStandards/Goals:
./ A.SSE.2: I can take the structure of an expression and identify different ways torewrite it
./ I can use the FOILmethod to multiply binomials that have radical expressions .
./ I can multiply binomial expressions using conjugates.We first want to focus on how to MULTIPLYbinomials that have radical expressions.What is the product of the following binomials?#1. (4 - .../3) (5 - 6.../3) #2. (1 + 2"fj) (4 - 3"fj)
We also want to focus on conjugates and how to multiply two radical conjugates, whendealing with rational numbers.
Conjugates:
What are the products for each pair of conjugates?#1. (8 + v'2o)(8 - vizO) #2. (S + 3v'2)(S - 3-v'2)
We also want to focus on how to 'rationalize' the denominator of an expression that hasradical expressions in the denominator.How can you write the expression with a rationalized denominator?
11 3J7#1. -'---;;; #2. r;; r;:;
6+v3 v7+v3
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We finally want to be able to relate radical expressions with some concepts we havepreviously learned in Geometry.#1. STEM: (Geometry) This wall-hanging is made of congruent 30-60-90 degree triangles withheight 6 cm and hypotenuse 12 cm. What is the perimeter of the wall hanging to the nearesttenth of a centimeter.
#2. STEM: (Geometry) This tile design is made of congruent right triangles with base 1 ft andheight 2 ft. Find the perimeter of the tile to the nearest tenth of a foot.
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HOMEWORK - Chapter 8 Day #3Multiply.
1. (3-F6)(2-J6) 2. (5+Fs)(1-Fs)Multiply each pair of conjugates.
4. (7 -h)( 7 + h) 5. (1+3J3)( 1-3J3)Rationalize each denominator. Simplify the answer.
7 3 8 7+$. 2+J6 . 6-Fs
3. (4+7r
6. (6 + 4.)7)( 6 - 4.)7)
8. 1-2.JlO4+JiO
9. 3-J16Fs-h
10. 2+MJ7+h12. 2+:f;3r;
13. A park in the shape of a triangle has a sidewalk dividing it into two parts.
300 ft 300 ..J3 ft
a. If a man walks around the perimeter of the park, how far will he walk?b. What is the area of the park?
14. A~Ioor tile is made up of smaller squares. Each square measures 3 in. on each side.Find the perimeter of the floor tile.
15. A section of mosaic tile wall has the design shown at the right. The design is madeup of eduilateral triangles. Each side of the large triangle is 4 in. and each side of a smalltriangle is 2 in. Find the total area of the design to the nearest tenth of an inch.
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Pre-AP Algebra II - Chapter 8 Day #8Topic: Graphing Radical Expressions & TransformationsStandards/Goals:
./' F.lF.7.b.: I can graph square and cube root functions .
./' Ican determine the transformations of the graph of a radical function.We first want to review how to determine the transformations of a radical transformation forboth square and cube roots.Summary of transformations:
Parent Function:Reflection in the x-axis:Stretch (a > 1), shrink (0 < a < 1) by the factor a:Translation: Horizontal by 'h'; vertical by'k':
Describe how to obtain the graph of 'g' from the graph of f(x) =
l.y=£+3 2. y=.Jx-l 3. y=.Jx+5
4. y=.Jx-3 5. y=-2.Jx-2 16. y=-.Jx-l+54
We also want to consider the graph of a square root function and what it looks like.Parent Function:
I
II
.~y,~
r
X
-1£1 - 0 10'
rJ
on
I/'
I
2
ExamPles:1 Match the function with its graph shown below:
1. f(x) = ~-1
4. f(x) = Vx + 2 - 1A. Y
I --l-
V 1 x
./- I, -I
2. j(x) = Jx-=1 + 1
5. f(x) = Vx + 2 + 1B. Y\
I ~I < ./'
l,.- I, 'V- 1 I
1 x
I II
3. j(x) = Fx+1 ~ 1
6. f(x) = Vx - 2 - L
C. Yi.-!oo.-I '
1 i'./
.....- I x
D. ,Y I !
, I- r-- I I
I ,-I /' x
./1..---
E. y II
! I- 1 -I ~I V2 x
!II
F.Y I '
I1 I - '.
--l- ;;;;;-0
v: 1 x
We finally, want to be able to use radical functions in some real-life situations.#1. The speed of a tsunami's' (in meters/ second) is approximated by s = .J gd, where 'g' is
the acceleration due to gravity (9.8 m/s2) and 'd' is the depth ofthe water (in meters). Ifthespeed of a tsunami is 200 mis, about how deep is the water?
#2. An exercise specialist has studied your exercise routine and says the formula t = 1.85--/c + 10express9s the amount of time t, in minutes, it takes you to burn c calories (cal) while exercising.
According to this formula, how long should it take you to burn 100 cal? 200 cal? 300 cal?
HOMEWORK - Chapter 8 Day #8I. Describe how to obtain the graph of 'g' from the graph of f(x) = ..;x.1. y=Jx-4+1 2. y=~-4
3. y=Jx+l
5. y=2~
7. y=-Jx-l
9. y=3~+2
II. Match the function with its graph.
1. f(x) = 4'x + 2
4. f(x) = Jx+IA. y ,.
1
<, 1 x...•.....r- -
D. y
~-""
_ •....- 1 /'
1 x
4. y=-.Jx+2-3
6. y=-2Jx+3+4
8. y=Jx+3-4
10. y =-Jx-2
2. f(x) = 4'x - 2
5. f(x) = - Jx+IB. y.
II--- I
J.-It'"'
1./ . x-~ -E. Y
2 --"" .-
V-I x
3
3. f(x) = .vx - 2
6., j(x) ~ .Jx - 1
C.
F. yl! .-1--J.-
v2
...-~ Ii x
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II. Word ~roblems#1. A center-pivot irrigation system can water from 1 to 130 acres of crop land.The length I in feet of rotating pipe needed to irrigate A acres is given by the function 1= 117.75JA.a. Graph the equation on your calculator. Make a sketch of the graph.b. What length of pipe is needed to irrigate 40,80, and 130 acres?
#2. A periscope on a submarine is at a height h, in feet, above the surface of the water. Thegreatest distance d, in miles, that can be seen from the periscope on a clear day is given by
d=/f.a. If a ship is 3 miles from the submarine, at what height above the water would the submarinehave to raise its periscope in order to see the ship?b. If a ship is 1.5 miles from the submarine, to what height would it have to be raised?
#3. You can use the equation t =!...,Jd to find the time t, in seconds, it takes an object to fall d4
feet after being dropped.a. Graph the equation.b. How long does it take the object to fall 400 feet?
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AlgebnliaII - Chapter 8 TestName Date _Multiple Choice
1. What is the real cube root of -64?a. 4b. -4c. -8
d. No real cube root
2. W~at is the real fourth root of- 16?81I a. 2/3b. -2/3c. -4/9
I d. No real fourth root
3. What is the simplified form of the expression.) 4x2y4?I a. 2xy2
b. 21xly2C. 4xy2d. 21xyl
4. What is the simplest form of .)SOX7y6?
a. 2X3y3..J20xb. 4x6y6J5x3C. 4.}SX7y6
d. 4x3y3-J5x5. What is the simplest form of "V25xy2 * V15x2?
a. SXV3y2
b. SxV3y
c. lSxyVYd. 5xy-v'15x
6. What is the simplest form of 2m - 3m?I a. 2m - 3mI b. 24.[2I c. -2..J2
d. 0
1
1. ®®©CID2. (~(ID © (Q)
3. ®OO©CID4. @@©®
5. ®@©@
6. ®@©@
2
-~---
J75xS7. What is the simplest form of ?
J12XYZ'
S-J3x4a. Z.j3y2
5x2b. -zy
5x..fXc. --
2y5x2yd.-
Z
8. What is the simplest form of (2 - v'7)(1 + 2v'7)a. -12 + 3V7b. 16+ SV7c. -12 - 3v'7d. 3 + v'7
79. What is the simplest form of -v's?
2 + 5a. -14 + 7..[5b. 14 + 7V5c. -14 - 7..[5d. 14 - 7..J5
1 Z- -10. What is X3 * y3 in simplest form?
a. x3JY3b. -}xy3
c. '!J (xy)Zd. '!Jxy2
11. Let f(x) = -2x + 5 and g(x) = x3. What is (g - f)(x)?
a. x3 - 2x + 5b. x3 + 2x - 5c. -x3 - 2x + 5d. -x3 + 2x - 5
12. Let f(x) = X2 - 2x - 15 and g(x) = x + 3. What is the domain of£. (x)?g
a. All real numbers
b. x ~ 5,-3c. x ~-3
d. x > 0
7. @®©®8. ®@©®9. ®®©@
10. ®®©®11. ®®©®12. ®®©®
13. Letlf(x) =.JX + 1 an g(x) = 2x + 1. What is (g 0 f)(x)?
a. z..JX + 3
b. 2x-IX + 2x .JX + 1
c. -JZx+ 1 + 1
d. 2x + -IX + 2114. What is the inverse r the function? F 5(x - 3)?
x+3a y=-. 5
1b. y=sx+3c. y = 5(x + 3)
1d. y = sX - 3
1
15. What is the graph 0r~.::= ~ +_ 4
a. _ .... I b.
0y
C
A
..• ~~1/- x
0
ny
v .-r-v ", ~
..•L
X
0
---- -
13. ®®©®14. @(ID(£)@
15. ®®©®
c.
0y
,.v
k-- --~x- - a
Free Response 1
1. Descrlbe the tra nsfirm ations for the followi ng function: y = 2";x - 1 + 3
2. What is the SOlutidn of the following equations?1
#2. 5x2 - 8::= 7#1. .J2x - 1 - 3 ::= 1
3
d.
ny
C
A ..-", .••... x- - 0
4
#3. -J2x - 6 = 3 - x #4. -V5x - 3 = -V2x + 3
3. A high school principal uses the formula y = 150x + 180 to predict a student's score on astate achievement test using the student's 11th grade GPA number x.
a. What is the inverse of the formula?
b. Using the inverse, what GPA does a student need to get a passing score of 510on the state exam?
1 ?4. What is the DOMAIN of g(x) = 4 -.Jx + 6·
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Algebra II - Chapter 8 TestName I Date _Multiple Choice
1. What is the real cube root of -64?a. -8b. No real cube rootc. 4
d. -4
2. What is the real fourth root of- 16?81
a. -4/9b. No real fourth root
c. 2/3d. -2/3
3. What is the simplified form of the expression.J 4x2y4?a. 4xy2b. 21xylC. 2xy2d. 21xly2
4. What is the simplest form of.J BOx7y6?a. 4.JSx7y6
b. 4x3y3v'SXc. 2X3y3..J20xd. 4X6y6..JSX3
5. WhatisthesimpiestformofVZ5xy2 * V15x2?a. 15xyVYb. 5xyv'15xC. 5x:}3y2
d. 5x:}3y6. What is the simplest form of 2m - 3m?
a. -2.[2b. 0
c. 2m-3md. 24v2
1. ®<ID(~)®2. (~OO©®3. OOOO©®4. ®OO(0®5. ®®(0@6. OO®C0®
7. What is the simplest form of ..[i50 7
sx.,fii J 1ZXy
2'
Zy
b. 5x2y
a.
Z
C. s·hx4
Z.J3y2
d. Sx2
2y
8. What is the simplest form of (2 - v7)(1 + 2v7)a. -12 - 3-17b. 3 + v7c. -12 + 3v7d. 16 + 5v7
79. What is the simplest form of .,;s?
2 + 5a. -14 - 7-J5b. 14 - 7-J5c. -14 + 7-J5d. 14 + 7-J5
1 2- -10. What is X3 * y3 in simplest form?
a. vCxy) 2
b. Vxy2C. x3.fY3d . .Jxy3
11. Let f(x) = -2x + 5 and g(x) = x3. What is (g - f)(x)?
a. -x3 - 2x + 5b. -x3 + 2x - 5C. x3 - 2x + 5d. x3 + 2x - 5
12. Let f(x) = X2 - 2x - 15 and g(x) = x + 3. What is the domain oft- (x)?9
a. x ~-3
b. x >0c. All real numbers
d. x ~ 5, -3
6
7. ®®©®8. ®®©®9. @@©@
10. ®®©@11. ®®©®12. ®®©®
13. Let f(x) = {X + 1 andlglx) = 2x + 1. What is (g 0 f)(x)?
a. .../Zx + 1+ 1
b. 2x + -IX + 2c. z{X + 3d. 2x.JX + 2x i{X + 1
14. What is the inverse of the function? y = 5(x - 3)?
a. y = 5(x + 3)1
b. y=-x-35x+3
c. y=-5
1d. y=-x+3
515. What is the graph of y = {X + 4
a. ----
n Y-0- -~
r-~v
/•..•-.. x-0 II>
b.
n Y
r:.I-- -r--.., x- - 0
13. -®<ID©®14. ®®©CID15. ®@©®
c.
0Y
r
-1+'/ ~ x
- - a
Free Response I1. Describe the transformations for the following function: y = 2.../x - 1 + 3
2. What is the SOlutiolnofthe following equations? 1
#1. .../2x - 1 - 3 = 1 #2. 5x2 - 8 = 7
7
d.
0y
c:...• --
/' xa
8
#3. -V2x - 6 = 3 - x #4. .v5x - 3 = .J2x + 3
3. A high school principal uses the formula y = 150x + 180 to predict a student's score on astate achievement test using the student's n" grade GPA number x.
a. What is the inverse of the formula?
b. Using the inverse, what GPA does a student need to get a passing score of 510on the state exam?
1 74. What is the DOMAIN of g(x) = 4 -.Jx + 6 .
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Algebra II - Cha Iter 8 TestName Date _Multiple Choice
1. What is the real cubs root of -64?
a. No real cube lroot
b. 4
c. -4d. -8
2. What is the real fouJth root of- 16?81
a. No real fourth root
b. 2/3c. -2/3d. -4/9
3. What is the simPlifi+d form of the expression) 4x2y4?
a. 21xylb. 2xy2
c. 21xly2d. 4xy2
4. What is the simple~~form of) SOx 7y6?
a. 4x3y3v'$XI
b. 2x3y3V20~
c. 4X6y6v'5X3/d. 4.JSX7y6
5. What is the simple~t form of "V25xy2 * V'15x2?a. 5XYV15x!b. 5xV3y2
c. 5xV3y
d. 15xyVY6. What is the slmplast form of 2m - 3m?
a. ab. 2m - 3~32c. 24{2d. -2..J2
1. ®CID©@2. OOOO©®3. ®CID©®4. @@(9@
5. @@C0®6. OOOO(0®
7. What is the simPlesj form of .J75;S -;>-J 12xy2'
5x2y ,a. --
Z
b S-V3x4
. Z-J3y2Sx2
c. 2yd. sx-!X
zy8. What is the simples~ form of (2 - -J7)(1 + 2-J7)
a. 3 +-J7b. -12 + 3-J7c. 16 + 5-J7d. -12 - 3-J7
79. What is the simplest form of2 + {5?
b.
d.1 2
10. What is X3 * y3 ih simplest form?
a. Vxy2b. x3-fY3C. ~xy3
d. V(xy)211.Let f(x) = -Zx + 5 at g(x) = x3
. What is (g - f)(x)?
a. _x3 + 2x 5b. x3 - 2x + 5C. x3 + 2x - 5d. _x3 - 2x 5
12.Let f(x) = X2 - 2x - 15 and g(x) = x + 3. What is the domain of L (x)?9
a. x > ab. All real nulnbers
c. x * 5,-3d. x *-3
10
---7_ ®®©®8. @®©@9. ®®©@10. ®®©®11. @@©@
12. @@©(ID
13. Let f(x) = -JX + 1 and g(x) = 2x + 1. What is (g 0 f)(x)?
a. 2x + Fx + 2
b. 2{X + 3c. 2x-JX + 2x -+ Fx + 1d. -v'2x + 1 +
14. What is the inverse bfthe function? y = 5(x - 3)?1
a. y=-x-35
bx+ 3
. y=-5
1C. Y = -x + 3
5
d. y=5(x+3)
15. Wh~t is the g~~Ph 0if Y = -JX ~ 4
a. I I I _t ~I I I b. c.oHT I Il£ I----<L I
., -./ x
0, 4 ~ ~
T3~-®®©® 11
14. ®<ID©®15. @®@®
d.- y !r Iv I
'...•..~ INx
- - 0 4.
0_y l
c~1
A /' I., i
xl
0 ~ f l
Free Response I1. Describe the transiormations for the following function: y = 2..Jx - 1 + 3
2. What is the SOlutiin of the following equations? 1
#1 . ..J2x - 1 - 3 = 1 #2. 5x2 - 8 = 7
#3. -JZx - 6 = 3 - x
12
#4. -JSx - 3 = V2x + 3
3. A high school princil'pal uses the formula y = 150x + 180 to predict a student's score on astate achievement est using the student's 11th grade GPA number x.
a. What is the inverse of the formula?
b. Using the i7verse, what GPA does a student need to get a passing score of 510on the state exam?
1 ?4. What is the DOMAIN of g(x) = 4 _ .J x + 6 •