Download INT Unit 4 Notes

Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-1 Homework Notes
4-1: Sorting data in Matrices
Information is often stored in __________________. The inventory of athletic clothing owned by a high school crosscountry team is shown in the matrix below:
Pants
Small
Medium
Large
x-large
 __
 __

 __

 __
Column
____
Shirts
__
__
__
__
Column
____
Shorts
__ 
__ 

__ 

__ 
Column
____
row ___
row ___
row ___
row ___
the element in the _______ row and the ______ column
This matrix has _____ rows and ______ columns. It is said to have the _________________________ 4 by 3, written
_____________. In general, a matrix with ____ rows and ___ columns had dimensions __________.
Example: The Matterhorn Company produced 1500 trumpets and 1200 French Horns in September; 2000 trumpets and
1400 French Horns in October; and 900 trumpets and 700 French horns in November.
a) Store the company’s products in a matrix.
b) What are it’s dimensions?
4-1 In-Class Notes
Points and polygons can also be represented by _____________________. The ordered pair (x, y) is generally
represented by the matrix
 __ 
 __  .
 
This is called a __________________________________.
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
Read through p. 8 in the pink Core Plus books that are on your tables. Complete questions 5-7 with your neighbors in
the space below:
If time, read through Investigation 3 on p. 10 and answer questions 1 & 2 together.
HW: Complete p. 207 #2,4-7,9,10 on a separate sheet
4-2 Homework Notes
4-2: Matrix Addition
There are many situations which require ______________ the information stored in matrices. For instance, suppose
matrix C represents the current inventory of the Chic Boutique.
dresses
suits
skirts
blouses






8
10
12
14
16






The quantities of new items received by the boutique are represented by the numbers stores in matrix D.
8
dresses
suits
skirts
blouses






10
12
14
16






Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
The new inventory is found by taking the _____ of matrices ____________. This ______________________ is
performed according to the following definition:
If two matrices ____ and _____ have the same ___________________________, their ______ A + B is the matrix in
which each element is the ______ of the corresponding matrix in which each element is the sum of the corresponding
elements in A and B.
Complete the matrix for the new inventory
8
10
12
14
16

suits 

skirts

blouses 







dresses
Addition of matrices is __________________, meaning that A + B = ___________. It is also _______________________
meaning (A + B) + C = ___________________. Subtraction works the same way.
Scalar Multiplication
The __________________ of a scalar k and a matrix A is the matrix _____ in which each element is ____ times the
corresponding element in A.
Example: Find the product
Complete p. 211 #1,2
7
5
4
2
9
1
11 
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-2 In-Class Notes
Read through p. 11-12 (under the car picture) in the pink Core Plus books. Complete questions #3-6 with your neighbors
in the space below:
Read through p. 13 (On Your Own) in the pink Core Plus books. Complete the questions with your neighbors in the
space below:
From p. 20 #3 “A square matrix is said to be _______________________ if it has symmetry about its ____________
_______________________. The main diagonal of a square matrix is the ____________________ line of entries running
from the ________ ________ to the ____________________ _____________ corner.” Answer question #3c together as
a class.
Monday Homework: p. 211 #5,6,7,10 on a separate sheet.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-3 Homework Notes
4-3: Matrix Multiplication
Matrix multiplication is much more complex than Matrix addition. You can not just multiply the individual elements.
Before you try to do the multiplication, it would be best to write down the definition first.
Definition of Matrix Multiplication
Suppose A is an ___________ matrix and B is an ____________ matrix. Then the product ______________________ is
the __________ matrix whose element in row ____ and column _____ is the product of _________________ and
___________________.
This means that the product of two matrices __________ exists only when the number of ___________________ of A
equals the number of ____________ of B. It also means that the dimensions of the answer matrix will have the same
number of ___________________ of A equals the number of ____________ of B.
Example: Multiply the following:
8 2  1
 4 1   0

 
5

2 
3
4
First write down the dimensions of the first two matrices to make sure that the product exists and to find the
dimensions of the answer matrix. Column 1
Column 1
Row 1
8 2  1
 4 1   0

 
3
4
5

2 
Row 1
_
_

_
_
_
_ 
To get the solution for the element in the 1st row and the 1st column, you multiply across the _____________ of Matrix A
and down the ____________________ of Matrix B. You then take the product of the 1st elements and the product of
the 2nd elements and add them together.
8  (1)  (2)  0 
Fill this into the answer matrix above. Then complete this process to fill in the rest. Show your work below.
Example: Can you multiply
Why or why not?
1
0

3
4
5  8 2 

2   4 1 
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
This indicates that, in general, multiplication of matrices is not ________________________________.
HW: Complete p. 217, #1-3
4-3 In Class Notes
When given a story problem, make sure that you set up your matrices so that they will be able to be multiplied.
Example: Costumes have been designed for the school play. Each boy’s costume requires 5 yards of fabric, 4 yards of
ribbon, and 3 packets of sequins. Each girl’s costume requires 6 yards of fabric, 5 yards of ribbon, and 2 packets of
sequins. Fabric costs $4 per yard, ribbon costs $2 per yard, and sequins cost $.50 per packet. Use matrix multiplications
to find the total cost of the materials for each costume.
Now I will show you how to do this on your calculator. Take notes below if needed.
Read through p. 30 (On Your Own) in the pink Core Plus books. Complete the questions with your neighbors in the
space below:
HW: Complete p. 217, #4-7, 10-12 on a separate sheet as well as WS Master 10
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
Geometry Transformation In-Class Notes
The Alhambra, a 13th-century palace in Grenada, Spain, is famous for the geometric patterns that cover its walls and
floors. To create a variety of designs, the builders based the patterns on several different transformations.
A ___________________ is a change in the position, size, or shape of a figure. The original figure is called the
_____________. The resulting figure is called the __________ A transformation maps the preimage to the image. Arrow
notation () is used to describe a transformation, and primes (’) are used to label the image.
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
Example 2: Drawing and Identifying Transformations
A figure has vertices at A(1, –1), B(2, 3), and C(4, –2). After a transformation, the image of the figure has vertices at
A'(–1, –1), B'(–2, 3), and C'(–4, –2). Draw the preimage and image. Then identify the transformation.
Check It Out! Example 2
A figure has vertices at E(2, 0), F(2, -1), G(5, -1), and H(5, 0). After a transformation, the image of the figure has
vertices at E’(0, 2), F’(1, 2), G’(1, 5), and H’(0, 5). Draw the preimage and image. Then identify the transformation.
To find coordinates for the image of a figure in a translation, add a to the x-coordinates of the preimage and add b to the
y-coordinates of the preimage. Translations can also be described by a rule such as (x, y)  _______________.
Example 3: Translations in the Coordinate Plane
Find the coordinates for the image of ∆ABC after the translation (x, y)  (x + 2, y - 1). Draw the image.
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Hour:
Example 4: Art History Application
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
The figure shows part of a tile floor. Write a rule for the translation of hexagon 1 to hexagon 2.
A’
A
A ___________ is a transformation that changes the size of a figure but not its shape. The preimage and the image are
always similar. A __________ ____________ describes how much the figure is enlarged or reduced. For a dilation with
scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b)  (ka, kb).
Helpful Hint
If the scale factor of a dilation is greater than 1 (k > 1), it is an
_______________. If the scale factor is less than 1 (k < 1), it is a _________.
Example 1: Computer Graphics Application
Draw the border of the photo after a dilation with scale factor 5/2 .
Check It Out! Example 1
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
What if…? Draw the border of the original photo after a dilation with scale factor ½.
Complete Worksheet
4-4 & 4-5 Homework Notes
4-4: Matrices for Size Change
A __________________________ is a one-to-one correspondence between the points of a ___________________ and
the points of an _______________. Consider ∆PQR with P = (3, 1), Q = (-4, 0), and R = (-3, -2). The size change with
center ________ and magnitude 3, denoted ________, can be performed by ________________________ each x- and ycoordinate on ∆PQR by ____. We write:
Recall that the symbol ________ is often used in mathematics to denote “_______________”.
For any k ≠ 0, the transformation that maps ________ onto ___________ is called the ________________________ with
center _________ and _______________________ k, and is denoted ________.
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
Size change images can also be found by multiplying _______________. When the matrix for the point __________ is
multiplied by ______________, the matrix for the point ___________ results:
Theorem: ___________ is the matrix for _______.
Example 1: Given ABCD with A = (0, 3), B = (-2, -4), C = (-6, -4), and D = (-6, 4), find the image of A’B’C’D’ under S4.
4-5: Matrices for Scale Change
For any nonzero numbers a and b, the transformation that maps ___________ onto ___________ is called the
________________________ with ___________________________________________ a and _____________________
____________________________ b, and is denoted _______.
Theorem:
is a matrix for ________.
Complete the work on p. 228 under “Matrices for Scale Change” to prove this.
Name:
Mrs. Gorsline
Integrated Math 2
Hour:
Unit 4 Notes
Example: Refer to the quadrilateral from the previous example. Use matrix multiplication to find its image under S2,5.
4-4 & 4-5 In-Class Notes
4-4: Matrices for Size Change
The transformation that maps each point __________ onto itself is called the __________________________________.
When a point matrix
is multiplied on the _________ by
each point __________ coincides with the
image. In other words, multiplying by the identity matrix is the same as performing a size change of magnitude _____.
Example: Refer to the quadrilaterals in the 1st example from the homework notes. Calculate each ratio.
D 'C '
DC
A' B '
AB
Find the mapping equations for both a size change as well as a scale change.
Homework: p.224 #4, 7-14, 18 and p.229 #1,9,11 on a separate sheet of paper.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-6 Homework Notes
4-6: Matrices for Reflections
A _____________________ is a transformation that maps a figure to its reflection image. Reflection over the y-axis can
be denoted as ____________ or ______. The mapping equation for this is ___________________. Reflection over the xaxis can be denoted as _____________ or ______. The mapping equation for this is ____________________. Reflection
over the line y = x can be denoted as _____________ . The mapping equation for this is ________________________.
List the matrices for these three below:
Example: If A = (1, 2), B = (1, 4), and C = (2, 4), find the image of ∆ABC under
graph.
ry , rx , and ry  x .
Show all 4 on the same
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-6 In-Class Notes
4-6: Matrices for Reflections
Remembering Matrices
At this point, you have seen matrices for some size changes, some scale changes, and three reflections. You may
wonder, is there a trick to remembering them? Here is one great way:
Pretend that you have two points, (1, 0) and (0, 1), giving you the matrix
 __
 __

__ 
__ 
(the identity matrix). If you can
figure out where these two points will go when you complete the given transformation, you will get the matrix you
need.
Example: Complete this for all of the transformations you have done so far.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-8 Homework Notes
4-8: Matrices for Rotations
_______________ are closely related to __________. The arcs used to denote angles suggest turns. Angles
with larger measures indicate greater ___________. The amount and direction of the turn determine the
magnitude of the _____________. The rotation of magnitude ___ around the ____________ is denoted ____.
Caution: A rotation is denoted with a ______________________, while a reflection is denoted with a
______________________________.
Rotations often occur one after the other. A _______________ of ______ following a rotation of ____ with
the same center results in a rotation of ____________. In symbols, ________________.
Use the trick you learned in class to figure out the matrices for 90˚, 180˚, 270˚, and 360˚.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-8 In-Class Notes
4-8: Matrices for Rotations
Prove that you can get R180 by multiplying R90 and R90 .
Prove that you can get R270 by multiplying R180 and R90 .
Prove that you can get R360 by multiplying R270 and R90 .
Homework: p. 248 #1-6, 11-14 on a separate sheet
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
4-10 Homework Notes
4-10: Matrices for Translations
The transformation that maps ___________ onto ____________________________ is a _________________________
of ____ units ______________________ and ____ units ____________________________ and is denoted ________.
Since we have to add numbers to x and y, we will have to add matrices instead of multiply. This means that the
dimensions of the transformation matrix must be ____________________ as the point matrix.
Example: A quadrilateral has vertices Q = (-4, 2), U = (-2, 6), A = (0, 5) and D = (0, 3). Put these points into a matrix.
What must the dimensions of the translation matrix be? ___________ Write the matrix for
Add
T3,5
T3,5
below.
to QUAD to find the matrix for Q’U’A’D’. Then graph both the preimage and the image.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
5-5 Homework Notes
5-5: Inverse of Matrices
Review:
Plot quadrilateral ABCD with A = (-2, 1), B = (1, 1), C = (1, 2), and D = (-2, -2). It will help to put the points in a matrix
first.
Next find the matrix for the image of that shape under
S2,5 . Then plot this on the same graph.
Now find the matrix for the image of the shape from #2
under S 1 1 . Plot this. How does this compare to the original shape?
,
2 5
Multiply the matrices from #2 and #3. Go back to your Chapter 4 Notes if you forgot how to multiply matrices.
What is this called?
Two matrices whose product is the _____________________________ are called
__________________________.
5-5 In-Class Notes
In general, only ________________ matrices can have inverses, and two matrices are only inverses if and only
if their product is the ___________________ matrix.
Name:
Hour:
Mrs. Gorsline
Integrated Math 2
Unit 4 Notes
 2 5  8 5
 is  3 2 
3
8

 

Example: Verify that the inverse of 
Write down some notes from the inverses from the previous examples that may help you figure out how to
find the inverse of a matrix.
Unfortunately, it is not always as simple as the examples above. Below is the Inverse Matrix Theorem:
a b 
 is
c
d


If ad  bc  0 , the inverse of 
 0 2 

2 1 
Example: Find the inverse of the matrix 
Not all square matrices have inverses. If ad  bc does in fact equal _____, then all of the numbers in the
inverse will be _____________________. Since ad  bc is so important, it has a special name, the
_________________________.
3 1 
 has an inverse or not.
6
2


Example: Determine whether 
HW: p. 3.2-3.3, #1-3, 5-9, 16 on a separate sheet