Lesson 1: Inverses of Functions Lesson 2: Graphs of Polynomial Functions Lesson 3: 3-Dimensional Space

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1 Table of Contents Introduction v Unit 1: Modeling with Matrices... 1 Lesson 2: Solving Problems Using Matrices Lesson 3: Linear Programming Lesson : Vertex Edge Graphs Unit 2: Polynomial Functions Lesson 1: Inverses of Functions... Lesson 2: Graphs of Polynomial Functions... 7 Lesson 3: 3-Dimensional Space Unit 3: Statistics Lesson 1: Mean and Standard Deviation Lesson 2: Empirical Rule Lesson 3: Collecting and Organizing Data Lesson : Sample Data from One Population Lesson : Comparing Distributions Lesson 6: Posing Questions Lesson 7: Random Sampling Lesson 8: Sampling from Non-Normal Distributions Unit : Right Triangle Trigonometry Lesson 1: Right Triangles Lesson 2: Problem Solving with Trigonometric Ratios Unit : Exponential and Logarithmic Functions Lesson 1: Exponential Functions Lesson 2: Geometric Sequences Lesson 3: Logarithmic Functions iii Table of Contents

2 Unit 6: Solving Equations and Inequalities Lesson 1: Exponential Functions Lesson 2: Solving More Complex Equations and Inequalities Unit 7: Conics Lesson 1: Circles Lesson 2: Equations of Conics... 3 Lesson 3: Planes and Spheres Unit 8: Data Analysis Lesson 1: Histograms Lesson 2: Normal Distributions Lesson 3: Producing Data Answer Key Appendix: z-scores Table iv Table of Contents

3 Unit 1 Modeling with Matrices Lesson 1: Operations with Matrices Georgia Performance Standards MA2A6: Students will perform basic operations with matrices. a. Add, subtract, multiply, and invert matrices, when possible, choosing appropriate methods, including technology. b. Find the inverses of two-by-two matrices using pencil and paper, and find inverses of larger matrices using technology. c. Examine the properties of matrices, contrasting them with properties of real numbers. Essential Questions 1. What are real world examples of matrices? 2. What is the purpose of organizing data into matrices? 3. What industries use matrices to organize and interpret their data? WORDS TO KNOW [ ] square brackets are used to enclose a matrix determinant of a matrix the product of the elements on the main diagonal minus the product of the elements off the main diagonal dimension of a matrix indicates the number of rows and columns in a matrix elements of a matrix the numbers within the matrix denoted by its row number and column number identity matrix has a 1 in the main diagonal with zeros everywhere else 1 0 The 2 2 identity matrix is. 0 1 The 3 3 identity matrix is

4 inverse of a square matrix The inverse of a square matrix S, written as S -1, makes the S(S -1 ) = Identity Matrix a true statement. matrix scalar matrix multiplication square matrix rectangular array of numbers Every element in a matrix is multiplied by scalar which is a regular number. number of rows equals the number of columns in the matrix 2 Unit 1: Modeling with Matrices

5 Guided Practice Addition and Subtraction Matrix addition and subtraction are performed on corresponding elements in each matrix. To perform matrix addition and subtraction, each matrix should have the same number of rows and columns. Show all steps of addition and subtraction to verify you have used corresponding elements in each matrix. Example 1 Add the following matrices Add corresponding elements = ( 7+ ) ( 3+ 7) ( ) ( 1+ 21) ( + 10) ( 9+ 1) ( + 12) ( ) ( 17+ 8) =

6 Example 2 Subtract the following matrices Begin by distributing the negative sign Add the two matrices by adding corresponding elements. ( 7+ ) ( 3+ 7) ( ) ( 1+ 21) ( + 10) ( 9+ 1) ( + 12) ( ) ( ) 12 3 = In order to add or subtract two matrices, the matrices must be the same size. Unit 1: Modeling with Matrices

7 Example 3 Add the following matrices Determine the size of each matrix. The first matrix is 2 2. The second matrix is Since the matrices are not the same size, they cannot be added = not possible Matrix Multiplication Matrix multiplication is trickier. When multiplying two matrices, you multiply each element in each row of the first matrix by each element in each column of the second matrix and find their sum. Example Find the product of the two matrices given below Determine if the matrices can be multiplied. The dimensions of the first matrix are 2 3. The dimensions of the second matrix are 3 2. The inner dimensions match, so the matrices can be multiplied. The product dimensions will be the outer dimensions of each matrix, 2 2.

8 2. Multiply the matrices. Multiply following the pattern of row by column and find the sum = ( 3 7) + ( 8 9) + ( 2 11) ( 3 ) + ( 8 ) + ( 2 20) ( 1 7) + ( 12 9) + ( 6 11) ( 1 ) + ( 12 ) + ( 6 20) = If the inner dimensions of the product matrices do not match, then the product is undefined. Example Multiply the following matrices Determine if the matrices can be multiplied. The dimensions of the first matrix are 3 1. The dimensions of the second matrix are 3 3. The inner dimensions do not match, so the matrices cannot be multiplied Inner dimensions do not match. 6 Unit 1: Modeling with Matrices

9 = undefined Scalar Multiplication Scalar multiplication for matrices is much easier. Simply multiply the scalar, which is a number, by each element in the matrix. Example 1 7 [ ] Multiply each element in the matrix by the scalar. The scalar is. 7 [ ] = 10 2 ( ) ( 7) ( 10) ( 2) 2. The answer is: [ ] =

10 PRACTICE Unit 1 Modeling with Matrices Practice Lesson 1.1.1: Operations with Matrices Perform matrix addition on the matrices Perform matrix subtraction on the matrices continued 8 Unit 1: Modeling with Matrices

11 PRACTICE Unit 1 Modeling with Matrices Perform scalar multiplication on each matrix. 8. [ 3] [ 2 ] Perform matrix multiplication on the matrices

12 Guided Practice The product of the original matrix and the inverse of the original matrix should equal the identity matrix. One of the steps to calculating the inverse of a matrix by hand is to find the determinant. The determinant is the product of the elements on the main diagonal minus the product of the elements off the main diagonal. See the example below. Example 1 Find the determinant of the following matrix Multiply the numbers in the main diagonal. 3 ( 2) = 6 2. Subtract the product of the elements off the main diagonal from the product you found above. 6 ( 1) = 6 ( ) = 10 The determinant is 10. Finding the inverse of a 2 2 matrix by hand involves several steps. First you must find the determinant. Then you divide each element by the determinant of your matrix after you make some changes to it. To change the matrix, switch positions of the elements of the main diagonal. Then, take the opposite of the elements off the main diagonal. The inverse matrix is the quotient of the elements in the changed matrix and the determinant. See the example below. Example 2 Complete the following steps to find the inverse of a 2 2 matrix: Given: Unit 1: Modeling with Matrices

13 1. Switch the elements in the main diagonal Find the opposite of the other two elements and leave the elements in their original position Find the determinant. Determinant = ()() ( 8)( 2) =. Divide every element in the matrix by the determinant and you will have the inverse of the original matrix. 8 2 = Finding the inverse of matrices larger than 2 2 can be time consuming. Therefore, use a graphing calculator to find the inverses of larger matrices. Example 3 Find the inverse of the matrix below

14 Follow these steps: 1. On your graphing calculator hit 2nd MATRX. Select EDIT and hit Enter on [A]. (This means that your matrix is called Matrix A.] 2. Enter the number of rows (3) and hit Enter. Enter the number of columns (3) and hit Enter. 3. Enter each element in the matrix, hitting Enter after you type in each number. If you make a mistake, use the arrows to go back to the element and change it.. Hit 2nd Quit so you are no longer in the matrix mode. Then turn the calculator back on.. Hit 2nd Matrix. The cursor should be on [A] under the Names menu. Hit Enter. Your screen will now show just [A]. 6. Hit the x -1 key. Your screen will show [A] -1. Then hit Enter. 7. The inverse of the original matrix will appear on screen. It may contain numbers with several decimal places. You can change these decimals to fractions by hitting Math, Enter, Enter. 8. Use the arrows to see all the elements in the matrix. Answer: Unit 1: Modeling with Matrices

15 PRACTICE Unit 1 Modeling with Matrices Practice Lesson 1.1.2: Inverses of Matrices Find the inverse of each 2 2 matrix using pencil and paper Find the inverse of each 3 3 matrix using your graphing calculator. Write your answer in fraction form continued 13

16 PRACTICE Unit 1 Modeling with Matrices Find the inverse of each matrix using your graphing calculator Write your answer in fraction form Write your answer in decimal form. Round to four decimal places Write your answer in fraction form. For problem 10, find the inverse of the matrix using your graphing calculator Write your answer in decimal form. Round to five decimal places. 1 Unit 1: Modeling with Matrices