Review for the Matrices Test

Today we reviewed for the matrix test - there are 14 questions and 2 bonus questions. If you want to bring a half-sheet of notes to your test, feel free to do so. You will get the whole period to complete your test.

The district told me that I have a math department meeting tomorrow, so you lucky ducks, you get one more day to study. The things that you need to know to succeed on the test are:

1. How to add/subtract matrices
2. How to multiply matrices
3. How to find the determinant of matrices and show work for a 3 x 3 matrix
4. How to find the inverse of a matrix
5. How to prove that two matrices are inverses of one another
6. How to solve matrix equations by both 1) adding/subtracting and 2) multiplying

Good luck! If you write your favorite color next to your name on the test I will add ten bonus points to your score!

Protein and Matrices

Today we served as health consultants and made matrices containing the protein and fat content for professional athletes. We walked around the room and looked at actual food labels to find this information - real world math! Then, we organized the data into matrices and found:

1) The determinant of the matrix
2) Solved an equation with the matrix
3) Expanded the matrix to include 9 clients rather than 1 client.

Shout out to Marcola Bobo, Brittany Ivory, and Nikki McKinney who impressed our class visitor very much! Another HUGE shout out to Aaron Houston for helping half of the class with their work - AMAZING JOB!

Solving Matrix Equations

Today was one of the longer days we've had in a while. First, we solved equations that involved adding and subtracting to solve for the missing variable. This was easy - just add or subtract the matrix on both sides to get the answer. Note that your answer represents an ENTIRE matrix, not just a number.

Equations involving multiplication were trickier. We know the opposite of multiplication is divison. However, we cannot divide matrices. So, as an alternative, we need to:

1. Find the inverse of the matrix next to the variable. To do this, recall that you must find the determinant, then multiply every element in the matrix by 1/determinant.

2. Once you find the inverse, multiply both sides of the equation by the inverse. The two matrices on the side with the variable cross each other out because they are the inverse of each other. It doesn't matter that they don't contain the same numbers - they're still inverses, so they still cross out. Bring the variable down.

Then, multiply the two matrices on the other side. Remember, you must multiply the first row with the first column, then first row with the second column, then the second row with the second column, then the second row with the second column. The resulting matrix will be your answer.

Review Sheet

I was in the library today doing a training for the ipads - you used today to work on your review sheets. Shout out to Marcola Bobo who was the only one excused from this assignment since she already finished it - nice work Marcola!

Inverse Matrices

We know from yesterday that, to prove that a matrix is an inverse, the matrix and it's inverse must be multiplied together, and the answer must be an identity matrix (1's on the main diagonal, 0's everywhere else).

Today, we calculated the inverse of a given matrix. As we know, the word "inverse" in math means the opposite, so the answer that we calculate will be a matrix that is, mathematically, opposite from the one that we started with.

To do this, we
1. find the determinant of the matrix (remember, the matrix must be a perfect square!)
2. multiply every element in the matrix by (1/determinant). This means that, in many cases, the inverse matrix will contain fractions/decimals in it.

The answer is your inverse matrix!
The only time that this DOESN'T work is when your determinant is zero, because (1/0) is undefined.

Proving Inverse Matrices

To prove an inverse matrix, we must multiply two matrices together. If their products form an identity matrix, then the two matrices are inverses to one another. If not, then Matrix B is not an inverse to Matrix A.

An identity matrix is one containing all 1's on the main diagonal and 0's everywhere else in the matrix. We can only solve for inverse and identity matrices if the matrices we are working with are perfect squares (example: 2 x 2 matrix or 3 x 3 matrix).

More practice on this tomorrow!

Multiplying Matrices

Multiplying matrices is a little different than adding/subtracting matrices. We must make sure that:

# columns in 1st matrix = # rows in 2nd matrix. 

Otherwise, we can't multiply them (and therefore, just write CAN'T SIMPLIFY). If we can multiply, then multiply each column element with each corresponding row element, then add them together. Repeat this process until you don't have any more columns or rows left.

This kind of problem requires a lot of practice to make perfect. If you need some extra examples I will be happy to give you some!

Matrices: Adding, Subtracting, and Solving for Missing Variables

Matrices order information into rows and columns. We always label matrices with a capital letter variable (such as, Matrix A). To add or subtract matrices, they must be the same size and shape. If there is a number in front of the matrix, such as 2A + B, we must first multiply everything in Matrix A by 2 before we add it to everything in Matrix B. Our answers should also be in the form of a matrix.

When solving for missing variables, we must set the element of the matrix with the variable equal to the number in the other matrix in the same corresponding position. This creates a new equation, which we can solve for the value of the variable.

Chapter 2 Test

The chapter 2 test was taken in the lecture hall today. Generally, the grades on this test were pretty high with a couple of exceptions. If you didn't take the test or you are dissatisfied with your grade, please tell me so that you can retake the test after school.

Our next topic is matrices!

Review for Chapter 2 Test

Your chapter 2 test on linear functions is tomorrow! The test contains 14 questions and 2 bonus questions. All answers are free response and you must show your work for everything.

The following will be on the test: be sure to check your notes or this blog to review anything that you're not 100% on. Bring your 1/2 page cheat sheet tomorrow too for extra help!

1. Graphing equations of lines - begin with the starting point, up/down and over by the slope
2. Graphing absolute value equations - plot the vertex, then graph both sides based on the slope
3. Absolute value translations - write the equation and graph the translation
4. Linear Inequalities - graph the line, dotted or solid depending on the sign, and test point (0,0) to shade
5. Piecewise Functions - graph the asymptote, determine which graph goes on which side, then graph
6. Step functions - graph each function on the y axis depending on what it can be between on the x axis.

Step Functions

Step functions get their name because they look like stairs. We graph one by first locating the y values. Then, we look to see which x values they are between. We typically see an open circle on one side of the graph and a closed circle on the other side of the graph.

Step functions have a domain of all real numbers - for every step function that you're going to see. The range of a step function can be found by locating the smallest and the largest number in the y-values that the problem gives to you.

Step functions only LOOK hard because there are many signs and numbers. Once you've gotten the hang of it - it's very easy!

Piecewise Functions, Day 2

We practiced more with piecewise functions today. They are, by far, the most difficult thing that we've done so far in algebra 2. Remember the steps for graphing:

1. Determine WHERE the asymptote of the function should be on the x-axis
2. Determine which function goes on what side of the graph: left or right.
3. Graph both of your functions ON THE CORRECT SIDES OF THE ASYMPTOTES. Remember, your line can only exist on one side of the graph - not both!

Shout out to everyone from 5th period who graphed some REALLY difficult functions today. I'm bringing you a surprise tomorrow in class. You all rock!

Piecewise Functions

Piecewise functions have more than one function within them. We first must determine the asymptotes of the graph - this helps us determine the sides that each function will go on. Then, we graph the function in the appropriate region of the graph. Sometimes, we cannot graph the function right away because the y-intercept is not in the correct region of the graph. When this happens, go up and over or down and over by your slope until you get into the correct region of the graph, then begin to draw your line. Remember - two points are needed to draw your line!

Also, don't forget about open circle for greater than, and closed circle for greater than or equal to.

Shout out to Jamal Turner for attempting to teach a student how to solve piecewise functions after finishing early in class - terrific work!

Thursday: Translating Absolute Value Functions

Knowing the base equation of absolute value is y = |x|, we can alter this function to change the slope, direction, or vertex of the graph. If the translation is horizontal, we put the number of units inside the absolute value and flip the sign. For a vertical translation: put the number of units outside the absolute value and keep the sign. Reflections are marked by a - in front of the slope, and shifts in the slope are placed in front of the x in the absolute value function.

Absolute Value Functions

We've been looking at absolute value functions for the past two days. They look like a "V" and contain a vertex, which is the starting point. Don't forget that when you look at the number inside the absolute value, you must flip the sign and go the opposite way on the x axis. We don't flip any numbers outside of the absolute value.

We also translated functions on the coordinate plane. We usually start with the base function y = |x| unless it specifies otherwise in the problem. Go over and backwards by your x, and up or down on the y.

The second review sheet was given out today! The due date is Sept 12 (Thursday) with the early bird deadline the day before. Shout out to Shulondria Smith for tutoring students in class today when they didn't understand, and shout out to Gabby Foster for staying after class to talk about math problems!