Chapter+05+Notes

We have skipped over Chapter 4 (my favorite subject of matricies) to pursue items that will be more commonly used. Here in Chapter 5 we will begin our study of non-linear relations with the subject of quadratics.

This section introduces us to how we can work with quadratic equations. Being able to identify a quadratic equation and the parts of a quadratic equation's graph (called a parabola) is very important. Also, we have a brief reminder of the ways to multiply algebraic expressions.

This is where we are taking our first major alteration from the book. We are using sets of at least three points to write equations of linear functions. The book suggests using the information and methods learned in Section 3.6. We did not do that section. In the interest of being able to complete this unit I have included a set of instructions for a Problem Solving technique called Finite Differences. The process takes some practice but it does work for all orders of polynomials.

We now take a look at how we can graph parabolas. The most important point to find on a parabola is the vertex. This lesson shows how to find the vertex and Axis of Symmetry. It then goes through the process of graping with this point as a basis as well as one other point: the y-intercept.

With an understanding of how to find the axis of symmetry and vertex, as well as how to graph the parabola, we are now looking at how and when we will use these items. This lesson focuses on the role that the vertex plays in the parabola.

Moving in parallel ideas to the translating (moving) of the absolute value equations we are now translating quadratics. To do this we are introduced to a new form of equation called the Vertex Form.

Now we are looking at methods for undoing our FOIL method of equation expanding. This method that is review today will work for quadratic equations where the a value is equal to 1. In the coming lessons we will look at cases when a is not equal to 1 and special cases.

In this lesson we look at cases of quadratic trinomials where a is not 1. The process taught here is called "Double Decker" and it is highly effective in factoring these expressions. There are other methods out there but this seems to be the most effective one that I have found.

Now that we have learned how to factor the general quadratic forms we are looking at two special cases. These cases are Perfect Square Trinomials and the Difference of Squares. For these you will need to recognize patterns and learn how to pull the pattern apart.

To day we begin to use our factoring abilities that we recently learned. This lesson works through how to solve quadratic equations by factoring.

This lesosn uses technology (graphing calculators) to find the solutions to a quadratic function. The steps for the TI-83 series are included here.

When we are solving for the zeros of a quadratic we sometimes need to take the square root of a negative number. In the Real number system this is not possible so we will need to enter the Complex numbers. This section introduces us to Complex and immaginary numbers.

With this section we are introduced to the operations that we can perform with Complex numbers including addition, subtraction and multiplication.

Now that we have a general background in the basics of quadratics we are looking at specific examples of ways that we can simplify expressions. This section looks at making quadratic equations into perfrct square trinomials.

Another tool of completing the square is that it can be used to easily convert from the standard form of an equation to the vertex form.

For our last section of the chapter we are going back to something that most are familiar with when they get to Algebra 2: the quadratic formula. This formula for finding the zeros of a quadratic equation is based on our ability to complete the square. This helpful tool will make it possible for us to solve all types of quadratics.