Project Overview
The term "quadratic" in algebra involves and relates to the square as well as the second and no higher power than an unknown variable or quantity. In class, we started bey reviewing the vocabulary of quadratic equations. A quadratic equation will always equal y, and that the highest exponent on an independent variable is 2, and this can be modeled as a square whose length and width are both x, which makes the area of the square x^2. A regular quadratic equation can have 1, 2, or 3 terms, so as we found in the previous statement, y=x^2 is a valid way of writing it. There are normally three parts to an equation, the first one being what is known as the squared term. This term can be either x^2 with a coefficient or just x^2 on its own. The next term is known as the linear term, which has an exponent of 1, or simply x. An example of a linear term would be 3x. The constant term of the equation is one that has no independent variable; an example would be 7. And so, in the quadratic equation of y=2x^2+3x+7, 2x^2 would be the squared term, 3x would be the linear term, and 7 would be the constant term. In addition, we reviewed the concepts of standard, factored, and vertex forms of quadratic equations, as well as families of equations. Standard form is written as y=ax^2 + bx= c, vertex form as y=a(x-h)^2 + k, and factored form as y=a(x-p)(x-q). When graphed on a coordinate pane, all three forms of the same equation make a parabola, that is, the graph of any of these functions. We also went over finding equations through area diagrams. However, one of the most important topics we covered in this unit was converting an equation from standard form to vertex form.
We also learnt how to solve quadratic equations algebraically with the aid of parabolas graphed on coordinate planes as well as in different forms. The extrema of parabolas were also covered in class; they are essentially the maximum points of parabolas that open downwards and the minimum points of those that open upwards. Expanding and simplifying expressions with linear terms was also a concept that we reviewed at length. When an expression is made up of two linear factors, we can expand them by lining up said factors along the length and width of a rectangle and then multiplying the values and later combining the like terms to come up with an equation in standard form. In factoring an equation, we essentially had to convert an expression in factored form to one in standard form. In the process, we learnt that the values p and q when multiplied equal the constant term in standard form and that when the same values are added together, they equal the value of b in standard form.
We also learnt how to solve quadratic equations algebraically with the aid of parabolas graphed on coordinate planes as well as in different forms. The extrema of parabolas were also covered in class; they are essentially the maximum points of parabolas that open downwards and the minimum points of those that open upwards. Expanding and simplifying expressions with linear terms was also a concept that we reviewed at length. When an expression is made up of two linear factors, we can expand them by lining up said factors along the length and width of a rectangle and then multiplying the values and later combining the like terms to come up with an equation in standard form. In factoring an equation, we essentially had to convert an expression in factored form to one in standard form. In the process, we learnt that the values p and q when multiplied equal the constant term in standard form and that when the same values are added together, they equal the value of b in standard form.
Quadratic Postcards Activity
For this capstone project, what we all had to do was come up with a thoughtful question about quadratics and how they relate to the real world. After doing this, we needed to find a way of going about solving said problem, after which we arranged this information in an aesthetically pleasing way on a 4'' by 6'' "postcard." The purpose of this project was to make us think beyond the worksheets and theory of quadratics in the classroom and wonder how they are important in everyday life, and why we would need it not only for high school but beyond it.
Reflection
Quadratics has always been a relatively easy concept for me, so I didn't struggle too much with anything in that sense. What I really liked about this project was the way that it made me think more about mathematics in terms of the world around me and how it has always been an important part of my life. I did my best to succeed on this project from beginning to end and always asked help from either my teacher or from my peers if something went askew, which greatly contributed to my success. I put in my utmost effort to make my piece worthy of being called beautiful work and always welcomed critique from others so I could further improve my work. However, I did take a while to figure out how to arrange everything on Google Drawings, as I am not too comfortable with using Photoshop or Illustrator on the MacBooks, so it took longer than I expected it to. I also had to research new information because I did not know that I would come to the conclusion that the Gateway Arch is in fact not a parabola.