Overview of Project
Over the course of a few short weeks, we've been overlooking quadratics in math class. This has been personally a very challenging few weeks because I haven't covered this type of math before. We covered kinematics, forms of quadratics, projectile motion, areas and volumes, the Pythagorean theorem, and economics. The objectives over the course was to understand and be able to identify what parabolas are and how they work. This helped develop our learning on quadratic functions, and methods for solving quadratic equations. We did about 25 worksheets that were done either independently or as a group. I worked with a few of my friends because I believe you learn better when you get a different perspective on how you can get the work done.
In the beginning, we were introduced to kinematics. Kinematics is the branch of mechanics that's concerned with the motion of objects with reference to the forces that cause the motion. Usually it ties in with the math of flying objects such as airplanes, or rockets, because of the curve from the take off to landing. We launched our quadratics project with the, "A Victory Celebration" handout. This paper dealt with the kinematics and physics of a firework being launched off a tower to celebrate the High Tech High sports team. The questions ask about the x-intercepts, which is foundational to the graphs of the quadratic functions. On the paper was a function that we had to solve for t, and this would help us answer the questions. |
The Vertex Form of Quadratic Equations
After doing the, "Victory Celebration,", we continued with learning with quadratic equations. We explored this by collaborating with our peers, and using an online graphing calculator called Desmos. This helped develop our learning of what the vertex form of a parabola is. We found the meaning behind a, h, and k, by figuring out how each of them effect the function on Desmos.
The vertex form is, "y=a(x-h)^2+k". Each part of the function plays an important part of a parabola. The variable a effects how opened up a parabola is. The bigger the number, the more closed the parabola. If it's positive, the parabola opens upward. If it is negative, it opens downward. The variable h in the function effects the horizontal shift of the parabola. Meaning how far left or right it the parabola is. The variable k represents a vertical shift, so depending on the number, it will effect how far up or down it it. On handouts four through eight, we developed our learning of what effects a parabola, and did several problems dealing with finding the formula for a parabola, and explaining how the variables effect what it looks like. |
Standard and Factored Form
The other two forms of a quadratic equation are standard, and factored form. The function for factored form is, "y=a(x-r)(x-s)". The factored form of the equation tells us the roots, x=r and x=s. These are the x-intercepts of the parabola. However, sometimes the factored form cannot be solved because there are no x-intercepts. Some examples of this could be; y=3/8(x-1)(x-9), y=(x+1)(x-5), y=(x+5)(x+10).
The standard form of quadratics is, "ax^2+bx+c". If the role of a is bigger than zero, the parabola opens upwards. If it is less the zero, the parabola opens downwards. Some examples of standard form is; x^2-4x+5, -x^2+4x+3, 3x^2-6x+5. In the photo on the top right of this section, is a picture of a graph with a forms of quadratics showing the same parabola. The first function is in standard form. It is, "y=x^2+8x+15". The second is that same function, but converted into factored form. It's shown as, "y=(x+5)(x+3)". The third function is in vertex form, and was converted to it from factored form. This is, "y=(x+4)^2-1". These all show the same parabola because there just converted to a different form of quadratics. |
Converting between Forms
Solving Problems with Quadratic Equations
There are three types of problems that you are able to solve using quadratics. In the first area, kinematics, you can definitely use quadratics equations to solve the problem.
In the photo on my right is an example of kinematics being solved using the standard form of quadratics. The problem is asking how long does it take the ball to hit the ground if the ball is shot into the air from the edge of a building 50 feet about the ground, and the initial velocity is 20 feet per second? Then it gives the equation, "h=16t^2+20t+50".
|
Vertex Form to Standard Form Handout
On the 16th handout, we converted from vertex form to standard form. There were eight questions asking to convert and make sure to combine like terms for our final answer. There's a pattern of for every two questions, only the positive sign is changed to a negative sign. For example, in the photo of the graph on the left, it shows two of the questions already converted from the handout. The only part changed from both of them, is the positive sign between x squared and 2x. This only effects where the parabola moved horizontally. If it's negative, it's behind the positive y and x axis.
In the two photos below the graph, are the examples from the graph, but shows how it was converted. I used an area diagram (the square with the multiplying) to help the multiplication part of converting. After the area diagram, I was left with the equation, "y=x^2+1x+1x+1". After combining 1x and the other 1x, and added 1 to 4, I was left with my final result of, "y=x^2+2x+5". |
Reflection
Before this project, I knew nothing of quadratics, and only a little bit about kinematics. At first I hard a very hard time understanding what the objective of the project was, but with collaboration and help from fellow peers, I was able to get comfortable with the math. In the beginning, before solving any form of quadratics, we went through methodically and systematic what each part of the equation did to a parabola. This really helped when it came to finding the vertex form of a parabola on the handouts, because I knew what each part of it meant and did.
|
Kinematics is something I still don't understand, but not everyone ends up being a rocket scientist. I tried my best to understand what the basics of it is by starting small, but I couldn't get past the Victory Celebration handout. However, with all the other handouts about parabolas, I believe I did good at completing them. Though there were some that I struggled with, I think I have an overall fundamental understanding of it. Having this skill of comprehending quadratics has impacted my thinking and readiness for the future. Now I take my time to take steps and be confident, patient, and persistent when solving equations. Before, I tried guessing or doing it in my head, but I know now that that doesn't always work.
|
After solving (or while working on) an equation, I tried to seek why the steps were taken, and used my work as evidence to prove it was correct. While solving for the same kind of formula, I looked for patterns in my work so that I could remember the same steps I took, and used those for the next problem. Furthermore, I conjectured and tested my solutions to see if they were right. I did this by checking in with other peers, and was wrong on a lot of things. Of course, this only meant I had to take apart what I was doing, and get a better understanding of how to do it before I put it all back together. I definitely kept my papers together in one part of my bag. I wanted to be as organized looking with my work on the paper, and keeping the actual paper.
|
Sometimes it was hard to stay motivated during this project, because at some points it was only a ton of handouts coming after the other. I felt like sometimes I was only trying to get the work completed more than really trying to understand it, which I'm a little disappointed about myself in. I felt like if the handouts were more spread out, then I wouldn't feel pressured to understand something in two days. With this, I had to generalize what I was thinking, and simplify what I was doing. Overthinking the problem only makes it harder, so I got help from my friends to make my life easier, and hear their way of explain the problems. Last but not least, I had to describe what I was doing in the problem by explaining my steps or ways of solving it.
|