Introduction
To begin this project we started with a worksheet called "Victory Celebration". This was our first introduction into quadratic equations and their different forms. This problem is about our school wanting to launch a rocket from a building that is 160 feet high. We had to figure out and answer the following questions; What is the maximum height of the rocket? When does the rocket reach its maximum height? How long is the rocket in the air? We were given a distance formula h(t) = d0 + v0 · t + 1/ 2 a · t². After writing out the formula we plugged in the information it gave us. After completing this worksheet, we moved onto many other ones that included parabolas, working with parabolic equations and seeing how they manipulate one another. We also used desmos to get a more hands on experience of how the manipulations between the equations work. We then moved onto vertex form for parabolas. The equation for that is y= a(x-h)^2+k. We learned about area diagrams that helped you change an equation from vertex form to standard form. There were also worksheets that took us back to the Pythagorean Theorem and we used the following equation a^2+b^2=c^2.
Exploring the Vertex Form of the Quadratic Equation
Using the online graphing site desmos we were able to graph the equations that we found. We were able to see what each of the parameters of the equation do. For example with the vertex form we were able to see what each part did it is y= a(x-h)^2+k. For example by putting in the equation y=ax^2 by messing with the a values, I was able to easily see what it did. The first 7 handouts had us do just focus on what each parameter did. I believe that these handouts really helped with my understanding of the quadratic equation. By doing these it really it proved to me what each parameter did. A affects whether the parabola concaves up or down and how narrow or wide it is. H represents what the x coordinate of the vertex is. K represents what the y value of the vertex is.
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Other Forms of the Quadratic Equation
There are two other forms of a quadratic equation. Standard form and Factored form. Standard form looks like y=ax^2+bx+c. The advantage of using this form is that you can compare it easily to other equations. Factored form looks like y=(x-p)(x-q). The advantage of using this form is that it tells you the X intercepts.
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Converting between Forms
Vertex to Standard: Vertex to standard is on the top. The steps you have to do is first you have to FOIL which stands for
First in the equation Outer in the equation Inner in the equation Last in the equation After you do that you need to distribute the a value into the equation that you now have and then you combine like terms and now you have standard form Standard to Vertex: To convert standard to vertex first you need to divide the bx value by a which you can tell by whats in front of x^2. After you do that you need to divide it by 2x which will give you a lone number which in this case is 1. after you do that you can add it to your equation which so far you know the a value. You can add the 1 to the equation which will give you -(x+1)^2. Now square the 1 and add it to 8 to get 9 and add that to the end of the equation to get -(x+1)^2+9. Factored to Standard: First you FOIL the equation then you distribute the negative into the equation and combine like terms.
Standard to Factored: First you take the standard form and find two numbers that when added up together equal -2 and when multiplied up together equal 8. Once you find those you have the two x-intercepts of the equation. Also another way to do it is by putting the standard form into desmos and looking at where the x-intercepts are. This is an example of using an area diagram to find the x-intercepts.
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Solving Problems with Quadratic Equations
Economics: We did a worksheet called Profiting from Widgets. For the problem we had to figure out what the most widgets they could sell using an equation they gave us. Then we had to make our own function that would include "d"(dollars). I created a table graph to see what the maximum revenue would be. By using a quadratic equation you will be able to look at the graph and see how many are going to sell, how much you're going to make, and how much to sell them for.
Kinematics: For kinematics we did a worksheets called Another Rocket. For this problem we had to find out the rocket's maximum height and we needed to find the coordinate point for where the rocket lands. We also had to find out how long it would take the rocket to reach maximum height and how long it would take the rocket to hit the ground. By looking at the graph you can tell how long it's going to take to reach its max height, how long it's going to take to hit the ground, and how far it goes. Geometry: For geometry we did a worksheet called Emergency at Sea. For this problem we had to find out things like setting up an equation that will tell us how far the boat is from tower A when it reaches the shore. By looking at it you can see what the maximum areas going to be and how big one side will be. |
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Reflection
Overall this project was very challenging for me and really made me think. For the first couple weeks I didn't really understand but I got help from my peers and they helped me with understanding it. The two Habits of a Mathematician I used the most were collaborate and listen and stay organized. I used collaborate and listen all throughout the project when we did worksheets my peers and I would help each other because there was always a point where one of us didn't understand something so we helped each other and listened. I also used stay organized all throughout the projects when we did worksheets because when doing math I like to keep everything in order to make sure that I made no mistakes. Especially when writing out equations I made sure it wasn't sloppy and I didn't rush through anything. I believe that I grew as a student with math because this project helped me with not becoming frustrated when I didn't know something. This will help me on my SAT because I know that most of these problems will be like the ones on the SAT and because it will help be with every question that I don't understand. It taught me to take a step back and look at problems as a whole and not focus on little details that might throw me off. The one thing that I really liked was learning about the area diagrams because it made working through some equations a lot easier and helped me understand it from an easier level before taking a step further and challenging myself.
Look for Patterns: Every new problem I got I looked for patterns that were the same as the last packet. This made it get easier and easier for me because I was able to remember what I did before and apply it to the new things I learned.
Start Small: I used this habit when we learned about different forms of quadratic equations. Each new thing I had to start small to be able to understand it and get a better grasp on it. After a few times of looking at the equation in this perspective I was able to figure it out easier and faster.
Be Systematic: I made quite a few small mistakes while working through equations so I used this habit when I would go back and see what I did wrong and fix it.
Take Apart and Put Back Together: In almost every packet there were multiple questions and for each question I like to take it apart to see what I need to do to solve it.
Conjecture and Test: When I was still confused on what formula or function to use I would test them all out and ask someone to help me with finding out which one was correct.
Describe and Articulate: Every time I ask for help I ask why is this the right answer. Rather than just flat out getting the answer I want to know why so if problems like this come up in the future I can do it on my own.
Seek Why and Prove: I think asking questions and understanding why the problem or answer is what it is, is very important with understanding the problem so that's what I did for every packet.
Be Confident, Patient, Persistent: In the beginning I would get frustrated a lot so I had to be patient with myself and let myself slowly learn quadratics. When I made a mistake I had to be confident that I would learn from my mistake and know that I won't make it again.
Generalize: I used generalizing to remember all of the variables and terms that were used in each related packet.
Look for Patterns: Every new problem I got I looked for patterns that were the same as the last packet. This made it get easier and easier for me because I was able to remember what I did before and apply it to the new things I learned.
Start Small: I used this habit when we learned about different forms of quadratic equations. Each new thing I had to start small to be able to understand it and get a better grasp on it. After a few times of looking at the equation in this perspective I was able to figure it out easier and faster.
Be Systematic: I made quite a few small mistakes while working through equations so I used this habit when I would go back and see what I did wrong and fix it.
Take Apart and Put Back Together: In almost every packet there were multiple questions and for each question I like to take it apart to see what I need to do to solve it.
Conjecture and Test: When I was still confused on what formula or function to use I would test them all out and ask someone to help me with finding out which one was correct.
Describe and Articulate: Every time I ask for help I ask why is this the right answer. Rather than just flat out getting the answer I want to know why so if problems like this come up in the future I can do it on my own.
Seek Why and Prove: I think asking questions and understanding why the problem or answer is what it is, is very important with understanding the problem so that's what I did for every packet.
Be Confident, Patient, Persistent: In the beginning I would get frustrated a lot so I had to be patient with myself and let myself slowly learn quadratics. When I made a mistake I had to be confident that I would learn from my mistake and know that I won't make it again.
Generalize: I used generalizing to remember all of the variables and terms that were used in each related packet.