For our final project in 10th grade math, we completed a project involving quadratic equations. When I heard about this project, I was very excited to learn about a new topic, one that I had never practiced before. We started off this project with learning with kinematics, which are the mechanics covering the motion of objects, not referencing the forced making the motion. To get introduced to this topic, we were given a problem that included a rocket launching from a raised platform, making a curved motion in the air, and coming back down:
From here we had to figure out how high the rocket flew, and how long it was in the air for. Also, if you look at the shop of the curve, it looks like a parabola. This is when we were introduced to parabolas, which is simply the graph of a quadratic equation.
After we figured out what parabolas were, we had to learn about all of their components. These being "A", "H", and "K". We figures out what these were by using a graphing app called "Desmos".
These variable come together to make the quadratic equation in "Vertex Form". Y=A(X-H)^2+K. To get in image of what these components do, I will give a brief example. Lets say you start with your standard Y=X^2 parabola. It looks like this:
After we figured out what parabolas were, we had to learn about all of their components. These being "A", "H", and "K". We figures out what these were by using a graphing app called "Desmos".
These variable come together to make the quadratic equation in "Vertex Form". Y=A(X-H)^2+K. To get in image of what these components do, I will give a brief example. Lets say you start with your standard Y=X^2 parabola. It looks like this:
Now, let's plug in a few values for "A".
As you can see, The larger the "A" value, the narrower the parabola is going to be. Now, let's plug in a few values for "H".
As you can see, the "H" value affects where the vertex lies along the X coordinate plane. Now, let's plug in a few values for "K".
As you can see, the "K" value affects where the vertex lies along the Y coordinate plane. Now that we know what these do, we can make combinations.
You can also have the parabola open downwards by using a negative "A" value
But not all quadratic equations are in "Vertex Form". You can also have "Standard Form" and "Factored Form". "Standard Form" Consists of the equation Y=AX^2+BX+C. "Standard Form" Is the best form for analyzing parabolas, giving the Y Intercept, and using the quadratic equation itself. An Example of where you would use this would be figuring out how high a rocket will go. "Factored Form" consists of the equation Y=(X+A)(X-B). "Factored Form" is the most useful form for finding the X intercepts of a parabola. You could use this to figure out where a rocket will land along the X intercept. "Vertex Form" is most useful for finding the vertex of the parabola. With this, you can figure out how high the rocket goes. All of these forms will give you a parabola. In fact, you can get the same parabola from all three of these forms. Here is one of those:
As you can see, these three equations are all making the same parabola.
After figuring out that you have these equations, which all make a parabola, you will need to figure out how to convert between them.
First up, let's convert from vertex form to standard form. To do this, you will need to use a process called "Completing The Square". This is when you have an area diagram, which looks like this:
After figuring out that you have these equations, which all make a parabola, you will need to figure out how to convert between them.
First up, let's convert from vertex form to standard form. To do this, you will need to use a process called "Completing The Square". This is when you have an area diagram, which looks like this:
Then you have to find the remaining numbers. In our case, lets take the equation Y=2(X+3)^2-22, and convert it to standard form; Y=AX^2+BX+C. First, lets complete the square. Take whats inside of the parenthesis, expand it, and factor in the exponent. It should look like this: (X^2+6X+9). What you are doing is squaring the x, multiplying the H value by 2x, and then adding on the H value squared. After this, you have to distribute the A value. Which in this case, is 2. So now you have the equation Y=2X^2+12X+18-22. Next, combine like terms, and then you are good to go. Y=2X^2+12X+4. Your process should look something like this:
Next, let's convert from standard form to vertex form. We will use the equation Y=3X^2-18X+5. The first thing we have to do is factor out the "A" value. Which gives us Y=3(X^2-6x)+5. Next, take half of the "B" value, which in this case would be 3, square it, add it to the equation, and subtract it. Now you have Y=3(X^2-6X+9-9)+5. Next, with your "BX" value, factor out the "X" value, and combine it with the positive value of the squared half of the "B" value. also, put that inside parenthesis with the "X" value, and square the parenthesis. This gives you Y=3((X-3)^2-9)+5. Next, multiply the negative 9 by the 3 in the beginning of our equation. Y=3(X-3)^2-27+5. Now, combine like terms. and your equation is Y=3(X-3)^2+22. Your process should look like this:
Next we will learn about how to convert from factored form to standard form. This conversion is fairly simple and easy. It's all about distribution. Let's use the equation Y=(X+2)(X-4). X*X=X^2, X*2=2x, X*-4=-4X, -4*2=-8. So tat leaves us with Y=X^2+2X-4X-8. Combine like terms, and you end up with Y=X^2-2X-8. Your distribution should look like this:
Next we will learn how to convert from standard to factored form. Start with a quadratic equation of Y=X^2+6x-8. Now we need to find 2 numbers that when multiplied equal 6, and when added equal 8. 2 and 4 fit that. So now we have (X+4) and (X+2). Y=(X+2)(X+4). Your process should look like this:
Quadratics can be used in real life situations. In problems such as kinematics, geometry, and economics. A kinematic example includes the problem we completed in the beginning of the project. We needed to figure out the vertex, and x intercept of a rocket launching from a tower. The problem looked like this:
For a real life use of geometry, we completed a problem called "A Corral Variation". This problem included a farmer who needed to figure out how much area the cattle pen he was building was going to be. It involved using quadratics and working with area. The problem looked like this:
For a problem regarding economics, we were given a problem called "Profiting From Widgets". This problem consisted of a large company planning to buy 1000 widgets. They had a set price of what they would charge, and how many they would sell. We needed to use quadratics in order to figure out what their profit was going to be.
There were a few problems that I was really confident with. No, they aren't complicated problems where you need to put skills to the test. I was very good at figuring out what the different parts of quadratic equations did. One specific problem being number 7, Vertex form for parabolas. In the beginning, we were given a number of handouts that were all based on vertex form, and all required sketching of parabolas. We were given a few days to figure out what each component of the quadratic equation did to the parabola. After we figured it out, we made small posters describing what they did. I always flew right through the "figuring it out" part. I was the first one the give the definition of each component to my group. I would always just plug in different numbers, see what happened, and then have my answer within a few seconds. As a habit of a mathematician, this is known as "Conjecture and Test". The problems would look like this:
There were a few problems that I was really confident with. No, they aren't complicated problems where you need to put skills to the test. I was very good at figuring out what the different parts of quadratic equations did. One specific problem being number 7, Vertex form for parabolas. In the beginning, we were given a number of handouts that were all based on vertex form, and all required sketching of parabolas. We were given a few days to figure out what each component of the quadratic equation did to the parabola. After we figured it out, we made small posters describing what they did. I always flew right through the "figuring it out" part. I was the first one the give the definition of each component to my group. I would always just plug in different numbers, see what happened, and then have my answer within a few seconds. As a habit of a mathematician, this is known as "Conjecture and Test". The problems would look like this:
Reflection:
I started off really strong with this project. In the beginning, I flew through the worksheets. Everything went by smoothly and I though that I would have no problem with this project at all. I was wrong. After being able to complete the work in the beginning fairly easily, I didn't give as much effort. After a few more worksheets, I started to loose a grip of retaining knowledge. I thought it was nothing. As the project progressed, I began to struggle. Towards the end of the project, I started to require lots of help on my projects. I just couldn't get an understanding. Even now, as this sentence is being read, there are things about this project that I am still shaky on. But just like anything else, I am still gaining understanding. I even learned things just from completing this DP update. The main thing that I can say is that I could've, and should've, put more effort into this project.
Looking into my future for 11th grade math, I can see that it is not going to be very easy. I used to thing that math was very easy. However, now I can see that I will need to put in much more effort than before. However, I am ready to put in that effort.
The main habits of a mathematician that I used in this project were generalize, conjecture and test, and stay organized. I already mentioned how conjecture and test helped me with parabolas. Generalize was used when converting. I realized that all of the different types of quadratics were, in a way, the same thing. Even though I am not an organized person, I did stay organized in my mind when completing this work. I had to make sure I was completing each step correctly. I had to write everything down and visually it. I would go over every step and make sure there was no mistake.
All in all, I think that this was a very interesting project.
I started off really strong with this project. In the beginning, I flew through the worksheets. Everything went by smoothly and I though that I would have no problem with this project at all. I was wrong. After being able to complete the work in the beginning fairly easily, I didn't give as much effort. After a few more worksheets, I started to loose a grip of retaining knowledge. I thought it was nothing. As the project progressed, I began to struggle. Towards the end of the project, I started to require lots of help on my projects. I just couldn't get an understanding. Even now, as this sentence is being read, there are things about this project that I am still shaky on. But just like anything else, I am still gaining understanding. I even learned things just from completing this DP update. The main thing that I can say is that I could've, and should've, put more effort into this project.
Looking into my future for 11th grade math, I can see that it is not going to be very easy. I used to thing that math was very easy. However, now I can see that I will need to put in much more effort than before. However, I am ready to put in that effort.
The main habits of a mathematician that I used in this project were generalize, conjecture and test, and stay organized. I already mentioned how conjecture and test helped me with parabolas. Generalize was used when converting. I realized that all of the different types of quadratics were, in a way, the same thing. Even though I am not an organized person, I did stay organized in my mind when completing this work. I had to make sure I was completing each step correctly. I had to write everything down and visually it. I would go over every step and make sure there was no mistake.
All in all, I think that this was a very interesting project.