Project description:
Digital Portfolio Update: The Quintessence of Quadratics
I. The Introduction
This project was launched when we had the example on a worksheet of launching a rocket from a tower 60 feet high then it had a certain amount of acceleration. We had to find how long it takes to get to it max height and also how long it will take from that point to the ground. This was of course going to give us a quadratic formula. From there we took what we knew and we put it into a online graph calculator. Once we did that we now knew that what we found was a quadratic equation. Once we knew what it was we could do example problems and find out how they work. I think that because I used the habit of a mathematician of looking for patterns it really helped with getting the work done fast.
II. Exploring the Vertex Form of the Quadratic Equation
To find how the Vertex form of the quadratic equation worked we had handouts that showed what the parameters did to the parabola. The vertex equation is y= a(x-h)^2+k. So for instance the h parameter tells us what the x coordinate is of the vertex of the parabola.I think that because Dr. Drew had us start small it helped us to learn the equations faster and get it to stick in our minds. Also by having the equations generalized it helps with us to understand how the equations are set up.
III. Other Forms of the Quadratic Equation
There are two other forms of a quadratic equation. One is standard form which is y= ax+bx+c. The advantage of using standard form is that you can convert it into other equations easier. The other one is factored form which is y=a(x+p)(x+q). The advantages of using factored form is that it shows where the x intercepts are.
IV. Converting between Forms Using your own examples
Here is the paper where I did the conversions
Digital Portfolio Update: The Quintessence of Quadratics
I. The Introduction
This project was launched when we had the example on a worksheet of launching a rocket from a tower 60 feet high then it had a certain amount of acceleration. We had to find how long it takes to get to it max height and also how long it will take from that point to the ground. This was of course going to give us a quadratic formula. From there we took what we knew and we put it into a online graph calculator. Once we did that we now knew that what we found was a quadratic equation. Once we knew what it was we could do example problems and find out how they work. I think that because I used the habit of a mathematician of looking for patterns it really helped with getting the work done fast.
II. Exploring the Vertex Form of the Quadratic Equation
To find how the Vertex form of the quadratic equation worked we had handouts that showed what the parameters did to the parabola. The vertex equation is y= a(x-h)^2+k. So for instance the h parameter tells us what the x coordinate is of the vertex of the parabola.I think that because Dr. Drew had us start small it helped us to learn the equations faster and get it to stick in our minds. Also by having the equations generalized it helps with us to understand how the equations are set up.
III. Other Forms of the Quadratic Equation
There are two other forms of a quadratic equation. One is standard form which is y= ax+bx+c. The advantage of using standard form is that you can convert it into other equations easier. The other one is factored form which is y=a(x+p)(x+q). The advantages of using factored form is that it shows where the x intercepts are.
IV. Converting between Forms Using your own examples
Here is the paper where I did the conversions
V. Solving Problems with Quadratic Equations
There are three different areas of which the quadratic equation can apply. The first one is kinematics. An example problem of that would be hitting a baseball. So what we know is that the ball is hit from the home plate 4 feet above the ground. We also know that it follows a perfect parabola that has a max height of 180 feet after 3 seconds. If we know that we can that then we can say that the equation is y=a(x-3)^2+180. Now we need to know the a value which tells us how wide the parabola is since we know that the y intercept is at 0,4 we can do the math to get the a value. To do that we need to just plug in the values for the y and x value so it looks like this. 4= a(0-3)^2+180. Now we can take 180 and subtract it from both sides and square 3 to get -176= 9a now we divide 9 from both sides to get 19.55555. That would be our a value so our finished equation would look like y=19.55555(x-3)^2+180. That looks like this in desmos a graphing app. |
The second way that it can apply is through geometry for this we had to find the maximum area of a rectangle with a set amount of perimeter. For instance with a fence length of 200 we have to find the maximum area of it. To start this we will say that x represent the width of the area. So we need to find the maximum area at first so from past problems we know that a square will give us the maximum area. So we can try 50 for each side length. Once we try that we can tell that it is the maximum size is. So if we do 50 x 50 which is 2500. So knowing that we can make a vertex form equation that is y= -(x-50)^2+2500. So that would be our equation and it looks like this:
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The third way is through Economics the problem says that a marketing director has 1000 widgets and the more that they sell for the less they will be able to sell. So we have to find an equation that will show how many widgets that will be able to sell and how much they sell for. After I used the habit of a mathematician of collaborating with my peers I came up with the equation
(1000-5d)d. This is just an add onto the already given equation of 1000-5d. Using the equation we can see how much you need to sell the widgets to get the most amounts of profits. I think that while doing this you really need to be systematic. I believe that because if you aren't it becomes very confusing very fast. |
VI. Reflection
I learned a lot in this project. I thought that this project really taught me a lot about how thing work and how when you throw something without considering wind/air resistance it follows a perfect parabola. I felt like I put in as much effort as I could. I think that because this came to me so easily it made it so I could do it with relative ease. I think that for this project I really needed to use organization. I think that in order to remember how to do each of the problems you needed to stay organized. Also if you didn't you would end up losing all of the papers that we got which would not be good. Overall I think that this was a fun project and it taught me a lot of things that I didn't know that you could do.
I learned a lot in this project. I thought that this project really taught me a lot about how thing work and how when you throw something without considering wind/air resistance it follows a perfect parabola. I felt like I put in as much effort as I could. I think that because this came to me so easily it made it so I could do it with relative ease. I think that for this project I really needed to use organization. I think that in order to remember how to do each of the problems you needed to stay organized. Also if you didn't you would end up losing all of the papers that we got which would not be good. Overall I think that this was a fun project and it taught me a lot of things that I didn't know that you could do.