Quintessence of Quadratics
Introduction:
We began this project by looking at a situation where an object starts at 0 and moves with a constant velocity of 5 ft/s. We created an in-out table to show this happening. As the table shows the velocity is constant. From this we were able to come up with a general rule for finding distance as a function of time: d(t)=vt. Next we did the same thing except we looked at acceleration instead of distance. Through this we found the rule for finding the velocity as a function of time: v(t)=at. After this we looked at a case where an object starts with an initial velocity, so its velocity at (t=0 is v0: v(0)=v0), it accelerates at a constant rate (a) and it travels for some length of time (t). To find the distance the object traveled we need to look at the area under the velocity time graph, to do this we need to break the graph into a triangle and a rectangle, the rectangle is shaded in blue and the triangle in purple. Since the area of a rectangle is length times width we get A= t*v0. The area of a triangle is 1/2 bh, so we would get A=1/2t*a*t, this then becomes A=1/2at^2. The total distance traveled is the sum of these two areas d(t)=v0t+1/2at^2. This function is better known as the displacement equation. After we found the displacement equation we looked at vertex form and explored what each part of the equation did. Next we talked about standard form, and worked on converting from vertex form to standard form. Finally we learned about factored form, and how to go from standard form to factored form. |
|
Exploring the Vertex Form of a Quadratic Equation:
One form of a quadratic equation is vertex form. When looking at an equation in vertex form you can almost instantly know the vertex of the parabola it creates, the vertex is either the highest or lowest point on the parabola depending on if it is concave up or concave down. Vertex form is y=a(x-h)^2+k. Each different letter in vertex form effects the parabola in some way or another. To learn what each letter did we filled out a series of worksheets that gave us equations that we had to put into desmos (an online graphing calculator) and figure out how this effected the parabola. |
a:
We began with the simplest equation: y=x^2, from there we were given the equation y=ax^2, different values were substituted for a and we had to determine what different a values did to the parabola. In an equation, the smaller the a value there is the wider the parabola will be, the larger the a value the narrower the parabola will be. The a value also determines the direction the parabola will open, if the a value is less then zero then the parabola will concave down. |
k:
K determines where on the y axis the vertex of the parabola will be. This is what makes vertex form so special, you can look at the equation and almost instantly know where the vertex will be. |
h:
H determines where on the x axis of the vertex of the parabola will be. |
Solving Problems with Quadratic Equations:
For this project we learned how to solve three real world problems using quadratics, kinematics, geometric, and economic.
For this project we learned how to solve three real world problems using quadratics, kinematics, geometric, and economic.
Kinematics:
There are some kinematics problems that can be solved with quadratic equations, in these problems an object is launched into the air in some way, this objects path will be a parabola of some sort. An example of a kinematics problem that can be solved with a quadratic equation is a situation where a sock is thrown across a room. The path the sock travels will be a parabola. |
Economics:
Quadratic equations can be used to solve economics problems as well. This can help companies price products in a way that would generate the most money. Say a company has created phone cases but the more they price them the less they will be able to sell. They have 500 phone cases, and are looking to sell them at $35 each, an equation that shows this is 500-4d, (d meaning price of a phone case) this shows how many they have, as well as represents that the higher the price the less phone cases they will sell. The vertex of the parabola will give the price that will maximize the profit. |
Geometry:
Quadratic equations can also be used to solve multiple types of geometry problems. They can be used with the Pythagorean Theorem to find the area of a triangle, as well as finding the maximum area of a shape. For example, someone is trying to build a rectangular planter with 50 feet of wood, they want it to have the maximum area possible. A parabola can help quickly find the maximum area. This can also be used for other shapes including triangles. |
How Much Can They Drink:
This problem was a geometry problem about Farmer Minh the worksheet said "Farmer Minh wants to build a drinking trough with a rectangular cross section. He has a metal sheet that is 40 inches wide and 80 inches long. He plans to bend the sheet along two lines parallel to the 80-inch side. This will make a square-cornered U-shape that will form the bottom and the long sides of the trough. He’ll use some other pieces of metal for the two ends." In the end we ended up creating a quadratic equation that could very quickly tell us the volume based on the size of the part that was folded up. This problem showed how a quadratic can be used in a real world situation. Solving:
This worksheet asked five different questions, each one got closer to finding the quadratic and making the parabola. |
1. Find the volume of the trough. It is 30 inches by 80 inches by 5 inches. That is, find out how much water this trough would hold if it were full.
To solve question 1 all that had to be done was calculating the volume of the trough. Since volume=length*width*height all that had to be done was 80*30*5, this gave an answer of 12,000in^3. |
2. Find the volumes for two other troughs farmer Minh could make this way. Use a value other than 5 inches for the width of the sections being bent up.
For question two it is just asking to find the volume of the trough with two other widths for the sides that are bent up. I picked to use 10 and 8 as my side lengths. It is important to remember to subtract two of the bent side lengths from 40, this is how the width of the trough is determined. |
3. Suppose the sections farmer Minh bends up each have a width of x inches. Find a formula for the volume of the trough.
Question 3 is asking you to create a formula for finding the volume of the trough. To do this x is used as a variable for the length of one of the bent up sides. The formula I came up with was: v=(40-2x)*80x. Since this is a project about quadratics I then converted the formula to standard form. First I rearranged the formula to be v=80x(-2x+40) after that I distributed the coefficient (80x) to the numbers and variables in the parentheses, this put the equation int standard form (y=-160x^2+3200x) |
4. Find the value of x that will make the volume a maximum. Find the volume that goes with that value of x.
Since the equation is already in standard form, all that is needed to answer this question is the equation in vertex form. First you distribute the coefficient. Next you complete the square. Then you factor out the coefficient, and that gives you the equation in vertex form (y=-160(x-10)^2+16000) Since it is in vertex form you automatically know the cordinates of the vertex of the parabola (10,16000) Ten in the x value and 16,000 is the maximum possible area. |
5. Suppose that instead of 80 inches, the trough length was increased to 120 inches. How would your answers to Question 4 change?
To solve the fifth and final question we just need to go back to the formula we created in question 3 and replace 80 with 120. After that you convert it to standard form, then to vertex form, like what was done in questions 3 and 4. Since the equation will now be in vertex form we can very quickly find the new vertex of the parabola, (10,24000) This means that the x value that gives the maximum volume will still be 10, but the new volume is 24,000. |
Habits of a Mathematician:
Look for Patterns:
During this project looking for patterns ended up being much more important then I originally thought it would be, sometimes in order to be able to identify a quadratic you need to find a pattern first, for example when looking at an equation it may not always be in the correct form, and you need to be able to find the pattern and identify that it is a quadratic equation.
Start Small:
Starting small is very important in all math, especially quadratics. An example of when we started small was when we did the set of worksheets that explored vertex form. We began with y=ax^2 and worked our way up to the full y=a(x-h)^2+k. This was really useful because it wasn't like there was a bunch of information getting thrown at you at once, each assignment built off of what was on the pervious one.
Be Systematic:
In this project it was quite important to be systematic, if not then it was very difficult to get the correct answer. For example when converting between forms it is very important to follow the correct steps, if not then you won't get a correct answer.
Take Apart and Put back Together:
Using area diagrams is a good example of take apart and put back together. You break apart (x+k)^2 into (x+k)(x+k) then you put it back together, another time we used this habit durning this project were for the two problems involving triangles, we split apart a triangle into two right triangles in order to find the height.
Conjecture and Test:
This habit was very useful when trying to create formulas. Once a formula is created it is necessary to test it to make sure it works properly.
Stay Organized:
An example of staying organized during this project was the use of area diagrams, they allowed me to keep track of all the distributing and factoring. It was also really helpful when I looked at my previous work for guidance during this DP update.
Describe and Articulate:
Multiple times in this project I went up in front of the class and presented my work, this habit helped me convey my answer and work to the rest of my classmates.
Seek Why and Prove:
I asked clarifying questions in order to find out why the steps were taken and gave the correct answer.
Be Confident Patient and Persistent:
I was very persistent when doing this project, even when I had no idea how to solve the problem, I still tried and ended up getting an answer.
Collaborate and Listen:
During this project I allowed myself to work with people I don't usually work with, this allowed me to see different perspectives and get insight into how other peoples brains work.
Generalize:
I looked at a problem in an overall context in order to better understand how each step I took helped to solve the problem.
Look for Patterns:
During this project looking for patterns ended up being much more important then I originally thought it would be, sometimes in order to be able to identify a quadratic you need to find a pattern first, for example when looking at an equation it may not always be in the correct form, and you need to be able to find the pattern and identify that it is a quadratic equation.
Start Small:
Starting small is very important in all math, especially quadratics. An example of when we started small was when we did the set of worksheets that explored vertex form. We began with y=ax^2 and worked our way up to the full y=a(x-h)^2+k. This was really useful because it wasn't like there was a bunch of information getting thrown at you at once, each assignment built off of what was on the pervious one.
Be Systematic:
In this project it was quite important to be systematic, if not then it was very difficult to get the correct answer. For example when converting between forms it is very important to follow the correct steps, if not then you won't get a correct answer.
Take Apart and Put back Together:
Using area diagrams is a good example of take apart and put back together. You break apart (x+k)^2 into (x+k)(x+k) then you put it back together, another time we used this habit durning this project were for the two problems involving triangles, we split apart a triangle into two right triangles in order to find the height.
Conjecture and Test:
This habit was very useful when trying to create formulas. Once a formula is created it is necessary to test it to make sure it works properly.
Stay Organized:
An example of staying organized during this project was the use of area diagrams, they allowed me to keep track of all the distributing and factoring. It was also really helpful when I looked at my previous work for guidance during this DP update.
Describe and Articulate:
Multiple times in this project I went up in front of the class and presented my work, this habit helped me convey my answer and work to the rest of my classmates.
Seek Why and Prove:
I asked clarifying questions in order to find out why the steps were taken and gave the correct answer.
Be Confident Patient and Persistent:
I was very persistent when doing this project, even when I had no idea how to solve the problem, I still tried and ended up getting an answer.
Collaborate and Listen:
During this project I allowed myself to work with people I don't usually work with, this allowed me to see different perspectives and get insight into how other peoples brains work.
Generalize:
I looked at a problem in an overall context in order to better understand how each step I took helped to solve the problem.
Reflection:
Overall I feel as if I did a very nice job on this project. I really understood the content that was being taught and I enjoyed the challenge of some of the problems quite a bit. It has showed me that I need to be challenging myself a bit more, and I will continue to try to challenge myself in 11th grade and beyond. I put a lot of effort into the whole project including this DP update. I did a much better job of keeping all of my papers together and not scrambling to complete things at the last minute. There are still a few things I could work on. I would like to continue to ask questions about why you get the correct answer. In 11th grade I hope to not do my math in pen, going through my work from this project I noticed how often I had to scratch something out. I really feel like I've grown quite a bit through this project, I honestly didn't realize I enjoyed math this much until this project. It really helped me broaden my knowledge. I found it very interesting how each different form of a quadratic equation tells you something about the parabola. I think its safe to say that I thoroughly enjoyed this project.
Overall I feel as if I did a very nice job on this project. I really understood the content that was being taught and I enjoyed the challenge of some of the problems quite a bit. It has showed me that I need to be challenging myself a bit more, and I will continue to try to challenge myself in 11th grade and beyond. I put a lot of effort into the whole project including this DP update. I did a much better job of keeping all of my papers together and not scrambling to complete things at the last minute. There are still a few things I could work on. I would like to continue to ask questions about why you get the correct answer. In 11th grade I hope to not do my math in pen, going through my work from this project I noticed how often I had to scratch something out. I really feel like I've grown quite a bit through this project, I honestly didn't realize I enjoyed math this much until this project. It really helped me broaden my knowledge. I found it very interesting how each different form of a quadratic equation tells you something about the parabola. I think its safe to say that I thoroughly enjoyed this project.