I. Introduction
Through projectile motion, geometry, and economics we learned about quadratic functions and how to solve quadratic equations. To kick off this project we used a problem that showed a rocket used to launch a fireworks display. The rocket's height was shown by a quadratic equation, then we had to answer questions involving vertices and x-intercepts. Knowing about these we were able to figure out how to put together quadratic equations.
We started trying to solve this problem by first looking at kinematic equations the distance formula: h(t) = d0 + v0 · t + 1/ 2 a · t^2
We plugged in the information we already knew, such as the starting height, acceleration due to gravity, and the rocket's initial velocity, and combined like terms to get our new equation: h(t) = 160 + 92t - 16t^2
At this point we had no previous knowledge that would help us so we paused this problem in order to learn how to complete it the rest of the way.Throughout the rest this project we worked with algebraic symbols to see how algebraic representations related to the answers of the problems. This helped us see how writing quadratic equations in different ways (i.e. Vertex form, standard form, and factored form) can help you find different things. (i.e. x and y intercepts and the vertex.) We also use graphing apps to help us see how the equation affects the parabola and connected geometry and algebra.
We started trying to solve this problem by first looking at kinematic equations the distance formula: h(t) = d0 + v0 · t + 1/ 2 a · t^2
We plugged in the information we already knew, such as the starting height, acceleration due to gravity, and the rocket's initial velocity, and combined like terms to get our new equation: h(t) = 160 + 92t - 16t^2
At this point we had no previous knowledge that would help us so we paused this problem in order to learn how to complete it the rest of the way.Throughout the rest this project we worked with algebraic symbols to see how algebraic representations related to the answers of the problems. This helped us see how writing quadratic equations in different ways (i.e. Vertex form, standard form, and factored form) can help you find different things. (i.e. x and y intercepts and the vertex.) We also use graphing apps to help us see how the equation affects the parabola and connected geometry and algebra.
II. Exploring Vertex form
To understand vertex form we first had to understand variables such as A,K, and H, To do this we used Desmos to help us look at quadratics in a more visual way. Then we were given handouts and we had to figure out how each variable affected the parabola using Desmos. We discovered that when you put together these variables you can create distance equations using geometry and graphing. Then we saw how adding a variable such as, a value to an equation: y=x^2, you get y=ax^2. Adding a affects the shape of the parabola since a higher sum of a creates a more narrow curve and a wider curve is caused by a lower sum of a. The next variable we added was k, which is equal to the y-coordinate value of the parabola's vertex when you put it into the equation: y=ax^2+k. The final variable we added was h, h shows the x-coordinate value. Our final vertex form quadratic equation look like this: y=a(x-h)^2+k
III. Other forms of the Quadratic Equation
There are three different types of quadratic equations vertex; Standard form, vertex form, and factored from. Vertex form is useful when identifying parabolas because you can find the exact coordinates of the vertex in the equation. Standard form (y=ax^2+bx+c), where a, b, and c can be any number. With an equation in this form, you can easily identify the y-intercept of the resulting parabola. Finally, factored form; y=a(x-q)(x-p). Is simply an expanded version of standard form. In this form, the constant a has the same effect as in the vertex form, while q & p are the x-intercepts of the parabola. This type of equation is important in that it allows you to directly see the x-intercepts of the parabola.
IV.
Convert from vertex form to standard form
-Using an area diagram, you multiply out the squared terms, distribute a, and combine like terms.
-Using an area diagram, you multiply out the squared terms, distribute a, and combine like terms.
Convert from standard form to vertex form
-To convert to vertex form you need to put the ax^2 and bx terms together and take a out. Then you need to complete the square by filling in the missing terms. Then take the subtracted term out of the parenthesis by multiplying it by a. Finally rewrite the parenthesis as a product of the square and combine like terms outside of the parenthesis. |
Convert from factored form to standard form
- Multiply the terms in the parenthesis, combine like terms, distribute a.
- Multiply the terms in the parenthesis, combine like terms, distribute a.
Convert from standard form to factored form
-To solve for factored form we have to work backwards. You know you want n and m values to multiply together to equal the c constant. You also know that if the n and m terms are added together they should equal the b constant, except that before they are added they need to be multiplied by values such that the values they are multiplied by are equal to the a constant. Then you are able to factor the a constant out of the equation. Then you have to guess and check to find the values you need. When you find the values, just write out the factored equation. |
V. Solving Problems with Quadratic Equations
Quadratics can be used in a lot of real world problems, here are a few examples we covered over the coarse of the project:
Kinematics (projectile motion)
An example of a kinematic equation being solved using quadratics is the central problem of the whole project, The Victory Celebration. Like I stated in my intro we converted a kinematic equation into a quadratic equation. To solve this problem we need the vertex coordinates and the positive x-intercept value. To do this we need to do is convert to vertex and factored form. Since this problem involves a lot of complex decimals, I used a calculator to convert to vertex form and then took the h and k constants as the time of maximum height and the maximum height. Since to solve for factored form you have to guess and check I to the conclusion that the n and m values would probably be decimals. This would make the conversion really hard figure out, so i decided to try another way. I started with Vertex form, and since there was only one x in that form, the calculations are easier. When I started to get bigger numbers again I used a calculator again. I ended up with two answers one positive and one negative. It was obvious that the positive answer was correct because the rocket could not of landed before it was launched. I checked my answer by making a graph in desmos.
Kinematics (projectile motion)
An example of a kinematic equation being solved using quadratics is the central problem of the whole project, The Victory Celebration. Like I stated in my intro we converted a kinematic equation into a quadratic equation. To solve this problem we need the vertex coordinates and the positive x-intercept value. To do this we need to do is convert to vertex and factored form. Since this problem involves a lot of complex decimals, I used a calculator to convert to vertex form and then took the h and k constants as the time of maximum height and the maximum height. Since to solve for factored form you have to guess and check I to the conclusion that the n and m values would probably be decimals. This would make the conversion really hard figure out, so i decided to try another way. I started with Vertex form, and since there was only one x in that form, the calculations are easier. When I started to get bigger numbers again I used a calculator again. I ended up with two answers one positive and one negative. It was obvious that the positive answer was correct because the rocket could not of landed before it was launched. I checked my answer by making a graph in desmos.
Geometry (triangle problems and rectangle area problems)
We worked on a lot of geometry problems during this project. The majority them involve a maximizing the area of a pen from a given amount of fencing. The catch is that one side of the fence is backed up against something else (someone else's fence). This creates a quadratic equation because if the side length of the side of the pen perpendicular to your neighbor's fence is x, and the given total fence length you have available is F , the area of the fence will be x(F-2x). You can re-write this as a quadratic equation in factored form: x(-2+F). |
Economics (maximizing revenue/profit or minimizing expenses/losses).
Economic problems often involve quadratic equations because, The more expensive you make a product the less people will buy it. If you want to know the best price of a product you need to take into account the price of the product and how many you will sell. Knowing how many you'll sell depends on the price of the object. This results in an equation in which the same variable appears twice, which almost always results in an x^2 term. This often shows a quadratic equation.
Economic problems often involve quadratic equations because, The more expensive you make a product the less people will buy it. If you want to know the best price of a product you need to take into account the price of the product and how many you will sell. Knowing how many you'll sell depends on the price of the object. This results in an equation in which the same variable appears twice, which almost always results in an x^2 term. This often shows a quadratic equation.
VI. Reflection
For me, the early stages of this project were very easy and I understood the worksheets very well. When we worked with desmos I understood how to use it and this benefited me a lot. As the project progressed and we moved into more complex problems I started to struggle. When I started to struggle more I had to make sure I started small, I made sure I was understanding smaller versions of the larger problem so I could take small steps towards understanding/ solving the problem. When we got into the last few weeks of this project I was starting to get a little lost with the complex math, I made sure to ask my peers around me to get help, by doing this I got to see different ways of understand quadratics and different ways to solve the problems. When I eventually understood the complex word problems we were working on I did a lot of being systematic, since the problems we were working on were often very similar I was able to get down the steps of how to solve for that type of problem. This helped me work through problems faster and still be able to understand how to do them. Overall, this project made me realize I am going to have to put in extra work to be able to do well next year and on the SAT. I think I am pretty good at understanding math in the early stages of the topic but when it progresses I start to get lost and can't connect what we started with and where we are now. Knowing this I know that next year and into the future I need to put in extra time to work on my math so I am ready for the SAT and college.
Habits of a Mathematician:
Look for Patterns- I looked for patterns throughout this project when I had to connect back to the simpler work we did at the beginning of this year. Also, we did a worksheet on solving equations but many of the equations were the same aside from their positive or negative signs so I had to look for patterns and when I realized that I was able to complete the worksheet much faster.
Start Small- Like I said above when I came to a bigger version of a more complex problem I went back and started from a smaller version with smaller numbers to be able to understand it better.
Be Systematic- I was systematic when I took notes on the steps of how to do a problem. Then I could go back and look at the steps so I could understand the new problem I was working on.
Take Apart and Put Back Together- For this Project I did a lot of taking apart and putting back together. When we worked on more complex problems I would have to break them down and do them step by step. This, also, helped me keep my work organized.
Stay Organized- I briefly mentions staying organized above, I made sure to write my problems step by step so I could refer back to them. Also I made sure I kept all my worksheets so I could look back at them especially for this DP update.
Conjecture and Test- I conjectured and tested the most in the project when working with Desmos. In the early stages of this project we used Desmos to help us understand how a,h, and k effect a probola. Since we just jumped in and gave it our best effort, I had to make guesses and check them with Desmos all the time.
Describe and Articulate- I used this habit most when drawing area diagrams. This helped me a lot because I was able to visualise my work and keep things straight in my head.
Seek Why and Prove- I had to seek why and prove during this project when I entered equations into Desmos without solving them first. Then I would look at how the equations affected the parabola and prove why they affected them in that way.
Be Confident, Patient and Persistent- Throughout this project I had to make sure I kept pushing through with my work even when I got lost. I knew that if i was just confident that I could do the math that the rest would fall into place.
Collaborate and Listen (with Modesty)- I used this habit a lot during this project. I worked a lot with Lauren on this project because she was really helpful when explaining the problems. I made sure I was working with her and not just copying the work, Also, worked with her on the problems so we could both get different perspectives.
Generalize- Generalizing is a great tool when working with quadratics, a lot of the work we did was connected so when I generalized I was able to see the overarching point of the problems.
Habits of a Mathematician:
Look for Patterns- I looked for patterns throughout this project when I had to connect back to the simpler work we did at the beginning of this year. Also, we did a worksheet on solving equations but many of the equations were the same aside from their positive or negative signs so I had to look for patterns and when I realized that I was able to complete the worksheet much faster.
Start Small- Like I said above when I came to a bigger version of a more complex problem I went back and started from a smaller version with smaller numbers to be able to understand it better.
Be Systematic- I was systematic when I took notes on the steps of how to do a problem. Then I could go back and look at the steps so I could understand the new problem I was working on.
Take Apart and Put Back Together- For this Project I did a lot of taking apart and putting back together. When we worked on more complex problems I would have to break them down and do them step by step. This, also, helped me keep my work organized.
Stay Organized- I briefly mentions staying organized above, I made sure to write my problems step by step so I could refer back to them. Also I made sure I kept all my worksheets so I could look back at them especially for this DP update.
Conjecture and Test- I conjectured and tested the most in the project when working with Desmos. In the early stages of this project we used Desmos to help us understand how a,h, and k effect a probola. Since we just jumped in and gave it our best effort, I had to make guesses and check them with Desmos all the time.
Describe and Articulate- I used this habit most when drawing area diagrams. This helped me a lot because I was able to visualise my work and keep things straight in my head.
Seek Why and Prove- I had to seek why and prove during this project when I entered equations into Desmos without solving them first. Then I would look at how the equations affected the parabola and prove why they affected them in that way.
Be Confident, Patient and Persistent- Throughout this project I had to make sure I kept pushing through with my work even when I got lost. I knew that if i was just confident that I could do the math that the rest would fall into place.
Collaborate and Listen (with Modesty)- I used this habit a lot during this project. I worked a lot with Lauren on this project because she was really helpful when explaining the problems. I made sure I was working with her and not just copying the work, Also, worked with her on the problems so we could both get different perspectives.
Generalize- Generalizing is a great tool when working with quadratics, a lot of the work we did was connected so when I generalized I was able to see the overarching point of the problems.