Introduction
This project is about quadratic equations and functions. We learned about different forms of quadratic equations and different ways to solve them. For our first activity we started with a worksheet called “Victory Celebration.” The problem involved shooting a rocket for a building that is 160 feet tall.We had to find out what the maximum height of the rocket will be, when the rocket reaches its maximum height, and how long the rocket is in the air. We then continued of with many other worksheets and learned about a site called desmos that helped us get a clearer visual of how manipulations between equations work.
Exploring the Vertex Form of the Quadratic Equation
To get some practice with Quadratics and Vertex form we completed many more worksheets. To help us complete these worksheets we used a website called Desmos which is an online graphing calculator. Desmos was used as a visual aid to help us better understand how the a, h, and k parameters affect the location and shape of the parabola. 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
Factored form and standard form are two other ways that quadratics can be written. Standard form is a function that can be written as ax^2 + bx + c = y. The advantages of using this form is that it is a more organized way and it gives you the y-intercept. Factored form is another function that is written like y=a(x-q)(x-p). The advantage of using this is that it gives you the x-intercept.
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Converting between Forms
Vertex to standard: To convert vertex form to standard form you start out by foiling the equation which means taking (x-h) and doubling it because it is squared then foiling it in the correct order which is first, outer, inner, last. After you foil the equation you distribute the a value. Finally you combine like terms and you will get your equation in standard form. y=ax^2+bx+c
Standard to vertex: To convert standard form to vertex the first step is completing the square which is to draw an area diagram and fill it in according to the equation you have, after this step you should have an equation that looks like this a(x^2+bx+d-d)+c. Then you change where your parenthesis are placed so your equation looks like this a(x^2+bx+d)-d+c. after this you take what you got when you completed the square and put it as your x and h value and combine like terms. Your final equation in vertex form would be y=a(x-h)^2+k. Factored to standard:To convert factored form to standard form the first thing you need to do is distribute using the foil method. Your next step would be to combine like terms and distribute the a value. After these steps your equation should be in standard form ax^2+bx+c=y. Standard to factored: To convert standard form to factored form the first step to converting standard form to factored form is doing the reverse of distributing which using an area diagram. After plugging in your numbers and solving the area diagram you simplify your equation and end up with the factored form a(x-f)(x-r)=y. The bottom area diagram represents Standard form to Vertex form. |
Solving Problems with Quadratic Equations
Throughout this unit we learned three types of real world problems;
1. Kinematics (projectile motion) One packet we used this for was "VIctory Celebration." We had to use our knowledge of quadratics to solve. We converted our equation from standard form into vertex form in order to find the vertex, which is what the problem asked for. 2. Geometry (triangle problems and rectangle area problems) One packet we used this for was "Emergency at sea." We used Pythagorean's theorem in order to find the side length of a triangle, which was a quadratic equation. 3. Economics (maximizing revenue/profit or minimizing expenses/losses) One packet we used this for was "Profiting from widgets." Using a formula they gave us for widget sales in terms of d dollars we determined the optimal price to sell widgets. |
Reflection
This was probably the hardest project of the year. Even though it was really hard I am proud of myself because I completed and turned everything in on time and learned quadratics at the same time. At first it took me a while to wrap my head around it but my peers helped me understand it better. One thing that help was the area diagrams because it gave me a shortcut to equations. This will help me a lot on the SAT because I know we will be doing problems like this on the SAT and it has prepared me for any form of quadratics that will be on the SAT. For next year I hope that it will help me with what we will be learning for next year and by me still knowing this it will give me a good head start on the math next year.
Look for Patterns - I picked up real quick when I saw a pattern and i feel like i used this one the most of all. Especially while playing with parabolas.
Start Small - To help myself get a better understanding of the math it was key to start small.
Be Systematic- When I made a mistake I was able to be systematic and go back and fix them.
Take Apart and Put Back Together-For each question on all of the packets I would take it apart and look at it that way to see what I needed to do to solve it.
Conjecture and Test- When I was confused on which formula to use I would test out all of them to see which one was correct
Stay Organized- I had to stay organized so that I didnt lose any packets when it came time to turn them in
Describe and Articulate- Talking through problems with the teacher and my peers helps me the most and when I ask someone to work me through it I ask a lot of questions so that I understand everything.
Seek Why and Prove- For all of these worksheets I always had a struggle with something but when someone helped me I always wanted to know why that was the answer so I would go through it even more.
Be Confident, Patient, Persistent- I had to be very patient with myself when I got frustrated and I had to be confident in myself when I made a mistake that I will find the mistake and fix it.
Collaborate and Listen- Collaborating with my peers is what helped me the most for this project. It always helps me when I see different takes on problems to see which one is easier for me so I go to mulitple peers for help.
Generalize- Generalizing helped me remember all of the formulas we needed for all of the worksheets.
Look for Patterns - I picked up real quick when I saw a pattern and i feel like i used this one the most of all. Especially while playing with parabolas.
Start Small - To help myself get a better understanding of the math it was key to start small.
Be Systematic- When I made a mistake I was able to be systematic and go back and fix them.
Take Apart and Put Back Together-For each question on all of the packets I would take it apart and look at it that way to see what I needed to do to solve it.
Conjecture and Test- When I was confused on which formula to use I would test out all of them to see which one was correct
Stay Organized- I had to stay organized so that I didnt lose any packets when it came time to turn them in
Describe and Articulate- Talking through problems with the teacher and my peers helps me the most and when I ask someone to work me through it I ask a lot of questions so that I understand everything.
Seek Why and Prove- For all of these worksheets I always had a struggle with something but when someone helped me I always wanted to know why that was the answer so I would go through it even more.
Be Confident, Patient, Persistent- I had to be very patient with myself when I got frustrated and I had to be confident in myself when I made a mistake that I will find the mistake and fix it.
Collaborate and Listen- Collaborating with my peers is what helped me the most for this project. It always helps me when I see different takes on problems to see which one is easier for me so I go to mulitple peers for help.
Generalize- Generalizing helped me remember all of the formulas we needed for all of the worksheets.