In this project we learned all about parabolas, quadratic functions and how to solve a quadratic equation. We started by learning about kinematics, acceleration, distance, time and revisiting area, volume and the Pythagorean Theorem. Then we started to practice using geometry and graphing with the online tool "Desmos". Desmos was a very helpful tool throughout this unit with helping us visualize the shapes, sizes, and locations of the parabolas on the graphs and using the equations to determine how the parabola will look and where it will be placed. It is very helpful to visualize the graphs, parabolas the quad technique when doing quadratics. This became very useful especially when we starting writing quadratic expressions. We used algebra and geometry in class to help us obtain the physics displacement/quadratic equation. The first big project we had in the quadratic unit was "Victory Celebration". This problem was about a High Tech High rocket used to launch a fireworks display from a platform. We were asked to figure out the time and height of the rockets apex. Also we were asked to find the total amount of time from launch off the platform to when the rocket hits the ground. We started by creating an equation with an input and output. The output being the height of the rocket and the input being the time from the rocket launch. We were already given the information that the rocket was launching off of a 160ft platform so we took the initial height of the rocket, the launch velocity which was 92m/2 and the downwards acceleration of -32m/s/s which is caused on the rocket due to gravity. We also took into account the angle of which the rocket is being launched at and how that will have an effect on it. We starting looking more into kinematic equations using the distance formula which is h(t) = d0 + v0 · t + 1/ 2 a · t^2. After we plugged in all of the information we were given and combined all of the like terms we got the equation h(t) = 160 + 92t - 16t^2. After looking deeper into this equation we soon realized that we were also working with a quadratic equation of y=ax^2+bx+c in this problem. After that step we found the vertex coordinates and the positive x intercept. The main objective and purpose of the project was not only to just understand quadratic functions but to understand their representations and how to use our specific set of skills and habits of a mathematician in order to solve quadratic equations. This whole unit involved being systematic throughout and I collaborated and listened a lot during this unit especially with my peer Jeffery who helped me a lot with understanding quadratics and converting forms. I also spent a lot of time taking the equations apart and putting them back together and I wouldn't stop until I figured out an answer.
Exploring Vertex Form
In order to understand vertex form we had to first understand variables such as A, K, and H. The equation we started out with was a great start to understanding the basics of solving quadratic equations but it made all the values we wanted to find not very easy to get right away. So to get even more information from the equation we started to convert to vertex form. Vertex form is written as y=m(x-h)^2 +k, with the vertex being (h,k).We also practiced using many handout packets and Desmos (Hand out packet images shown below). When we plugged the equation into Desmos, we immediately realized that quadratic equations create parabolas when you graph them. The parabolas change using many factors in the equations such as negative and positive numbers ( concave up or concave down) and the number range ( how wide or narrow the parabola will be). Next we saw how adding a variable like a value to the equation: y=x^2, you get y=ax^2 also has an affect on the shape of the parabola. The higher the time of A the more narrow the parabola become and a lower sum of A creates a wider curve of the parabola as well.The next variable out of the three we plugged into the equation was k. K is equal the y-coordinate value of the parabola's vertex, when you plug or add it into the equation of y=ax^2+k. The last variable we added was h. H shows the x-coordinate value. Our finished vertex form equation came out to be this y=a(x-h)^2+k.
Other Quadratic Forms
We learned all about the three different forms of writing quadratic equations. Not only Vertex form. The other two forms are Standard form and Factored form. Standard form written as an equation looks like y=ax^2+bx+c. This is also where a, b, and c can be absolutely any number. The b constant in standard form was very difficult for me to understand at first. But once I figured out that the b constant is where all of the confusing stuff from the other forms gets stuffed into when you are converting into standard form I felt much better and understood more of what was happening in the equation.Using standard form can be very helpful especially because you can find the y-intercept or slope of the parabola very easily. Factored form looks like y=a(x-q)(x-p) and is the expanded version of standard form. Using factored form the constant a has the exact same effect as in the vertex form. While q & p are the x-intercepts of the parabola. Factored form allows you to see the x-intercepts if there are any in the parabola directly. Factored form is also great for if you are trying to find the zeros of a parabola. I had to seek why and prove a lot when I was learned about each form and what each form is best used for. Like how I looked for the pattern during an SAT warm up to figure out what the answer was based on what form each answer was in and what the question was looking for.
Converting Between Forms
Vertex to Standard Form: Simplifying the equation In order to convert from vertex form which is y=a(x-h)^2+k to standard form you need to expand the equation so it doesn't have any parentheses first and it's important to simply as much as you can until you get what you want or are trying to find. You can use FOIL by using the (x-h)(x-h) term and then multiplying/distributing that by the a variable. Then you combined all of the like-terms and the equation should turn into the form y=ax^2+bx+c. Once you get rid of the ()^2 you need to start multiplying every number by every other number there is and then you can also use distributive property for the "a"/ constant. Then you continue to simplify everything again. Standard to Vertex Form: Completing the square In order to convert from Standard form or a quadratic in the form of y=ax^2+bx+c to y=a(x-h)^2+k first you need to finish the square by filling in all of the missing terms in the square. Then leave the c value alone and factor out the a that is from ax^2+bx then add and subtract a value from the equation so that way the equation looks like y=a(x^2+bx+d-d)+c. Where x^2+bx+d also can be written as a squared term such as (x-h)^2. Then multiply the -d by a and after you've done that add that to c to get k. To finish the problem rewrite the parenthesis as a product of the square and then combine all the like terms that are located outside of the parenthesis. The equation is now in Vertex form y=a(x-h)^2+k. I used be confident, patient and persistent habits of mathematician when I was learning how to convert these forms. It is important to use the habit of a mathematician stay organized when you are converting between standard to vertex. It is very helpful to right down and visualize all the numbers and what you are doing instead of trying to remeber everything and trying to convert in your head. Factored to Standard Form Whenyou are converting from factored form to standard form you are basically step by step, multiplying all the terms that are in the parenthesis, combining all the like terms and distributing a. Another way you could describe factored to standard form would be by multiplying every number by every other number and then simplifying the equation. Standard to Factored Form Personally this was the hardest conversation for me to understand. In order to convert from standard form to factored form you have to work backwards.You want to multiply the n and m values to get the c constant value. If the n and m values are added together you'll get the b constant. But before they are added you also need to multiply them by values such as the values that are multiplied by are equal to the a constant. Afterwards you now are are able to factor the a constant out of the equation. Then you have to conjecture and test to find all of the values you will need. Then after you find all of the values, write out the factored equation and you are finished converting from standard to factored form.
Solving problems with quadratic equations
Here are some examples of what different kind of problem solving with quadratics we covered over the coarse of this unit Kinematics The very first project we did about rockets "The Victory Celebration" was all about kinematic equations being solved using the quadratic formula. In this project we converted a kinematic equation into a quadratic equation, the project overview is written in my introduction. In order to solve this problem the first thing you need to do is get the vertex coordinates and the positive x-intercept value. Those two things are very important and essential to starting. In order to do this you need to convert from vertex form into standard form which is written step by step above. Because this problem has a lot of different decimals that can get really confusing if you aren't organized it's helpful to use a calculator when you convert to vertex form. Then take both the h and k variables and use them as the time of the rockets maximum height. Because in order to solve for factored form there is some guessing and checking to try and figure out that the n and m values will most likely be decimals and would make the conversion difficult to figure out I collaborated and listened with my peers to try and find another way to solve. We started small and with vertex form, then since there was only one x in vertex form. My friends and I did a lot of generalizing during this part and found one positive answer. We immediately knew we had done this correctly because there is no way it would be possible for the answer to be negative because a rocket can not land before it has even been launched. At the end of this project I checked my answer by plugging it into desmos and I was very happy the answer was correct. Geometry We spent a lot of time learning about the geometry parts of quadratics during this project. Area diagrams became very useful as well. We spent a lot of time trying to maximize the area of a certain amount of fencing or a pen. This problem creates a quadratic equation because of the side length of the side of the pen perpendicular to your neighbor's fence which is x. The total fence length you have available is F and the area of the fence is x(F-2x). You can then re-write this as a quadratic equation in factored form: x(-2+F). You can use quadtrcis to solve this problem we did in class by using the equations above. Economics The problem we did that involved economics was all about finding the best price for "widgets" based on a guess of approximately how many would be sold based on the price of the widgets. If the company starts off with a thousand widgets it is predicted that they will sell a thousand for every five widgets where the price is represented as d. What this means is that the amount of money that the company will earn based on the price of each widget M(d)=d(1000-5d). In order to find the best price for the widgets we conjecture and test again and re-write the equation in vertex form as M(d)=-5(d-100)^2+50,000. Now we know that we should sell each widget at 100 dollars each and so all 500 will add up to 50,000 total. Overall the amount of widgets bought based on the price the widgets are being sold for is most likely not linear and the equation does not include the information that we need to know about the price each widget costs to make. If we knew how much the manufacturing price of each widget was we could make this problem one step more challenging to solve. Economic problems involve quadratic equations a lot because, for example the more expensive you make a product the less people will buy it. And the less expensive you make a product the more people will buy the product and more of it will sell. If you are trying to figure out what the best price of a product is the first thing you need to do is think about the price of the product and how many you are wanting to sell. If you want to sell less product for a higher price of more product for a lower price. The price of the product will determine how it sells. This then leads to an equation where the same variable appears two times. This can be written as a x^2 squared term. Economics is a really cool topic in math to learn about because it connects directly to real world situations and money.
Reflection
I am very proud of the work I did in this project and how I challenged myself a lot. I knew absolutely nothing about any of what we learned this semester before I was taught. Many of my other peers already has experience using FOIL, quadratic equations and parabola work. I didn't even know was a parabola was or what quadratics was all about before this unit. I spent a majority of my time learning collaborating with others in class and learning from them as well. I really challenged myself and pushed myself to fully understand this content because I know it will be very important and be a base for the rest of high school math, moving into junior year and for the SAT coming up as well. I made sure to be systematic throughout this whole entire unit and it really helped me with math and learning how to specifically convert between different forms. I also am a very visual learner and love to draw and express my work and thinking through drawing. I describe and articulate myself best when I can write or draw what I am thinking. This was amazing for this project because area diagrams and demos made it much easier to understand what I was doing in a way I could clearly see it and it made it very fun for me to learn about all of this intimidating new math problems I had never seen before. This was defiantly one of the most challenging but also my favorite projects I have ever done in math class. I feel like I really learned so much which sometimes I don't feel like I fully understand but I move on. That was never the case this year, I spent many days staying in at lunch and making sure I never was behind and understood all that was being taught. Especially considering the fact I want to get into some colleges that are definitely out of my reach right now but I don't want it to stay that way. I have also don't lot's of SAT prep work out of the book I ordered and that has helped me grow on my own time with this topic as well. I know learning this now will help me with the SAT and help me get into the colleges of my dreams like Stanford. Now that I am moving into 11th grade I want to continue to challenge myself in math and reach for the colleges, SAT scores and further better myself as a mathematician. I am also very grateful for my school and all the help my peers and teachers have been for me this year and helping me grow a lot as a person.