Introduction
The purpose of this project overall was to understand and solve Quadratics. We learned about the three different forms of quadratic equations: standard, vertex, and factored form. We learned how to convert from one form to another, and we use a method called "completing the square" to help us visualize what we were doing. These appear in several contexts, including motion (e.g., rockets launching), areas and volumes, and even economics. These showcase how quadratics can be used to solve real world problems. This project connected algebra and geometry, both concepts that we had learned previously.
Essentially, a quadratic equation is a rule that determines the distance from a focus point and a directrix (a horizontal line). The distance from the focus point and the directrix must always be the same, so this creates several points in a curved shape. This rule is what creates the parabolas that we saw throughout this project.
To start off this project, we did a problem called "The Victory Celebration". This problem involved a rocket being launched off a platform. We used quadratics to discover the path that the rocket took. As it turns out, quadratics can be used to accurately predict the highest point of the rocket as well as the ending point.
Essentially, a quadratic equation is a rule that determines the distance from a focus point and a directrix (a horizontal line). The distance from the focus point and the directrix must always be the same, so this creates several points in a curved shape. This rule is what creates the parabolas that we saw throughout this project.
To start off this project, we did a problem called "The Victory Celebration". This problem involved a rocket being launched off a platform. We used quadratics to discover the path that the rocket took. As it turns out, quadratics can be used to accurately predict the highest point of the rocket as well as the ending point.
Exploring the Vertex Form of the Quadratic Equation
There are several forms of quadratic equations; vertex form is one of them. In full, the equation is f(x)=a(x-h)^2+k. In this form, the parameters a, h, and k, affect the location and shape of the parabola. h and k affect the vertex of the parabola, acting as the x and y values, respectively. a affects the concavity of the parabola. If a is less than zero, the parabola opens downward. If a is greater than zero, the parabola opens upward.
Before being formally introduced to these concepts, we were given a program called Desmos, that graphs any line made by an equation you give it. I played around with different formulas, and eventually figured out the rule of what a, h, and k determine in a vertex form equation. This shows how experimenting with equations by yourself can actually be useful, show how they work better than without a visual representation.
Before being formally introduced to these concepts, we were given a program called Desmos, that graphs any line made by an equation you give it. I played around with different formulas, and eventually figured out the rule of what a, h, and k determine in a vertex form equation. This shows how experimenting with equations by yourself can actually be useful, show how they work better than without a visual representation.
Other Forms of the Quadratic Equation
Aside from vertex form, the other two forms of quadratics are standard form and factored form. The equation for Standard Form is ax^2+bx+c, and the equation for factored form is a(x-p)(x-q). An advantage of vertex form is that is gives the vertex of the parabola, which can be useful for finding trajectory. The advantage of standard form is that the monomials are added in decreasing order, which makes things simpler. Factored form is easier to solve, because each factor is set to zero.
As you can see above, the three forms of the quadratic here are making the exact same parabola. I had to use decimals here instead of fractions, but 0.333 is equivalent to 1/3.
Converting Between Forms
To convert from standard form to vertex form, we use a process called "completing the square". To do this, we first isolate the x and y terms by adding or subtracting. Then, we factor out any leading coefficient, so that our x^2 value has a coefficient of 1. Then we take half of the coefficient of the x term, square it, and add it to the end. At this point, we just simplify the equation, and we're done.
Converting from vertex form to standard form is much simpler. We first factor out (x+h)^2 into (x+h)(x+h). Then, multiply that by the number outside the parenthesis. Finally, we foil the remainder, and then simplify.
Area diagrams, like this one here, can help people visualize what is happening when they convert between forms. This is very similar to the geometry based problems that we did in this project.
Solving Problems with Quadratic Equations
There are three types of real-world problems you can solve using quadratics. These are: kinematics, geometry, and economics. For kinematics, quadratics can be used to calculate projectile motion. In geometry, quadratics can be used to find the area of triangles and rectangles. In economics, quadratic equations can be used to find profit and revenue.
Here is an example of a kinematics problem I solved recently. In this problem, a rocket is launched off a 160 foot platform at a velocity of 92 feet/second^2. I turned this information into the equation h(t)=160+92t-16t^2. Then, I converted this into a vertex form equation. This gave me the vertex at 292 feet. Solving this same equation with y set to zero showed that the final time of the rocket in the air was 4.2 seconds.
Reflection
Overall, I feel like this project taught me a lot of new things. I took the SAT recently, and I feel like my learning in this project prepared me for that test a lot. I saw a few concepts on the test that I wouldn't have learned if it wasn't for this project. I feel like this project has made sense of the struggles I've had with SAT math before. Despite learning quadratics with a tutor, I still got intimidated and confused when I saw problems with them on the actual SAT. I feel like the next time I take it, I will be more comfortable with these types of problems.
In this project, I used the Habits of Mathematicians to help me in doing my work. I used the habit "Starting Small" to get started on bigger quadratic equations. This usually meant simplifying the equations and formatting so that they were less visually confusing to me. I also used the habit "Seek Why and Prove", when I graphed the equations that I created. Looking at the actual parabola an equation makes makes it a lot more clear whether or not I did a problem correctly. I "Looked For Patterns" when I was converting between equations. I used "Be Systematic" when I was looking back on my work. I "Took Apart and Put Back Together" when I was converting between vertex and standard form. I "Conjectured and Tested" when I tested out my equations in Desmos. I "Stayed Organized" by keeping all my math work in order. I "Described and Articulated" when I shared my work with my peers. I used "Be Confident, Patient, and Persistent" when I was stressed out by a hard question. I used "Collaborate and Listen" when I was working with my peers. Finally, I used "Generalize" to invent rules that applied to the quadratic equations.
I still struggle a little bit with figuring out what to do when given an equation. I sometimes get the forms mixed up, or forget the steps involved in solving them. Equations that look really messy and complex stress me out sometimes, and I feel like I'm not capable of solving them. I feel like these are things that I still need to work on over the summer so that I can increase my learning get the best score on the SAT possible. Overall, I think that this was a good project to wrap up the year with.
In this project, I used the Habits of Mathematicians to help me in doing my work. I used the habit "Starting Small" to get started on bigger quadratic equations. This usually meant simplifying the equations and formatting so that they were less visually confusing to me. I also used the habit "Seek Why and Prove", when I graphed the equations that I created. Looking at the actual parabola an equation makes makes it a lot more clear whether or not I did a problem correctly. I "Looked For Patterns" when I was converting between equations. I used "Be Systematic" when I was looking back on my work. I "Took Apart and Put Back Together" when I was converting between vertex and standard form. I "Conjectured and Tested" when I tested out my equations in Desmos. I "Stayed Organized" by keeping all my math work in order. I "Described and Articulated" when I shared my work with my peers. I used "Be Confident, Patient, and Persistent" when I was stressed out by a hard question. I used "Collaborate and Listen" when I was working with my peers. Finally, I used "Generalize" to invent rules that applied to the quadratic equations.
I still struggle a little bit with figuring out what to do when given an equation. I sometimes get the forms mixed up, or forget the steps involved in solving them. Equations that look really messy and complex stress me out sometimes, and I feel like I'm not capable of solving them. I feel like these are things that I still need to work on over the summer so that I can increase my learning get the best score on the SAT possible. Overall, I think that this was a good project to wrap up the year with.