The Quintessence of Quadratics
Introduction
In this project, we explored a variety of topics related to quadratics. Specifically, converting from standard form to vertex form (and vice versa), converting from vertex form to standard form, solving problems with quadratics, and more. Because of this exploration, I have gained much more knowledge about quadratics as a whole, as well as the in depth math that quadratics entail. I improved on multiple habits of a mathematician, such as Describing and Articulating, Staying Organized, and Being Systematic. These habits of a mathematician will help me in future math classes because they are the essence and foundation of solving problems in math, but especially in quadratics for me.
This project was first introduced by kinematics, when we worked with equations to determine the distance, motion, and acceleration. The worksheet that was given for practice in kinematics was the Distance, Velocity, and Acceleration Practice Problems paper. This paper gave me in depth practice with kinematics, as well as a better understanding.
We were then given another worksheet called the Victory Celebration, which was about a firework show and the launch of the rocket. The questions asked about the height of the building the rocket is shot from, how high the rocket goes, how long it takes to reach the ground, and where it will reach the ground. This gave us a very important equation: h(t) = d0 + v0 · t + 1^2 a · t^2
This equation gives us the initial distance, d0, initial velocity, v0, and acceleration, a. With this in mind, we are then given the equation a = g = -32 ft / s^2 which gives us the acceleration formula of the rocket mentioned in the Victory Celebration. This process in kinematics, algebra, and geometry will help us with quadratics because quadratics are an important part of algebra because they link geometry, economics, and algebra altogether.
The learning objectives in this project were mainly for a more in depth look into algebra and some preparation for the SAT. Quadratics are quite hard, but once you understand them, it is easier to comprehend and solve. My teacher wanted to provide us with that help in order for us to succeed on his worksheets and overall understanding of quadratics, algebra, geometry, and kinematics.
This project was first introduced by kinematics, when we worked with equations to determine the distance, motion, and acceleration. The worksheet that was given for practice in kinematics was the Distance, Velocity, and Acceleration Practice Problems paper. This paper gave me in depth practice with kinematics, as well as a better understanding.
We were then given another worksheet called the Victory Celebration, which was about a firework show and the launch of the rocket. The questions asked about the height of the building the rocket is shot from, how high the rocket goes, how long it takes to reach the ground, and where it will reach the ground. This gave us a very important equation: h(t) = d0 + v0 · t + 1^2 a · t^2
This equation gives us the initial distance, d0, initial velocity, v0, and acceleration, a. With this in mind, we are then given the equation a = g = -32 ft / s^2 which gives us the acceleration formula of the rocket mentioned in the Victory Celebration. This process in kinematics, algebra, and geometry will help us with quadratics because quadratics are an important part of algebra because they link geometry, economics, and algebra altogether.
The learning objectives in this project were mainly for a more in depth look into algebra and some preparation for the SAT. Quadratics are quite hard, but once you understand them, it is easier to comprehend and solve. My teacher wanted to provide us with that help in order for us to succeed on his worksheets and overall understanding of quadratics, algebra, geometry, and kinematics.
Exploring the Vertex Form of the Quadratic Equation
Later on in the project, we explored parabolas and the equations, as well as each meaning of the variables in the equations, before using full quadratic equations. The important variables in parabolas and equations are a, h, and k.
The vertex is the highest or lowest point on the graph in the parabola. Two of the variables also control where it is on the axes.
The variable a controls the concave of the parabola. If it is positive, it opens upward, and if it is negative, it opens downward. Here are some examples:
The vertex is the highest or lowest point on the graph in the parabola. Two of the variables also control where it is on the axes.
The variable a controls the concave of the parabola. If it is positive, it opens upward, and if it is negative, it opens downward. Here are some examples:
The variable h, controls where the vertex is on the x-axis. Numbers that represent h greater than zero are on the positive side of the graph on the x-axis, and numbers less than zero that represent h are on the negative side of the graph on the x-axis. Below are some examples:
The variable k represents the point on the y-axis in the equation. Numbers that represent k that are greater than zero are above the x-axis, and numbers that represent k that are less than zero are below the x-axis. Below are some examples:
The relationship between a parabola and an equation is that the parabola is the result of the equation. Without the equation, there would be no x-intercept, y-intercept, or vertex, so it would just be a straight line up or down, dependent on the sign (positive or negative).
Other Forms of the Quadratic Equation
There are two other forms of the quadratic equation: standard form and factored form.
Standard form is y = ax^2 + bx + c
Standard form is a product of a constant and some variables. This form is important because it allows us to distribute in order to get into another form or solve for a quadratic problem. A pro of standard form would be that it is easier to convert into factored form, which is explained below. An example of standard form would be 2x^2 + 5x + 3 or y = 2x^2 - 4x + 5.
Factored form is another form of a quadratic equation.
Factored form is a⋅(x-p)⋅(x-q) or a⋅(x-p)^2.
Factored form is the product of two constant linear terms. This form is important because it is the conversion of a quadratic equation, called factoring. It is the sum of a product of a sum, which means that it is a representation of logic within quadratic equations. An example of factored form would be y = 2(x - 4)(x - 8) or y = 6(x - 3)(x - 2).
Below are three pictures of graphs and parabolas that have the same equation, but in different forms, such as vertex, standard, and factored form.
Standard form is y = ax^2 + bx + c
Standard form is a product of a constant and some variables. This form is important because it allows us to distribute in order to get into another form or solve for a quadratic problem. A pro of standard form would be that it is easier to convert into factored form, which is explained below. An example of standard form would be 2x^2 + 5x + 3 or y = 2x^2 - 4x + 5.
Factored form is another form of a quadratic equation.
Factored form is a⋅(x-p)⋅(x-q) or a⋅(x-p)^2.
Factored form is the product of two constant linear terms. This form is important because it is the conversion of a quadratic equation, called factoring. It is the sum of a product of a sum, which means that it is a representation of logic within quadratic equations. An example of factored form would be y = 2(x - 4)(x - 8) or y = 6(x - 3)(x - 2).
Below are three pictures of graphs and parabolas that have the same equation, but in different forms, such as vertex, standard, and factored form.
Converting Between Forms
Converting between the different forms of quadratic equations is important to find different answers and form different solutions.
Below includes the variety of conversions between vertex form, standard form, and factored form.
Below includes the variety of conversions between vertex form, standard form, and factored form.
Converting Standard Form to Vertex Form
When converting from standard form to vertex form a part of the process includes an area diagram. An area diagram is used to "complete the square", which essentially means to make it complete by adding and subtracting until it equals 0 or the number you need to make it equal in the equation. In the picture to the left, the area diagram provides us with a number that helps us solve the equation. Converting from standard form to vertex form first starts by bringing back the hidden 1 in front of x. After this, you combine like terms. Then, you complete the area diagram or the square. This then gives you numbers to put into an equation. Once you subtract and add the equation, you are given a final equation that means the same as the first equation, but it is written differently; here it is in vertex form. |
Converting Vertex Form to Standard Form
Converting from vertex form to standard form is much more simpler than the opposite.
First, you take the original vertex form equation and distribute the a variable out to the parentheses in the equation. Then, combine like terms, and the result is an equation in standard form that means the same in vertex form. |
Converting Standard Form to Factored Form
Converting from standard form to factored form is quite simple as well. First, take the original standard equation, and distribute the numbers from the variables apart. This should result in a longer equation with parentheses.
Then, by looking at the two numbers in the parentheses, calculate what the product of the first number in the original standard equation would be (what number times what number equals the first after the equals sign) and calculate the sum of the second number in the original equation (what number plus another number equals the second number after the equals sign). The two numbers in the parentheses should be the same numbers, because they have to be the ones multiplied and added together to get to factored form. |
Converting Factored Form to Standard Form
Converting from factored form to standard form just implies a few steps in one acronym, F.O.I.L. This acronym stands for First, Outside, Inside, Last. This means that once given the factored equation, you multiply the first number or variable in the first set of parentheses by the first number or variable in the second set of parentheses, followed by the first by the last or outside number, then the inside by the first in the second set of parentheses, and the second number by the last number. In the example, we would multiply x by x, then x by -8, and then 9 by x, and lastly 9 by -8. This gives us x^2 - 8x + 9x - 72. |
Solving Problems with Quadratic Equations
In solving problems with quadratic equations, there are three different real-world types of problems. Each of them can be solved by using quadratic equations. The three areas are kinematics, which includes projectile motion, geometry, which involved triangle problems and rectangle area problems, and lastly economics, that had maximizing revenue and profit or minimizing expense or loss problems. An example of each are below, and a detailed step-by-step of Leslie's Flowers is also below.
Kinematics
Kinematics is the section of mechanics that focuses on the motion of objects that doesn't relate to the forces that actually cause the motion. In math, we answered questions and took notes for worksheets called Simple Motion 1, 2, and 3. In the problems in the Simple Motion questions, we explored how to find the x-intercept when y was set as zero. An example is below.
To find the x-intercept, first set y as 0.
This will allow for subtracting a positive 13 from both sides and move it to the side where y was, which turned it into -13. From there, put plus or minus in front of the square root of 13 equals the square root of (x-4)^2. Both sides were given the square root so the squared on (x-4) will be gone after we find the square root of +/- 13. 13 is both positive and negative because when the equation is graphed and x is solved, the answer is both positive and negative. After the square root is found, the x-intercept is given by using both positive and negative signs in the answer. |
Economics
Economics is the section of studies in which production, transfer of wealth, and consumption are researched and considered. In math, we did problems involving quadratics to find what profit was made, such as worksheets called Profit of Widgets, where we explored how much money each widget should be sold in order to make a certain amount of profit. Below is an example of an economics problem, as well as an explanation.
In this example problem, it asked about profit from mowing neighbors lawns for a certain amount of their yard per amount of money.
P = profit and R = revenue. With that in mind, we wanted to find the profit, so we took the equation in standard form and solved for P, which would normally be x. After combining like terms, we're left with -1/10x^2 + 1,880x. This then is combined with the 8,000 from the second equation, and we are given our final equation, which gave us x = (0, 18800). |
Geometry
Geometry is the section of mathematics that focuses on relations of surfaces, points, solids, lines, and different higher dimensional shapes. In math, we used triangle problems and rectangle area problems to practice and learn more about geometry. A worksheet that was given was called Leslie's Flowers, and this worksheet explored solving x through the area of a triangle. Below is a step-by-step instruction of Leslie's Flowers and another example on how to solve triangle and rectangle area geometry problems.
This triangle area problem is quite simple, but when I practiced it, it helped me better comprehend how to continue on to much more complicated problems.
Since finding the area of a triangle is 1/2 times the base times the height, the formula for this triangle would become 1/2⋅(22⋅26.8). That equation would simplify down to 1/2⋅589.6 Half of 589.6 is 294.8, therefore that is the area of this triangle in inches squared. |
The worksheet Leslie's Flowers asks about the Pythagorean Theorem and how it applies to right triangles. In his problem, we review how to do that.
The first question asked to find an equation for x. For this problem, I found x by using a to determine the equation. As shown in the picture, I solved for an equation for a and was left with a^2 = -x^2+13^2. I then took the two numbers and applied them to find an equation, which was y = (x + 5)^2 + 21. The next question asked for us to solve for x. I did this by finding the height, and then solving for x. I found the height by combining two of my earlier equations and solving. I was left with x = 12.14, but I had to be sure so I continued to find the height so I could plug it into my equation. h ended up resulting in 12, because I applied the formula for the area of a triangle. From there, I plugged in 12 and was left with 5^2, which equals 25, so x = 5. |
Reflection
Throughout the past few months we have been working on quadratics, I have come to realize that my key experiences have been practically formed through every worksheet and new topic in quadratics that we covered. I struggled with converting from standard form to vertex form because I did not understand how the area diagram gave us any new information to create and solve another equation for the converted format. I honestly felt very lost, but once I used Khan Academy to practice and asked questions to anyone and everyone, I understood more of quadratics than I thought I could. I knew how to explain it to other people and felt comfortable doing it when others asked for my help, which was rare until I knew how to do it on my own. I think this helped me become well rounded in this area of math and prepared me for 11th grade because it showed me what hard work I would have to do in the coming weeks as well as next year, and how to stay on task and study so I could succeed, which is also incredibly helpful for the SAT. I believe that I put in as much effort as I could, always trying my hardest to challenge myself and question what I am doing and why. In conclusion, I feel like I ended 10th grade on a high note, and I am proud to say I can keep practicing quadratics and equations on my own in preparation for the SAT. Meanwhile, I also used a variety of Habits of a Mathematician, yet some I was not fully aware of when conducting this project. Below is a reflection and description on how Habits of a Mathematician relate to quadratics and this project as a whole.
Habits of a Mathematician
Look for Patterns |
In Quadratics, looking for patterns is important because it helps you find the exponential growth in an equation. When graphing a quadratic equation to make a parabola, by looking at the average you can see a pattern form through the exponential growth in the parabola.
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Start Small |
When using standard form, you are starting small by looking only at the variables in the equation. From there, you can determine what you need to find and how to find it.
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Be Systematic |
When converting from one form to the other or solving a quadratic equation, we have to follow certain steps in order to get the correct answer. Being systematic while working with quadratics is crucial to the understanding and solving of equations and quadratics.
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Take Apart and Put Back Together |
In many forms of quadratics, specifically standard form and vertex form, you have to take apart the equation in order to change it into another. This is also essential to quadratics because without separating the equation, you wouldn't be able to convert it into another form.
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Conjecture and Test |
In some problems dealing with quadratics, the answer is easy to tell if you understand each form of each type of equations. For example, when a question asks for the factored form of a standard form equation, the answer is the choice that is in factored form (if multiple choice), which is a⋅(x−p)⋅(x−q). This means that on every problem you can guess at first, but then you have to go back and work through the problem to check your answer.
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Stay Organized |
Without staying organized while working with quadratics, the work can get messy. By staying organized while solving problems, it allows you to go back to the important information needed to complete or review the problem.
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Describe and Articulate |
Explaining what we did and how we did it is vital to grasp the concept of quadratics, and it can help our peers if one understands and others do not.
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Seek Why and Prove |
Every quadratics problem has a purpose, but they are all different. By looking at the fundamental question in the problem, we can comprehend what it is asking and further completing the problem.
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Be Confident, Patient, and Persistent |
When we first started to explore quadratics, I wasn't very confident, but that held me back. Over time and a lot of practice, I came to realize that I did learn and follow what I had to do to solve quadratic problems and equations.
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Collaborate and Listen |
Collaborating with peers is crucial as well because they can help you figure out where you went wrong and how to fix it. Listening is imperative because if the teacher explains how to solve or convert an equation or problem, you will want to know how to do it by taking notes and listening to the process.
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Generalize |
Generalizing is critical all throughout quadratics because in order to solve a problem or equation, you need to be able to generalize the equation to a form that you understand. For example, if a problem is asking you to solve an equation, you need to generalize and remember which format for an equation it is in to know how to take the first step and solve it.
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