Project Overview
Intro to Quadratics: We kicked off our unit by looking at kinematics, and from there used variables we were already familiar with (distance, acceleration, time, etc.) to derive the distance equation with geometry and graphing. Using this equation, we were able to begin solving problems that involved parabolas, such as "Victory Celebration."
Vertex Form: Once we began to get more familiar with the idea of parabolas, we began to look at the equations for these graphs. To accomplish this, we completed a series of handouts, each covering a new aspect of the main topic. We started with the most basic of quadratic equations: y=x^2. From there, we added a- a variable which when added into the equation (y=ax^2) effects the curve of the parabola. A higher value of a results in a narrower curve, while a lower value will make it wider. Next we looked at k, which in the equation (y=ax^2+k) is equal to the y-coordinate of the parabola's vertex. Lastly, we learned about h, which is equal to the x-coordinate of the vertex. A quadratic equation in vertex form looks like this: y=a(x-h)^2+k
Other Forms: Equations in vertex form are especially useful when identifying parabolas because you can clearly see the exact coordinates of the vertex in the equation, but there are other forms as well. For example, 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. The last type of quadratic equation is called factored form, and looks like this:
y=a(x-q)(x-p) and 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 (if any) of the parabola.
Vertex Form: Once we began to get more familiar with the idea of parabolas, we began to look at the equations for these graphs. To accomplish this, we completed a series of handouts, each covering a new aspect of the main topic. We started with the most basic of quadratic equations: y=x^2. From there, we added a- a variable which when added into the equation (y=ax^2) effects the curve of the parabola. A higher value of a results in a narrower curve, while a lower value will make it wider. Next we looked at k, which in the equation (y=ax^2+k) is equal to the y-coordinate of the parabola's vertex. Lastly, we learned about h, which is equal to the x-coordinate of the vertex. A quadratic equation in vertex form looks like this: y=a(x-h)^2+k
Other Forms: Equations in vertex form are especially useful when identifying parabolas because you can clearly see the exact coordinates of the vertex in the equation, but there are other forms as well. For example, 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. The last type of quadratic equation is called factored form, and looks like this:
y=a(x-q)(x-p) and 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 (if any) of the parabola.
Converting Between Forms:
Vertex to Standard: Simplify the equation
y = (x-2)^2 + 3
y = (x-2)(x-2) + 3
y = x^2 - 2x - 2x + 4 + 3
y = x^2 - 4x + 7
Standard to Vertex: Complete the square
y = x^2 - 6x + 5
y = (x^2 - 6x) + 5
y = (x^2 -6x +9) +5 -9
y = (x-3)^2 -4
Factored to Standard: Simplify the equation
y = 2(x+3)(x+2)
y = 2(x^2 +5x + 6)
y = 2x^2 + 10x + 12
Standard to Factored: Reverse FOIL
y = x^2 + 7x + 12
{3+4 = 7 3x4= 12}
y = (x+3)(x+4)
Area Diagrams: Area diagrams are a method of simplifying and converting quadratic equations that provide a visual to those who learn visually. With an area diagram, you are able to write out and label every step of the problem. They help you to see very clearly where each term should go, and keep your work organized. It helps you to break down the problem into simpler, more straightforward steps.
Solving Problems with Quadratic Equations: There are many real world connections that can be made with quadratic equations. The main three that we worked with throughout this unit were kinematics, geometry, and economics. An example of a problem that represented kinematics was the "Another Rocket" handout. In this handout, we looked at the arcs of different projectiles. For geometry, the handout "Leslie's Flowers" applies, in which we used quadratic equations to find missing side lengths in a triangle. For economics, "Widgets" was a worksheet where we were tasked to use a quadratic equation to look at the relationship between the price of an object and the amount of that object you will sell.
Solving a Problem: An example of a problem I solved is the "Another Rocket" problem. You can see my work for the problem in the two pictures below. For parts 1 and 2 of the problem, the equation for the height of the rocket is given in standard form (h(t) = 80 + 64t - 16t^2). Part 1 asks you to put it in vertex form, so I completed the square to get (h(t) = -16(t-2)^2 + 144). Part 2a asks for the time it took for the rocket to reach its maximum height, and part 2b asks for the rocket's maximum height. On a graph of the rocket's height v.s. time, the rocket's height is a parabola, and it reaches its maximum height at the vertex of the parabola. The amount of time it took is the x-coordinate of the vertex, and the maximum height is the y-coordinate of the vertex. Both values can be found easily from the vertex form of the equation: the vertex is (2 , 144). So, it took 2 seconds to reach the its maximum height, and its maximum height was 144 feet. Part 2c asks for the amount of time it takes for the rocket to hit the ground. When the rocket hits the ground, its height is 0. I found the time it takes the rocket to hit the ground by setting the vertex form of the equation equal to 0 and solving for t. It takes 5 seconds for the rocket to hit the ground. Part 3 asks for the coordinates of the point where a different rocket, whose path is given by the parabola (y = -0.0625x^2 + 5.5x), hits a hill that can be modeled with the line (y = 0.5x). From the graph given in the problem, it's apparent that the point where the rocket hits the hill is the point where the parabola and the line intersect. I found the x-coordinate of this point by setting the two equations equal to each other (-0.0625x^2 + 5.5x = 0.5x) and solving for x. Then, I plugged the value of x into the equation (y = 0.5x) to get the y-coordinate. The rocket hits the hill at the point (80 , 40).
Vertex to Standard: Simplify the equation
y = (x-2)^2 + 3
y = (x-2)(x-2) + 3
y = x^2 - 2x - 2x + 4 + 3
y = x^2 - 4x + 7
Standard to Vertex: Complete the square
y = x^2 - 6x + 5
y = (x^2 - 6x) + 5
y = (x^2 -6x +9) +5 -9
y = (x-3)^2 -4
Factored to Standard: Simplify the equation
y = 2(x+3)(x+2)
y = 2(x^2 +5x + 6)
y = 2x^2 + 10x + 12
Standard to Factored: Reverse FOIL
y = x^2 + 7x + 12
{3+4 = 7 3x4= 12}
y = (x+3)(x+4)
Area Diagrams: Area diagrams are a method of simplifying and converting quadratic equations that provide a visual to those who learn visually. With an area diagram, you are able to write out and label every step of the problem. They help you to see very clearly where each term should go, and keep your work organized. It helps you to break down the problem into simpler, more straightforward steps.
Solving Problems with Quadratic Equations: There are many real world connections that can be made with quadratic equations. The main three that we worked with throughout this unit were kinematics, geometry, and economics. An example of a problem that represented kinematics was the "Another Rocket" handout. In this handout, we looked at the arcs of different projectiles. For geometry, the handout "Leslie's Flowers" applies, in which we used quadratic equations to find missing side lengths in a triangle. For economics, "Widgets" was a worksheet where we were tasked to use a quadratic equation to look at the relationship between the price of an object and the amount of that object you will sell.
Solving a Problem: An example of a problem I solved is the "Another Rocket" problem. You can see my work for the problem in the two pictures below. For parts 1 and 2 of the problem, the equation for the height of the rocket is given in standard form (h(t) = 80 + 64t - 16t^2). Part 1 asks you to put it in vertex form, so I completed the square to get (h(t) = -16(t-2)^2 + 144). Part 2a asks for the time it took for the rocket to reach its maximum height, and part 2b asks for the rocket's maximum height. On a graph of the rocket's height v.s. time, the rocket's height is a parabola, and it reaches its maximum height at the vertex of the parabola. The amount of time it took is the x-coordinate of the vertex, and the maximum height is the y-coordinate of the vertex. Both values can be found easily from the vertex form of the equation: the vertex is (2 , 144). So, it took 2 seconds to reach the its maximum height, and its maximum height was 144 feet. Part 2c asks for the amount of time it takes for the rocket to hit the ground. When the rocket hits the ground, its height is 0. I found the time it takes the rocket to hit the ground by setting the vertex form of the equation equal to 0 and solving for t. It takes 5 seconds for the rocket to hit the ground. Part 3 asks for the coordinates of the point where a different rocket, whose path is given by the parabola (y = -0.0625x^2 + 5.5x), hits a hill that can be modeled with the line (y = 0.5x). From the graph given in the problem, it's apparent that the point where the rocket hits the hill is the point where the parabola and the line intersect. I found the x-coordinate of this point by setting the two equations equal to each other (-0.0625x^2 + 5.5x = 0.5x) and solving for x. Then, I plugged the value of x into the equation (y = 0.5x) to get the y-coordinate. The rocket hits the hill at the point (80 , 40).
Reflection
Throughout this project, I've learned a lot of new material. Quadratics were very new to me, and so this unit gave me some more insight into new methods of solving problems. While I did face quite a few challenges, I was lucky in that I tend to learn quickly, and therefore I managed to grasp many of the concepts easily. This unit did teach me some new math strategies and concepts, but I wish that I had challenged myself more and been able to increase my learning. However, this project has helped me to look into my future of math classes here at High Tech High and I understand that if I want to continue advancing, I'm going to have to challenge myself a lot more. I do believe that this project was put in place to help prepare us for eleventh grade, and I think that it will be helpful to have like sort of foundation in quadratics in the future.
Look for Patterns: This habit was extremely applicable to our quadratics unit, and it's always helpful to keep this in mind when starting a new problem or set of problems. Here, it proved very helpful to remember certain aspects of previous problems and utilize the skills gained in solving a new, similar problem.
Start Small: Starting small was consistently a useful skill to have during this quadratics unit, since many of the problems were very complicated and it was helpful to work with a simpler version first.
Be Systematic: Taking the various forms of quadratic equations and playing with the variables individually to see what each one did to the parabola helped me to gain a better understanding of each form.
Take Apart and Put Back Together: This project helped teach me how the different forms of quadratic equations are useful in various types of problems. I now have a better understanding of when I want to use the vertex form (when I need to find the vertex of a parabola), the factored form (when I need to find the x-intercepts of a parabola), or the standard form (when I need to find the y-intercept of a parabola).
Conjecture and Test: Conjecture and test proved useful mainly during factoring, when you occasionally had to make an educated guess about which numbers worked.
Stay Organized: Since the problems could get so complicated, it was essential to stay organized so that if something went wrong, you could easily identify and correct mistakes.
Describe and Articulate: In this project, it was useful to use graphing as well as numbers and formulas to check my work, especially by graphing multiple forms of an equation in Desmos to make sure that I had converted between the forms correctly.
Seek Why and Prove: I had never seen completing the square before, but I now understand how the process works and why the process is the way it is.
Be Confident, Patient, and Persistent: Before this unit, I had a very limited understanding of quadratics, but I worked hard to learn, and I'm now confident in my ability to do complicated things like completing the square.
Collaborate and Listen: Collaborating was especially useful when I wasn't sure how do go about starting a problem. It helped to be able to ask my table members how they started the problem, and from there get inspired to solve it in a way that made sense to me.
Generalize: I not only understand how to manipulate numbers, but I understand the structures of the various forms as well and what the variables represent. This allows me to use the forms effectively in all types of problems.
Look for Patterns: This habit was extremely applicable to our quadratics unit, and it's always helpful to keep this in mind when starting a new problem or set of problems. Here, it proved very helpful to remember certain aspects of previous problems and utilize the skills gained in solving a new, similar problem.
Start Small: Starting small was consistently a useful skill to have during this quadratics unit, since many of the problems were very complicated and it was helpful to work with a simpler version first.
Be Systematic: Taking the various forms of quadratic equations and playing with the variables individually to see what each one did to the parabola helped me to gain a better understanding of each form.
Take Apart and Put Back Together: This project helped teach me how the different forms of quadratic equations are useful in various types of problems. I now have a better understanding of when I want to use the vertex form (when I need to find the vertex of a parabola), the factored form (when I need to find the x-intercepts of a parabola), or the standard form (when I need to find the y-intercept of a parabola).
Conjecture and Test: Conjecture and test proved useful mainly during factoring, when you occasionally had to make an educated guess about which numbers worked.
Stay Organized: Since the problems could get so complicated, it was essential to stay organized so that if something went wrong, you could easily identify and correct mistakes.
Describe and Articulate: In this project, it was useful to use graphing as well as numbers and formulas to check my work, especially by graphing multiple forms of an equation in Desmos to make sure that I had converted between the forms correctly.
Seek Why and Prove: I had never seen completing the square before, but I now understand how the process works and why the process is the way it is.
Be Confident, Patient, and Persistent: Before this unit, I had a very limited understanding of quadratics, but I worked hard to learn, and I'm now confident in my ability to do complicated things like completing the square.
Collaborate and Listen: Collaborating was especially useful when I wasn't sure how do go about starting a problem. It helped to be able to ask my table members how they started the problem, and from there get inspired to solve it in a way that made sense to me.
Generalize: I not only understand how to manipulate numbers, but I understand the structures of the various forms as well and what the variables represent. This allows me to use the forms effectively in all types of problems.