Introduction to Quadratics
To begin this project, we were given problems involving distance, velocity, and acceleration. We learned the steps to creating equations with these factors. This paved the way for a new problem, which was asking us to find the formula for various questions with quadratics. We had to find the height of a rocket using a given formula. Little did we know, we were going to be continuously asked these questions throughout the year. We began this problem with a given function of time.
The height of a rocket by a function of time is: h(t) = d0 + v0 · t + 1/2a · t^2
We used this function to solve for a new one.
That being: h(t) = 160 + 92t - 16t^2
Over the the rest of our year in math, we were unaware that we would revisit this problem. After the completion of this problem, we focused specifically on quadratic equations. From there, we learned more advanced ways to create and solve quadratics using parabolas and equations.
The height of a rocket by a function of time is: h(t) = d0 + v0 · t + 1/2a · t^2
We used this function to solve for a new one.
That being: h(t) = 160 + 92t - 16t^2
Over the the rest of our year in math, we were unaware that we would revisit this problem. After the completion of this problem, we focused specifically on quadratic equations. From there, we learned more advanced ways to create and solve quadratics using parabolas and equations.
Exploring Vertex Form of the Quadratic Equation
We were first introduced to standard form. We did this by learning what a parabola is and how to create one. We began this introduction by using a computer program called ‘Desmos’ and explored how to create a parabola.
We began with the simplest given equation: y = x^2
The needed variables to be able to create and understand parabolas were a, h, and k.
From there, we moved to more advanced equations: y = a (x - h)^2
Using parabolas, we began to focus primarily on vertex form. We were given a parabola and we were asked to find the equation used to make it (Vertex Form)
Vertex form is the equation used to create a parabola. To better understand this topic, we created posters reflecting our knowledge. In groups, we made three collectively, each being a different variable.
We began with the simplest given equation: y = x^2
The needed variables to be able to create and understand parabolas were a, h, and k.
From there, we moved to more advanced equations: y = a (x - h)^2
Using parabolas, we began to focus primarily on vertex form. We were given a parabola and we were asked to find the equation used to make it (Vertex Form)
Vertex form is the equation used to create a parabola. To better understand this topic, we created posters reflecting our knowledge. In groups, we made three collectively, each being a different variable.
The first poster explained what a does to a parabola. a determines how narrow or wide the parabola is.
The second poster explained what k does to the parabola. k is equal to the y-coordinate value on the vertex. Which determines the path it follows up and down.
The third and final poster explained what h does to the parabola. h is equal to the x-coordinate value on the vertex. Which determines the path it follows right and left.
Other Forms of the Quadratic Equation
We moved on from the traditional vertex form to other forms of a quadratic equation. Those included, standard form and factored form.
Standard form: is very similar to vertex form, despite the final equation. Meaning that the variables, a,b, and c act as the a,h, and k.
A standard form equation looks like: Y=Ax^2+Bx+C
It is obviously very different than the first form we learned, which was vertex form. This equation shows the y-intercept value of c.
Factored Form: There are variables included in this form of a quadratic equation, as well. The two variables are p and q.
A factored form equation looks like: y=a(x-p)(x-q)
In this equation, p and q are x-intercept values.
We can be sure that a variable is an x-intercept value because the corresponding y-value is zero.
You can find factored form from vertex form by finding the center point in between the x-intercepts and use that to find a value for y.
Standard form: is very similar to vertex form, despite the final equation. Meaning that the variables, a,b, and c act as the a,h, and k.
A standard form equation looks like: Y=Ax^2+Bx+C
It is obviously very different than the first form we learned, which was vertex form. This equation shows the y-intercept value of c.
Factored Form: There are variables included in this form of a quadratic equation, as well. The two variables are p and q.
A factored form equation looks like: y=a(x-p)(x-q)
In this equation, p and q are x-intercept values.
We can be sure that a variable is an x-intercept value because the corresponding y-value is zero.
You can find factored form from vertex form by finding the center point in between the x-intercepts and use that to find a value for y.
Converting Between Forms
In order to convert between forms, you must use an area diagram.
Vertex to Standard
You must first begin with your equation in vertex form, (y=a(x-h)^2-k)
From there, you multiply the squared terms, distribute the variable a, and combine like terms.
standard form to vertex form
You first take out and combine the ax^2 and bx together. Using an area diagram, you must fill in the missing terms. After this, you take the subtracted term out of the parentheses by multiplying it by the variable a. To finish, you write rewrite the parentheses and combine the like terms.
factored form to standard form
You must combine the like terms using parenthesis. To do this, you multiply the the terms within the parenthesis, combine like terms, and distribute a.
Standard Form to Factored form
To complete this conversion, you must work backwards. If you have the variables a and b, if you add these together, they equal c (Constant). You multiply these by the values that equal the constant. After this, you factor the constant out.
Area Diagram
An area diagram is beneficial to understand distribution of multiplication over addition, completing the square, and factoring a quadratic. It allows you to visualize the equation. Doing this, helps people that are visual learners. It guided me to be able to better understand the problem in order to solve it.
Vertex to Standard
You must first begin with your equation in vertex form, (y=a(x-h)^2-k)
From there, you multiply the squared terms, distribute the variable a, and combine like terms.
standard form to vertex form
You first take out and combine the ax^2 and bx together. Using an area diagram, you must fill in the missing terms. After this, you take the subtracted term out of the parentheses by multiplying it by the variable a. To finish, you write rewrite the parentheses and combine the like terms.
factored form to standard form
You must combine the like terms using parenthesis. To do this, you multiply the the terms within the parenthesis, combine like terms, and distribute a.
Standard Form to Factored form
To complete this conversion, you must work backwards. If you have the variables a and b, if you add these together, they equal c (Constant). You multiply these by the values that equal the constant. After this, you factor the constant out.
Area Diagram
An area diagram is beneficial to understand distribution of multiplication over addition, completing the square, and factoring a quadratic. It allows you to visualize the equation. Doing this, helps people that are visual learners. It guided me to be able to better understand the problem in order to solve it.
Solving Problems With Quadratic Equations
Due to this quadratics project, there are three types of problems we can now solve.
Kinematics
We used kinematics the most during our victory celebration we had been working on throughout the year. This rocket project involved motion as it was asking us to figure out the rocket’s height. Like I have previously stated, we began with the equation h(t)=-16t^2+92t+160.
We converted our equation from standard form to vertex form to find the vertex.
We simplified the equation by first attempting to find the coordinates of the vertex.
If we convert this equation from vertex form to factored form it becomes: h(t)=-16(t-2.875)^2+292.25.
The final maximum height of the rocket: 292.25
The amount of time it took to reach the maximum height: 2.875
The x-intercept: 7.14883
Kinematics
We used kinematics the most during our victory celebration we had been working on throughout the year. This rocket project involved motion as it was asking us to figure out the rocket’s height. Like I have previously stated, we began with the equation h(t)=-16t^2+92t+160.
We converted our equation from standard form to vertex form to find the vertex.
We simplified the equation by first attempting to find the coordinates of the vertex.
If we convert this equation from vertex form to factored form it becomes: h(t)=-16(t-2.875)^2+292.25.
The final maximum height of the rocket: 292.25
The amount of time it took to reach the maximum height: 2.875
The x-intercept: 7.14883
Geometry
We used geometry the most during the Leslies Flowers Problem. The problem asked us to help a gardener with the shape of her planting box. She was using triangular planter boxes, which allowed us to practice Pythagorean theorem.
We used geometry the most during the Leslies Flowers Problem. The problem asked us to help a gardener with the shape of her planting box. She was using triangular planter boxes, which allowed us to practice Pythagorean theorem.
Economics
We used economics the most during the Widgets problem. In this problem, we were asked to find for the sales of a company. We were given various problems in order to find how much profit they would gain.
We used economics the most during the Widgets problem. In this problem, we were asked to find for the sales of a company. We were given various problems in order to find how much profit they would gain.
Reflection
I found this project to be very challenging. To begin, the first worksheet given was very hard for me to complete as I did not fully understand the project. The parabolas using desmos was not particularly challenging, but I did struggle with it a bit. As the following assignments increased in difficulty I found myself falling behind. Quadratics was a topic that took me a very long time to grasp and fully understand. The most difficult part of the project for me was attempting to understand the different forms of a quadratic equation. This project was a bit of a preview for 11th grade math. It was a difficult problem, which may be what I will receive next year. Especially if I choose to take honors math, which I know as a fact is very advanced. This problem definitely prepared me for the SAT and beyond. As I previously stated, it is an advanced problem and I was not used to it. After finishing the project, however, I feel prepared and ready for future tests and learning.
During this project, we used all of the Habits of Mathematician:
Look for patterns: I had to use this habit periodically as various problems were extremely tedious. I used this to attempt to make the problems simpler and easier for me to understand.
Start Small: I used this habit when exploring Desmos. Before going into all of the equations given, I attempted to get to know the program before and begin with small equations before moving on to the packet.
Be Systematic: I used this habit when doing the corral variation problems. This habit was making small changes to look for change and permanence. This problem was extremely difficult for me, but using this habit helped me to better understand and complete the problem.
Take apart and Put back together: I used this habit the most when we were focusing on parabolas. When using Desmos, I had to look for repetition in order to find correct vertex form.
Conjecture and Test: I used this habit the most when we were given the handout, ‘Square it’. This habit is the process of asking, “What if?” I used this to attempt to guess what the answer would and checking my answer.
Stay Organized: I used this habit throughout the entire project, as I didn’t want to get confused with all of my paperwork. I had to stay on top of my work and routinely check back to be prepared for the next handout.
Describe and Articulate: This habit is the process of drawing visuals (Graphs, art pieces, etc.) I used this when we were focusing on parabolas. I drew out many graphs and area diagrams for this entire project.
Seek Why and Prove: I used this habit the most during the rocket project. I asked many questions as to why the answers were correct. This project was very difficult to for me and doing this helped me a lot.
Be Confident, Patient, and Persistent: I used this habit throughout the course of this project. Despite the difficulty of this project, I continued to thrive and have confidence in my work. I never faltered, even when I did not get answers I disagreed with.
Collaborate and Listen: This was the habit I used the most during this project as I needed lots of aid with my work. It was quite difficult and having my peers there to help me along was very beneficial to me.
Generalize: I used this habit the most when we were beginning to learn vertex form. I supposed many answers and checked to make sure my answers were actually correct.
During this project, we used all of the Habits of Mathematician:
Look for patterns: I had to use this habit periodically as various problems were extremely tedious. I used this to attempt to make the problems simpler and easier for me to understand.
Start Small: I used this habit when exploring Desmos. Before going into all of the equations given, I attempted to get to know the program before and begin with small equations before moving on to the packet.
Be Systematic: I used this habit when doing the corral variation problems. This habit was making small changes to look for change and permanence. This problem was extremely difficult for me, but using this habit helped me to better understand and complete the problem.
Take apart and Put back together: I used this habit the most when we were focusing on parabolas. When using Desmos, I had to look for repetition in order to find correct vertex form.
Conjecture and Test: I used this habit the most when we were given the handout, ‘Square it’. This habit is the process of asking, “What if?” I used this to attempt to guess what the answer would and checking my answer.
Stay Organized: I used this habit throughout the entire project, as I didn’t want to get confused with all of my paperwork. I had to stay on top of my work and routinely check back to be prepared for the next handout.
Describe and Articulate: This habit is the process of drawing visuals (Graphs, art pieces, etc.) I used this when we were focusing on parabolas. I drew out many graphs and area diagrams for this entire project.
Seek Why and Prove: I used this habit the most during the rocket project. I asked many questions as to why the answers were correct. This project was very difficult to for me and doing this helped me a lot.
Be Confident, Patient, and Persistent: I used this habit throughout the course of this project. Despite the difficulty of this project, I continued to thrive and have confidence in my work. I never faltered, even when I did not get answers I disagreed with.
Collaborate and Listen: This was the habit I used the most during this project as I needed lots of aid with my work. It was quite difficult and having my peers there to help me along was very beneficial to me.
Generalize: I used this habit the most when we were beginning to learn vertex form. I supposed many answers and checked to make sure my answers were actually correct.