Part One: An Introduction
In this project, we learned much about Quadratics. Quadratics, according to the Merriam-Webster dictionary, means "involving terms of the second degree at most." Therefore, most quadratic equations involve squares and square roots. Over the course of this project, we learned about Quadratics to a very large degree. From all the various forms of a quadratic equation to how to apply them over many different forms of problems, I am proud of what I have learned, and I am ready to go over this in finer detail.
In this project, we learned much about Quadratics. Quadratics, according to the Merriam-Webster dictionary, means "involving terms of the second degree at most." Therefore, most quadratic equations involve squares and square roots. Over the course of this project, we learned about Quadratics to a very large degree. From all the various forms of a quadratic equation to how to apply them over many different forms of problems, I am proud of what I have learned, and I am ready to go over this in finer detail.
Part Two: Exploring the Vertex Form of the Quadratic Equation.
Before we begin, I would like to go into detail about one of the most important subjects discussed in these last few months: Vertex Form. Vertex form is as follows.
Before we begin, I would like to go into detail about one of the most important subjects discussed in these last few months: Vertex Form. Vertex form is as follows.
We will now begin an in-detail dissection of Vertex Form.
Y is the value of any point on the parabola on the "Y" value. X is the value of any point on the parabola on the "X" value. This is fluid, and has infinite potential values for each position on the parabola. The following screenshot of a graphing calculator will demonstrate possible locations on a parabola.
Y is the value of any point on the parabola on the "Y" value. X is the value of any point on the parabola on the "X" value. This is fluid, and has infinite potential values for each position on the parabola. The following screenshot of a graphing calculator will demonstrate possible locations on a parabola.
"A" determines the shape of the parabola that is generated by the equation. If the number representing “A” is larger, the parabola will be much “thinner”. If the number is smaller, the parabola will be much more “wide”. If the number is a positive number, the parabola will face “open” upwards. If the number is a negative number, the parabola will face "open" downwards.
"H" represents the location of the vertex on the X axis. It is very important to note that this is inverted: if it is -X, we result with +X. On the other hand, "K" represents the location of the vertex on the Y axis.
This can be seen on the screenshot of a graphing calculator below.
This can be seen on the screenshot of a graphing calculator below.
Part Three: Other Forms of the Quadratic Equation
The other major form for graphing quadratic equations is Standard Form. Standard Form proceeds as following.
The other major form for graphing quadratic equations is Standard Form. Standard Form proceeds as following.
The major benefit of standard form is that it can be converted to any other form. Also, I will mention that you can simply plug it into a graphing calculator for the same effect as any other form. Therefore, personally, Standard Form is amongst my favorites due to my constant close proximity to a graphing calculator.
Another form that we were taught is factored form.
Another form that we were taught is factored form.
The major benefit of factored form is that it includes the locations of the X intercepts in the form. X^1 and X^2 both represent X intercepts.
Part Four: Converting Between Forms
Part Five: Examples
In order to begin, we would need to make an equation in standard form.
H(T)=-16t^2+335t+2
From here, we need to convert it into vertex form. If you follow the procedure I laid out earlier, this results in...
-16(t-10.46875)2+1755.515625
We now know the following. The vertex, also known as the peak of the rocket's flight, locates itself at (10.46875, 1755.516). We now know that the maximum height of the rocket is 1755.515625 meters. We also know it took about 10.46875 seconds to reach this point. We now know two of the three problems we set out to find. All that remains is the third: How long is the rocket in the air?
This is a problem that is remedied extremely easily. We know that at the highest point, and exactly half way through the flight of the rocket, the rocket has flown for 10.46875 seconds. Therefore, if we multiply it by two, we can find out where it is at the end of its flight.
10.46875 * 2 = 20.9375
And there we have it: the rocket flew for 20.9375 seconds. That is the last of the problems that we set out to solve.
H(T)=-16t^2+335t+2
From here, we need to convert it into vertex form. If you follow the procedure I laid out earlier, this results in...
-16(t-10.46875)2+1755.515625
We now know the following. The vertex, also known as the peak of the rocket's flight, locates itself at (10.46875, 1755.516). We now know that the maximum height of the rocket is 1755.515625 meters. We also know it took about 10.46875 seconds to reach this point. We now know two of the three problems we set out to find. All that remains is the third: How long is the rocket in the air?
This is a problem that is remedied extremely easily. We know that at the highest point, and exactly half way through the flight of the rocket, the rocket has flown for 10.46875 seconds. Therefore, if we multiply it by two, we can find out where it is at the end of its flight.
10.46875 * 2 = 20.9375
And there we have it: the rocket flew for 20.9375 seconds. That is the last of the problems that we set out to solve.
In conclusion, I have learned much about quadratics this year. The knowledge i have accumulated has allowed me to develop skills in kinematics, geometry, economics, and more. At times I may have struggled, but I am now confident as a student in my ability to tackle problems with the use of quadratics.