Quadratics
I. Introduction: Project Overview
In this project we learned how to solve and use Quadratic Equations, specifically using the Vertex Form. Throughout the entire learning process we learned about all the elements that went into a Quadratic equation. It began with a series of handouts that helped us first understand the basics, such as what a parabola is, and later went into how to rearrange and manipulate equations. Our whole objective of the project was to keep advancing in our knowledge of what a Quadratic is.
We began by studying parabolas in our worksheet packets to fully understand what Quadratic Equations look like on a graph. With this type of introduction to Quadratics we proceeded by learning about how parabolas on the graph can change form in terms of width, vertex point, and whether it is concave up or down. Parabolas change form when different values are plugged in for the Vertex Form of the Quadratic Equations, which I will go more in depth on in a different section.
After we were introduced to what the Vertex Form of a Quadratic Equation is, we were given another set of worksheet packets that established our understanding of creating a Standard Form Quadratic Equation and a Vertex Form Quadratic Equation. Throughout the worksheets of that packet we were able to begin learning about the different aspects within each type of Quadratic equation and by using the Completing the Square Method (which will be discussed in a different section) we could convert between Standard and Vertex Quadratic Equations.
Near the end of that packet we began applying our knowledge of the different forms of Quadratic Equations and the Completing the Square Method to real world problems, as well as fully practicing our skills of Completing the Square. Such as in picturing different objects in the real world and putting them on a coordinate plane to understand how to use equations to figure out the different minimums and maximums of the objects.
Near the end of our homework assignments and packets, we had already established our knowledge of using the Completing the Square Method and how to use Vertex Form, we then finally applied those methods to more math problems, some specifically involving the triangles and its sides. Along with that we learned about how to find Minimums and Maximums in Vertex Form Quadratics Equations, as well as understanding how the values plugged into a Vertex Form Equation correlate with the value of the Vertex.
In this project we learned how to solve and use Quadratic Equations, specifically using the Vertex Form. Throughout the entire learning process we learned about all the elements that went into a Quadratic equation. It began with a series of handouts that helped us first understand the basics, such as what a parabola is, and later went into how to rearrange and manipulate equations. Our whole objective of the project was to keep advancing in our knowledge of what a Quadratic is.
We began by studying parabolas in our worksheet packets to fully understand what Quadratic Equations look like on a graph. With this type of introduction to Quadratics we proceeded by learning about how parabolas on the graph can change form in terms of width, vertex point, and whether it is concave up or down. Parabolas change form when different values are plugged in for the Vertex Form of the Quadratic Equations, which I will go more in depth on in a different section.
After we were introduced to what the Vertex Form of a Quadratic Equation is, we were given another set of worksheet packets that established our understanding of creating a Standard Form Quadratic Equation and a Vertex Form Quadratic Equation. Throughout the worksheets of that packet we were able to begin learning about the different aspects within each type of Quadratic equation and by using the Completing the Square Method (which will be discussed in a different section) we could convert between Standard and Vertex Quadratic Equations.
Near the end of that packet we began applying our knowledge of the different forms of Quadratic Equations and the Completing the Square Method to real world problems, as well as fully practicing our skills of Completing the Square. Such as in picturing different objects in the real world and putting them on a coordinate plane to understand how to use equations to figure out the different minimums and maximums of the objects.
Near the end of our homework assignments and packets, we had already established our knowledge of using the Completing the Square Method and how to use Vertex Form, we then finally applied those methods to more math problems, some specifically involving the triangles and its sides. Along with that we learned about how to find Minimums and Maximums in Vertex Form Quadratics Equations, as well as understanding how the values plugged into a Vertex Form Equation correlate with the value of the Vertex.
II. Exploring the Vertex Form of the Quadratic Equation
We first began by going through the basics of understanding a Quadratic Equation on a graph and the core of the equation. In one of our earlier handouts, we went over how Quadratic Equations come out on graphs and how they are different from linear equations. Those earlier handouts forced us to ask ourselves questions about what Quadratic Equations looked like on graphs.
We did find out that the form that a Quadratic would make on a graph would be a parabola. As shown below--
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A quadratic equation would create a figure, known as a parabola, such as the one above. After learning how a Quadratic looks on a graph, we practiced our skills in creating equations where the parabola would change form, in further worksheet packets. Throughout that learning process of adjusting parabolas, we were able to identify how to form a quadratic equation to alter the parabola on the graph. It was then that we were taught about the Vertex Form of a Quadratic Equation.
At this stage in our learning, because of the changes in parabolas’ forms, we were introduced to the Vertex Form:
At this stage in our learning, because of the changes in parabolas’ forms, we were introduced to the Vertex Form:
Y = A(X-H)^2 + K
What we learned is by plugging in different values for A, H, and K you are able to adjust the parabola to:
By plugging in different specific values for the A, H, and K in Y = A(X - H)^2 + K you are able achieve any of the shown aspects in a parabola, shown above.
- become wider or narrower
- become concave up or down
- go below or above the x-axis
- go to the right or left of the y-axis
By plugging in different specific values for the A, H, and K in Y = A(X - H)^2 + K you are able achieve any of the shown aspects in a parabola, shown above.
Y = A(X - H)^2 + K
In order to create a wide parabola you need to have a significantly low value of A (the value X is being multiplied by) in the Vertex Form of the Quadratic Equation. A value lower than 1.
In order to create a narrow parabola you need to have a higher value of A in the equation. A value higher than 1.
For example, in the graph shown below from desmos, the Vertex Form Equation with the higher value of A has a narrower parabola, while the wider parabola has a lower value of A.
In order to create a narrow parabola you need to have a higher value of A in the equation. A value higher than 1.
For example, in the graph shown below from desmos, the Vertex Form Equation with the higher value of A has a narrower parabola, while the wider parabola has a lower value of A.
The decision of whether a parabola is concave up or down, is based on whether the value of A in the Vertex form (Y = A(X - H)^2 + K) is negative or positive number. If it is negative it is concave down, if positive it is concave up.
As you can see in the graph below, the parabola concave up is positive, while the parabola concave down is negative.
The next aspect of a parabola we are going to inspect is whether it is placed below or above the x-axis. The letter that determines this in the equation, Y = A(X - H)^2 + K, is K. If the value of K is zero, the parabola's vertex will stay on the x-axis. If the value of K is a negative number the parabola's vertex will be below the x-axis. If the value of K is a positive number, the parabola's vertex will be above the x-axis.
In the graph below, you will see the specific placement of the vertex demonstrated by what it's equation is.
The final aspect of a parabola, that the equation Y = A(X - H)^2 + K can control, is whether the parabola's vertex is to the right or left of the y-axis. The letter in the Vertex Form of the Quadratic Equation that determines if it is on the right or left side of the y-axis is the value of H.
If you plug in a positive number for H, the vertex of the parabola will be to the right of the y-axis. If you plug in a negative number for H, the vertex of the parabola will be to the left of the y-axis. As you will see in the graph below.
Note: Even though the numbers plugged in for H appear to be the opposite of what their value is, it is only because of the negative sign within the equation. The value of the number plugged in for H will retain it's value.
III. Other Forms of the Quadratic Equation
Besides Vertex Form, which was our main focus during this project, we were introduced to Standard Form and Factored Form. Standard Form is the most simplified version of a quadratic, while Factored Form is a Quadratic Equation in a FOIL structure.
For example, a Quadratic Equation in Standard Form is seen as:
ax^2 + bx + c
(Values would be plugged in for a, b, and c)
Factored Form is seen as:
a(x - p)(x - q)
(Values of the Equation would be plugged in for a, p, and q)
(Values would be plugged in for a, b, and c)
Factored Form is seen as:
a(x - p)(x - q)
(Values of the Equation would be plugged in for a, p, and q)
The advantages of using Standard Form and seeing the equation in it's most simplified form is you are able to see the exact values that are going into the equation. It may not be the easiest equation to solve but at least you know what's going into the equation.
The advantages of using Factored Form and seeing it in a Foil Structure is that you are able to find out the roots of the function (the x-intercepts) just by looking at the values of P and Q within the equation.
The advantages of using Factored Form and seeing it in a Foil Structure is that you are able to find out the roots of the function (the x-intercepts) just by looking at the values of P and Q within the equation.
IV. Converting Between Forms
In order to convert between different forms of Quadratic Equations, one must follow specific steps to achieve each form.
Vertex Form to Standard Form
Y = A(X - H)^2 + K → Y = ax^2 + bx + c
In order to convert from Vertex Form to Standard Form, one must simply simplify to reach the most simplified form.
For example: Here is a Vertex Form Equation being converted to Standard Form.
Y = A(X - H)^2 + K → Y = ax^2 + bx + c
In order to convert from Vertex Form to Standard Form, one must simply simplify to reach the most simplified form.
For example: Here is a Vertex Form Equation being converted to Standard Form.
Simply by using the FOIL method to obtain the completely reduced and simplified answer.
Standard Form to Vertex Form
Y = ax^2 + bx + c → Y = A(X - H)^2 + K
Converting from Standard Form to Vertex Form is a bit different than doing it the other way around.
When doing this, you begin by working with the variables with an X attached to them.
Step 1: Creating the (x-h) by isolating the X^2 and the X from X^2's coefficient, and
after doing so, using the Completing the Square Method to find the (x-h) section of the Vertex Form.
Step 2: Set up the Vertex For of the problem by writing up the value of A with the value of (x-h.)
Step 3: The leftover number in the bottom right-hand box of the Square, from the
Completing the Square Method, should be subtracted from the value of K in the Standard
Form Equation to create the Value of K in the Vertex Form Equation.
For example: (Standard Form Equation being Converted to Vertex Form)
Factored Form to Standard Form
a(x - p)(x - q) → ax^2 + bx + c
This is a fairly simple conversion, in order to solve a Factored Form equation to make a Standard Form equation you use the FOIL Method to achieve the most simplified version of a Quadratic equation
For example:
Standard Form to Factored Form
ax^2 + bx + c → a(x - p)(x - q)
If you were to convert from Standard Form to Factored Form you would need to first begin by taking your Standard (Most Simplified) Form and factoring it into a FOIL Structure of the equation.
ax^2 + bx + c
When Factoring it, the Equation Will Become
a(x - p)(x - q)
When Factoring it, the Equation Will Become
a(x - p)(x - q)
For example:
If you have a Standard Equation such as
4x^2 + 4x + 1
You Factor it by Using the Completing the Square Method Backwards
Step 1: First fill out the square with the values given.
Step 2: Find out what the Factored Form would be by looking at the sidesof the square, which
will reveal the numbers. (This part is circled in the image of the square below)
Your Answer is(according to the sides of the square): (2x + 1)(2x + 1)
4x^2 + 4x + 1
You Factor it by Using the Completing the Square Method Backwards
Step 1: First fill out the square with the values given.
Step 2: Find out what the Factored Form would be by looking at the sidesof the square, which
will reveal the numbers. (This part is circled in the image of the square below)
Your Answer is(according to the sides of the square): (2x + 1)(2x + 1)
After learning about all of these conversions I realized how helpful it is to use diagrams for something such as the Completing the Square Method. When you are able to see the equation's values spread out on a graph you are better equipped to understand how to form new equations.
V. Solving problems with Quadratic Equations
There are three different types of real world problems that you are able to solve using Quadratics. Those problems involve Kinematics, Geometry, and Economics.
A problem we were given in the packets was one called Leslie's Flowers which involves geometry. In the problem we were given the task of finding out the length of half of one of the sides of Leslie's triangular flowerbed. The problem gave us the side lengths of the triangle and with that information, I plugged in the value of the side with the half of a missing side length into the Pythagorean Theorem.
By solving for the value of X I was able to find the length of the half of a missing side length. The problem was also a good first teacher for me, because it taught me how to deal with squared values inside equations. That knowledge at the time would come in handy for future problems to come.
Another problem we were given in our worksheet packets was an Economics problem that evaluated how much you will receive based on the certain values. The problem was called "How Much Can they Drink?" and it gave you the task of figuring out the best way to get the most water out of a drinking trough Based on it's volume.
Another problem we were given in our worksheet packets was an Economics problem that evaluated how much you will receive based on the certain values. The problem was called "How Much Can they Drink?" and it gave you the task of figuring out the best way to get the most water out of a drinking trough Based on it's volume.
It was our task to figure out what the volume was of the water trough and where the two folds should be in order to get certain volumes. We would then create an equation for the water tough, with the X's value equally measuring the width of the trough's middle and the Y's value measuring the total volume of the tough (how much water it can contain.
The final problem I will be going over was our Kinematics problem that we received from one of worksheet problems. The problem was called "Another Rocket" and in the problem we were given an equation that measured how high a rocket would go after a certain number of seconds. The equation was y= -16x^2 + 64x + 80 and we were asked to change it into a Vertex Form Quadratic Equation in order to answer a few questions.
The final problem I will be going over was our Kinematics problem that we received from one of worksheet problems. The problem was called "Another Rocket" and in the problem we were given an equation that measured how high a rocket would go after a certain number of seconds. The equation was y= -16x^2 + 64x + 80 and we were asked to change it into a Vertex Form Quadratic Equation in order to answer a few questions.
The questions we were asked to answer were:
- How long did it take for the rocket to reach it's Maximum Height?
- What was the rocket's maximum height?
VI. Reflection
For the past month we have been learning about Quadratics and we have gone through a lot of different content in Quadratic Equations. We now know what they look like, what forms they come in, and how to solve for them. Some Habits of a Mathematician I had to utilize this past month was definitely looking for patterns and taking apart and putting back together. Because we were going through so many packets with many worksheets in them, we got a lot of repeated content in order to practice our skills and in those moments we had to look for patterns of information, repetitive content, and being able to take apart to identify content that we had learned before. In almost all cases with the math problems we received, I had to start small and think systematically in order to stay organized with all of the numerical knowledge I was pointing down on paper. After awhile when working on a problem you could get caught up in all the small work and not realize the big question you had to solve. That was an issue that I struggled with a lot, I would get lost in all of the numbers, letters, and equations I was putting down on paper. For the times we had to answer the big question, it took a lot of time doing trial and error after trial and error, when you did find a way to solve you had to really think "why?" am I solving it this way versus another way. Those times took a lot of patience and persistence on my part. However after some collaboration and some visualization on my part I was able to work through it and solve for the answer on all the word problems. I now can say going out of this project, I know a lot about Quadratics and the specific ways to solve for Quadratics.