The Quintessence of Quadratics
Introduction
In this project we used algebra and geometry to learn to solve a quadratic equation. We used standard form, vertex form and graphing equations. We started learning how to create every single type of form, then we practiced solving them by having an example from another, for example be able to convert an equation from vertex form to standard form.
Exploring the Vertex Form of the Quadratic Equation
We received handouts and we had to solve them by using Desmos to explore different letter that have an important meaning in a quadratic equation y= x². For example a < 0 the parabola is concave down, it's below the x-axis and the 0-intercept. in the graph a<1 it opens down, negative, and if is larger is closer to the 0 and if is smaller then it far away from 0.
The equation y=x² + k is another family of graphs, where k is a particular number. For example k>0 the vertex is above x-axis and k<0 vertex is below the x-axis. The k is the y-coordinate of the vertex.
The equation y=(x-h)² is a graph where h is a special number. h>0 is the vertex to the right of the y-axis and h<0 is the vertex to the left of y-axis. h is the x-coordinate of the vertex.
Other forms of the Quadratic Equation
There is two other forms of the quadratic equation, standard and factored. In the factored form you write the multiplication as the product and in the standard form you at least contain a term that is squared. For example I have y= (x=4)(x+4) in factor form and x²+8x=16 in standard form, both of us have the same parabola but expressed in a different way.
Converting between Forms
In this project of quadratics I put effort to begin trying hard and finish it with effort also. In this project I had little bit of review at the beginning but once I was putting attention in class it could be easier to go home and work in my homework and finish it on time. I love multiplying and working with figures and this project was full of that type of activities, so I really enjoyed working all this type of problems. Last projects I didn't put as much effort as I did this one and it was because I wanted to give myself a chance and accomplish my goal to learn math and not give up. Once I started working with Vertex Form, Standard Form and Factored Form it was like a piece of cake because I love multiplying and working with variables. I am really proud of myself because I accomplished my goal of a math mathematician which was to stay organize and I am happy that I am here utilizing it.
Introduction
In this project we used algebra and geometry to learn to solve a quadratic equation. We used standard form, vertex form and graphing equations. We started learning how to create every single type of form, then we practiced solving them by having an example from another, for example be able to convert an equation from vertex form to standard form.
Exploring the Vertex Form of the Quadratic Equation
We received handouts and we had to solve them by using Desmos to explore different letter that have an important meaning in a quadratic equation y= x². For example a < 0 the parabola is concave down, it's below the x-axis and the 0-intercept. in the graph a<1 it opens down, negative, and if is larger is closer to the 0 and if is smaller then it far away from 0.
The equation y=x² + k is another family of graphs, where k is a particular number. For example k>0 the vertex is above x-axis and k<0 vertex is below the x-axis. The k is the y-coordinate of the vertex.
The equation y=(x-h)² is a graph where h is a special number. h>0 is the vertex to the right of the y-axis and h<0 is the vertex to the left of y-axis. h is the x-coordinate of the vertex.
Other forms of the Quadratic Equation
There is two other forms of the quadratic equation, standard and factored. In the factored form you write the multiplication as the product and in the standard form you at least contain a term that is squared. For example I have y= (x=4)(x+4) in factor form and x²+8x=16 in standard form, both of us have the same parabola but expressed in a different way.
Converting between Forms
- From Vertex to Standard Form: you have an equation that is in vertex form y= a(x-h)² +k to standard form y= ax² + bx+ c. You solve the vertex form equation by multiplying, using factors, or creating a chart that is made out of four spots, and you place the axis in the top and in the side. When you have them in the right place you multiply the number to other from the side to the top and place your answer inside the square.
- Standard Form to Vertex Form: To convert a quadratic from y= ax²+ bx+ c form to vertex form, y= a(x-h)² + k, you use the process of completing the square. You create a square that has four spaces where you place your number, putting each one of the boxes and once you have them finish the numbers in the top are going to become your solution.
- Convert Factored Form to Standard Form: To convert from factored to standard from, multiply the factors. If the factored is in the form ax(x+d) distribute. How you changed is setting up your equation and multiplying or distributing.
- Convert Standard Form to Factored Form: To convert from standard form y= ax+bx=c to factored form y=a(x-p)(x-q). First multiply a(x-p) and then multiply the product by (x-q).
- Kinematics: the branch of mechanics that deals with pure motion, without reference to the masses or forces involved in it. We worked in the project trying to figure out the distance that rocket travels in a certain time. For example "Another Rocket"
- Geometry: treats the properties, measurement, and relations of points, lines, angles, surfaces, and solids. How we worked with Geometry in this class was by using triangles and rectangle problems. For example "Leslie's Flowers" we used the triangle and solved the problem using quadratic equations.
- Economics: An advantage claimed for the approach is its allowing formulation of theoretical relationships with rigor, generality, and simplicity. How we worked with economics in this class was using graphs and use quadratic equations. For example in the SAT practice #25 packet we used an example of economics.
- The problem that I choose to give an example of all the work that I did through the year is "Emergency Sea". In this problem we had to solve show that where the boat reached the shore wasn't 400 meter from the tower. In this problem I had to use the quadratic equation to show work that 400 wasn't the answer and It took me a while but at the end I could understand.
In this project of quadratics I put effort to begin trying hard and finish it with effort also. In this project I had little bit of review at the beginning but once I was putting attention in class it could be easier to go home and work in my homework and finish it on time. I love multiplying and working with figures and this project was full of that type of activities, so I really enjoyed working all this type of problems. Last projects I didn't put as much effort as I did this one and it was because I wanted to give myself a chance and accomplish my goal to learn math and not give up. Once I started working with Vertex Form, Standard Form and Factored Form it was like a piece of cake because I love multiplying and working with variables. I am really proud of myself because I accomplished my goal of a math mathematician which was to stay organize and I am happy that I am here utilizing it.