Introduction to Quadratics
The main objective of this project is to understand quadratic equations. This was introduced to us by Dr. Drew through a series of problems including different forms: vertex, factored and standard. We learned how to apply quadratic equations to geometric problems and we learned how to solve real world problems.
Exploring the Vertex Form of the Quadratic Equation
Through the worksheets handed out to us in class we used a helpful website called Desmos. In class and on the websites we explored how the equation a(x-h)+k effected the location of the shape and the parabola. All quadratic equations form a parabola. The vertex form is a(x-h)+k. The leading coefficient of the vertex form is a. This determines how wide or narrow the parabola will be. If it is negative it will face down if it is positive it will face up.The vertex is represented by (h,k) which is used in the vertex form. The coefficient h represents where on the x axis the vertex will fall and k represent where on the y axis the vertex will fall.
Other Forms of the Quadratic Equation
The other two forms are standard and factored. The the factored form which isy=a(x-r)(x-s)this helps you because it helps you find the x-intercepts and it also gives you and tells you where on the x-axis the parabola crosses. The standard form which is (y=ax^2+bx+c) that is very useful because the variable c represents the y-intercept. The y-intercept is where on the y-axis the parabola crosses.
Converting between forms
The first thing you need to do in solving this problem is squaring (x+5). You can write this out like this y=(x+5)(x+5)+1 next you read the problem and find 2x so you square the x2 and add and multiply 4 and then you have an equation that looks like that y=(x2+10x+7)+1. Now you need to find like terms or subtract 1 from 10 now your new final equation is y=x2+7x+9.
Solving Problems with Quadratic Equations
Kinematics
One example of a kinematic problem is Another Rocket. This measures kinematics because it uses quadratics to answer questions related to motion for this specific problem we converted standard form to vertex form to get the max height of the rocket.
Geometry
One example of a geometric problem is Leslie's Flowers. This uses pythagorean theorem and quadratics to determine the height of the triangle. We did this by setting the two equations equal to one and other and solving it. Also the Emergency at Sea is a real world geometric problem. We needed to determine the altitude of the triangle formed by the boat and the towers. We set two quadratic equations equal to one another in order to find the altitude.
Economics
Profiting From Widgets is a world economic problem. We found the max profit by setting a quadratic equations equal to one another.
One example of a kinematic problem is Another Rocket. This measures kinematics because it uses quadratics to answer questions related to motion for this specific problem we converted standard form to vertex form to get the max height of the rocket.
Geometry
One example of a geometric problem is Leslie's Flowers. This uses pythagorean theorem and quadratics to determine the height of the triangle. We did this by setting the two equations equal to one and other and solving it. Also the Emergency at Sea is a real world geometric problem. We needed to determine the altitude of the triangle formed by the boat and the towers. We set two quadratic equations equal to one another in order to find the altitude.
Economics
Profiting From Widgets is a world economic problem. We found the max profit by setting a quadratic equations equal to one another.
What I learned in during this project was the actual quadratics, standard and vertex form problems and with help from my classmates , I could know how to fully dissect the equations down to the basics so that I can start to grow in every small part. And when I would understand all of the small steps in the equation and I would finally understand solving the whole equation and having a correct answer. This project has impacted my thought process of math because I do feel better in the way that I do the problems that are provided and given to me for whatever math given to me. Now that I know how to actually do the problems and with the habits of Mathematician of start small and finding patterns while doing all the problems. So this was an educational project and I learned a ton and this will help me with what had in store for me in the future.