I.Introduction
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.
II. Exploring the Vertex Form of the Quadratic Equation
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.
III. Other Forms of the Quadratic Equation
The other two forms are standard and factored. The the factored form which is y=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.
IV. Converting between forms
The first thing you need to do in solving this problem is squaring (x+5). When you look at this problem you know 5 is being multiplied by 2 because its squared 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.
V. 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.
(pic at the bottom of the page)
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.
(pic at the bottom of the page)
Economics
Profiting From Widgets is a world economic problem. We found the max profit by setting a quadratic equations equal to one another.
VI. Reflection
What I learned in during the duration of this project was the in’s and out’s of quadratic equations, and how to fully dissect the equations down to the basics so that I can start to excel in every small part. And when I would understand all of the small steps in the equation I would finally be 100% confident in solving the whole equation and having a confident answer. This project has impacted my thoughts for the 11th grade because I do feel more confident in the way that I approach the problems that are provided and given to me for next year. Now that I know how to dissect the problems and start small with larger equations I do feel way more confident going into the 11th grade because I feel like I can tackle and solve the problems a lot easier now. And for the SAT and college readiness I feel more faster and more efficient in solving the problems faster because of all the practice that I have gotten during this project.
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.
II. Exploring the Vertex Form of the Quadratic Equation
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.
III. Other Forms of the Quadratic Equation
The other two forms are standard and factored. The the factored form which is y=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.
IV. Converting between forms
The first thing you need to do in solving this problem is squaring (x+5). When you look at this problem you know 5 is being multiplied by 2 because its squared 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.
V. 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.
(pic at the bottom of the page)
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.
(pic at the bottom of the page)
Economics
Profiting From Widgets is a world economic problem. We found the max profit by setting a quadratic equations equal to one another.
VI. Reflection
What I learned in during the duration of this project was the in’s and out’s of quadratic equations, and how to fully dissect the equations down to the basics so that I can start to excel in every small part. And when I would understand all of the small steps in the equation I would finally be 100% confident in solving the whole equation and having a confident answer. This project has impacted my thoughts for the 11th grade because I do feel more confident in the way that I approach the problems that are provided and given to me for next year. Now that I know how to dissect the problems and start small with larger equations I do feel way more confident going into the 11th grade because I feel like I can tackle and solve the problems a lot easier now. And for the SAT and college readiness I feel more faster and more efficient in solving the problems faster because of all the practice that I have gotten during this project.
Kinematics ^
Converting between forms
Geometry
Exploring the Vertex Form of the Quadratic Equation
Other Forms of the Quadratic Equation