Introduction
This project was about us learning Quadratics and the steps of solving the equations. There are multiple forms of quadratics. The forms are:
1. Vertex form y= (x-h)^2+k
2. Standard form y=ax^2+bx+c
3. Factored form y= a(x - f )(x - r)
We learned how to create, solve and convert quadratic equations. This was done through a series of worksheets provided by Dr. Drew. We were also able to use quadratic equations and apply them to geometric problems. We did this in the Corral Variation problem. We used the quadratic formula to solve area. quadratics gives us the opportunity to find height volume area we can also use Pythagorus theorem and we can use this in economics to find out the prices and quantic sold. This gives us the chance to try are own methods of how we want to solve these equations. The original question that was given to use to makes us learn about quadratics was about a rocket that would be launching fire works at the peak of the rockets trajectory. This relates to the x-intercept as well as vertices. This gives us the opportunity to strengthen are understanding of algebraic expressions. Throught are own work we had to conjecture and test to see if what we knew was done correctly.
1. Vertex form y= (x-h)^2+k
2. Standard form y=ax^2+bx+c
3. Factored form y= a(x - f )(x - r)
We learned how to create, solve and convert quadratic equations. This was done through a series of worksheets provided by Dr. Drew. We were also able to use quadratic equations and apply them to geometric problems. We did this in the Corral Variation problem. We used the quadratic formula to solve area. quadratics gives us the opportunity to find height volume area we can also use Pythagorus theorem and we can use this in economics to find out the prices and quantic sold. This gives us the chance to try are own methods of how we want to solve these equations. The original question that was given to use to makes us learn about quadratics was about a rocket that would be launching fire works at the peak of the rockets trajectory. This relates to the x-intercept as well as vertices. This gives us the opportunity to strengthen are understanding of algebraic expressions. Throught are own work we had to conjecture and test to see if what we knew was done correctly.
Vertex Form
All quadratic equations form a parabola. In this section I will be talking about Vertex form y= a(x-h)^2+k. The a represents the upward or downward direction of the parabola depending on whether or not the a is positive or negative. If the a value is positive it will open upwards, if the values a is negative it will open downwards. It also represents how wide or narrow a parabola will be the larger the number being hole numbers makes the porabla shreank smaller the number the larger the porabloa ends up making. The h represent where on the x coordinate the vertex, or bottom/top of the parabola, will fall. The k represents where on the y axis the vertex will fall.
Other forms of the Quadratic Equation
The other forms of the quadratic equation are standard form (y=ax^2+bx+c), factored form y= a(x - f )(x - r). You can convert one form the the next form and they will all have the same parabola. The advantage of standard form is that it gives you the opportunity to use any of the equations by making a stable platform. The standard form provides us with the y- intercept(c), or where the parabola crosses the y-axis. Factor form is important because it gives us the x-intercepts, or where the parabola crosses the x-axis. This step really makes me step back and generalize the hole make up of the equations involved with find a parabola and the way that the parabola open and closes.
Converting between Forms
1. Convert from vertex to standard form
Multiply what is in the parenthesis and then add the coefficients. Being able to use Foil can help you solve the equation Foil stands for First, outer, inner, and last. We are able to look at the equation and just move parts of it around by adding, and multiplying variables. 2. Convert Standard to Vertex form Step 1 Divide terms by a Step 2 Move c to the right side of the equation Step 3 Complete the square on the left side of the equation Step 4 Balance this by adding the same value to each side of the equation.This allows us to go through and see what the equation equal to at its reduced stage. 3. Convert factored form to standard form One this equation we are able to just reduce Foil and solve are question. We just distribute straight across and reduce it into stander form from factored form. while working on the part of the equation you can look for patterns int the way the numbers are being multiplied to find the end resolte of the eqaution. 4. Convert Standard to factored form First you need to write out your equation and make a area diagram. Ounce done labe each side with the missing variables and fill in the rest of the are diagram that you know. Then we know what ever the missing numbers are have to multiply to 12 and then we can just look for the missing number. Personally I put mine in factored form and then went from there on reducing it and finding the end of the equation. When looking back on the topics as a whole the where really similer and made me realize that I just needed to start small and look at the little differences between the two of them. I also noticed there is a lot of patters going on in the 4 forms that we are takes into finding are equations. |
Solving Problems
There are three types of real world problems that we practiced using quadratics.
Kinematics Is it a Homer is one example of a real world problem that we solved using kinematics or motion. We first solved for a using vertex form and then used the formula to solve for y. I found my self to start working small on this side of the qeutions because I had a hard time understanding the way we came to the anwser. Geometry One example of a geometric problem would have been emergency at sea. We had to find the altitude of the triangle the is created by the boat and towers. To find the distance we set to equations equal to one another. Economics We set them equal to each other to help find the maximum value. The profiting from wiglets work sheet allows us to see a real word view on how economics is used. Looking at Geometry it was one of those ones that made better scene to me. Ounce I was able to see what was going on I was able to go through and see what was going on in the equation. Working on the hand out Leslie's Flowers, this is where we first introduced into Geometry. To solve this equation we had to what specific side lengths where. While working on the geometry side of are work I found my self wanting to seek why and prove because it was one of the topics that really drawed me into the work. 1. We started with labeling all the sides by a,b,c. This allows us to pug it into a equation. Then we puged the information that we already new into the equation from there. (14-x)^2+y^2=x^2-y^2 then we worked out the equation where we ended up with 13^2-x^2=15^2-(14-x)^2 then we made a area diagram and came out with 225-(196-14x-14x+x^2) The we combined liked terms, then reduced everything down to x=5. |
Reflection
When I look back on this project I really found it to be challenging personally I had a really hard time with learning Quadratics. I Really found my self asking fellow piers to show me what I needed to do because I wasn't grassing the concept very will. While i was writing this DP I found my self to understand the concept of Quadratics significant more. I Really think the begging of this project I was understanding it but as more and more went on I was having a hard time continuing I really think we went through this math course a little to quick. I know that know I have a great recourse for me to go back and look at how to do quadratics and the kinds of quadratics. As I look at my work I really saw how I had be systematic because I had to look at each little piece of the equation to see what ended up effecting are resolt. I also noticed how I needed to draw out area diagrams to help with some of the questions that I didn't know how to enter. IN total this project really helped with more then just learning what is going on in math it really helped me stay organized because we in total had 25 piece of paper that we are going be getting turned in that had all of are work and to lose that would not be a very good thing.