Introduction
The most obvious thing we were supposed to learn during this project was how to solve and understand quadratics. Besides that, we were also expected to improve our knowledge of algebraic symbols and equations, and be able to connect algebra to problem situations. We were supposed to obtain an understanding on how rewriting quadratic expressions, varying from factored or vertex form, to provide insight to different graphs that connect to these functions. The main problem we were given to help understand quadratics, was centered around launching a rocket for the use of creating a firework display. We later used quadratics, and graphs to solve different pieces of this problem. Overall we had two learning objectives. One, to get a better understanding to the relations between geometry, and two, to learn how to use quadratic equations to solve real word problems. This project was started by gradually teaching us about parabolas, which I will get into more of later.
Exploring the vertex form of the quadratic equation
In the equations we used in Desmos, they contained a,h, and k variables, which all did different things to the parabola depending on the numbers inserted into them. The "a" variable, determines if the parabola is concave up or down, as well as how wide it opens. For example, if "a" is positive the parabola will concave up, and it will concave down when negative. Also, the smaller number "a" is, the wider, and more stretched the parabola will be.
The "k" determines how far the parabola moves up or down. For example, if the "k" value is -3, the vertex on the parabola will be on the -3 below the x-axis, and if its +3, it will be at the positive 3 above the x-axis. |
*"k" variable above**"H" VARIABLE above* |
The "h" value changes where the vertex of the parabola is on the y-axis. So, when the "h" is negative, the vertex is on the left side of the y-axis, and if its positive, it will be on the right side of the y-axis.
Quadratic equations are written as, y=a(x-h)^2+k, and to use the equation to graph a parabola, you just input what you know about each "letter value." For example, say the equation is y=1(x-5)^2+7, then your parabola will be opening upwards, because the "a" value is positive, it will be at the -5 on the left side of the y-axis because its negative, and it will be 7 above the x-axis because the 7 is positive.
Quadratic equations are written as, y=a(x-h)^2+k, and to use the equation to graph a parabola, you just input what you know about each "letter value." For example, say the equation is y=1(x-5)^2+7, then your parabola will be opening upwards, because the "a" value is positive, it will be at the -5 on the left side of the y-axis because its negative, and it will be 7 above the x-axis because the 7 is positive.
other forms of the quadratic equation
Two forms of the Quadratic equation include the Standard, and Factored and vertex forms. Standard form looks like this; y=ax^2+bx+c. For example, y=x^2+4x+3, and we can graph it to look like the top graph on the right. Factored form looks like this; y=(x+r)(x+s) For example, y=(x+4)(x+4)+7. Vertex form looks like this: y=a(x-h)^2+k.
|
In summary, all of these equations are the same thing, just written differently, but each have their advantages.
The standard form advantage is that it can be used to write every single linear equation, as it can measure vertical and horizontal lines.
The factored form advantage is that it is overall, the easiest to solve, but this method is not useful for plotting.
Last, the vertex form advantage is that it is good for trying to plot quadratics on a graph.
The standard form advantage is that it can be used to write every single linear equation, as it can measure vertical and horizontal lines.
The factored form advantage is that it is overall, the easiest to solve, but this method is not useful for plotting.
Last, the vertex form advantage is that it is good for trying to plot quadratics on a graph.
As proof that all of these equations are basically the same thing, I can show three different equations, (one of each of the equations I have went over) and graph them all on a graph. They will all create the same parabola.
Vertex form: y=-4(x-2)^2+4 Factored form: y=-4(x-1)(x-3) Standard form: y=-4x^2+16x-12 (Graph they all form on the right) |
converting between forms
Vertex Into Standard Form:
Vertex form looks like: y=a(x-h)^2+k, and we need to turn this into Standard form, which looks like this: y=ax^2+bx+c. For this example, we will use this equation: y=5(x+6)^2+19. First we need the squared Binomial, and for our equation it will look like this: (x+6)^2 = (x+6)(x+6) Then we can use an area model to make it into a grid (on the right) |
From the grid, we can put the numbers back into an equation:
X^2+6x+6x+36
x^2+12x+36
Next we need to distribute the "a" factor. and we do this by multiplying it into the new equation.
5(x^2+12x+36)
and we get,
5x^2+60x+180)
Last we need to add in our "k" factor to the end.
5x^2+60x+180+19
5x^2+60x+199 <------This is our answer.
X^2+6x+6x+36
x^2+12x+36
Next we need to distribute the "a" factor. and we do this by multiplying it into the new equation.
5(x^2+12x+36)
and we get,
5x^2+60x+180)
Last we need to add in our "k" factor to the end.
5x^2+60x+180+19
5x^2+60x+199 <------This is our answer.
From the area model, we can easily find that the missing square is 12.25, which is
(x+3.5)(x+3.5)= (x+3.5)^2
Now we need to add and subtract 12.25 to our equation:
4(x^2+7x+12.25-12.25+3.5)
4((x+3.5)^2+12.25+3.5)
4(x+3.5)^2+8.75
(x+3.5)(x+3.5)= (x+3.5)^2
Now we need to add and subtract 12.25 to our equation:
4(x^2+7x+12.25-12.25+3.5)
4((x+3.5)^2+12.25+3.5)
4(x+3.5)^2+8.75
We first take the x^2
Then we add the 8x and the 4x together to make 12x
Next we combine what we have so far, which is x^2+12x
and we add our last number which was obtained by multiplying 8x and 4x together which is 32.
we end up with: x^2+12x+32 as our answer.
Then we add the 8x and the 4x together to make 12x
Next we combine what we have so far, which is x^2+12x
and we add our last number which was obtained by multiplying 8x and 4x together which is 32.
we end up with: x^2+12x+32 as our answer.
Solving problems with quadratic equations
Their are three types of real world problems that can be solved using quadratic equations. These include, Kinematics, which can be used to solve projectile motion problems, Geometry, which can be used to solve rectangle, and triangle area problems, and Economics, which can be used to solve profit based problems.
Kinematics Problem Example:
An arrow is shot into the air on a farm covered in small trees, and is surrounded by a fence. The arrow reaches its maximum height at 60 feet. The place on the ground underneath the arrow is 180 feet away from a small tree. The fence is 360 feet away from where the arrow was originally shot, and its 13 feet tall. Will the arrow hit the fence? |
Geometry problem example:
You have a rectangular area of space consisting of 4 smaller rectangles. The largest rectangle has a length of 6 units, and a height of 5 units. The rectangle to the right of the largest, has a length of 2 units, and a height of 5 units. The rectangle on the bottom of the largest rectangle, has a length of 6 units, and a height of 3 units. Last, the last rectangle has a length of 2 units, and a height of 3 units. What is the overall area? |
Reflection
Throughout this section of learning about quadratics, I feel like I learned a lot, and improved on my style of studying, even though I still struggle with remembering the equations. Some things I think I can take away from this section, as something I fully understand now, would include understanding the standard, factored, and vertex equation forms, as well as the geometry section of this project. Some things I still very much need to improve on are kinematics and economics problems, as well as remembering the correct directions for equation values, "a," "k," and "h." After finally completing this math section, I have a better understanding of the math difficulty of 11th grade, but I believe that I have developed enough new math skills to be able to handle 11th grade math enough. Though, I know It will be easier for me to understand this, only once I've come upon the level of difficulty in math next year.
How I used habits of a Mathematician during this project:
Look for patterns:
I used this habit most when solving and creating area graphs, as well as solving the equations that went with them. Eventually I noticed that factored form equations can just be added together and squared to get most of the answer to a lot of questions. This ended up being useful when solving problems quicker.
I used this habit most when solving and creating area graphs, as well as solving the equations that went with them. Eventually I noticed that factored form equations can just be added together and squared to get most of the answer to a lot of questions. This ended up being useful when solving problems quicker.
Start small:
I used this habit for basically every single problem and thing we did, because i'm not the best as visualizing in my head how everything works, so I need to draw problems out anyways. When I do this, I can see what parts of each problem I already understand, and start solving those pieces first. These eventually just blend with the rest of the problem allowing me to easily solve it.
I used this habit for basically every single problem and thing we did, because i'm not the best as visualizing in my head how everything works, so I need to draw problems out anyways. When I do this, I can see what parts of each problem I already understand, and start solving those pieces first. These eventually just blend with the rest of the problem allowing me to easily solve it.
Be Systematic:
I didn't use this habit that much, but the very few times I did, it was on problems I didn't yet understand. I used this habit on those problems because sometimes the way problems are created or worded aren't in a way I can understand them at first. By changing small pieces of these problems I usually come to a way to understand them better, and what their asking, and even sometimes how to properly solve them.
I didn't use this habit that much, but the very few times I did, it was on problems I didn't yet understand. I used this habit on those problems because sometimes the way problems are created or worded aren't in a way I can understand them at first. By changing small pieces of these problems I usually come to a way to understand them better, and what their asking, and even sometimes how to properly solve them.
Take apart and put back together:
This habit is basically describing the main use of the area graphs, in my eyes at least. What I mean is, the area diagrams make you pull apart the problem into a diagram so solve for missing pieces. Then you put everything back together, back into a new equation.
This habit is basically describing the main use of the area graphs, in my eyes at least. What I mean is, the area diagrams make you pull apart the problem into a diagram so solve for missing pieces. Then you put everything back together, back into a new equation.
Conjecture and test:
I used this habit when we were introduced to the graphing site, Desmos. This is because we were given the freedom to input our own made up equations to create parabolas, using the equation formats we were assigned.
I used this habit when we were introduced to the graphing site, Desmos. This is because we were given the freedom to input our own made up equations to create parabolas, using the equation formats we were assigned.
Stay organized:
I still need to work on this habit a lot, because the way I solve math problems given to me is extremely messy, and I do better using this habit for word problems. This is because when given word problems, I can draw out images of what the problem is asking, and later I can label my images accordingly. I used this most with kinematics problems.
I still need to work on this habit a lot, because the way I solve math problems given to me is extremely messy, and I do better using this habit for word problems. This is because when given word problems, I can draw out images of what the problem is asking, and later I can label my images accordingly. I used this most with kinematics problems.
Describe and articulate:
I also rarely used this habit, but the few times I did use it was for kinematics problems, because I struggle most with these, and it makes them slightly easier to explain and draw out in diagrams what these problems are asking. I also used this habit when we were working with parabolas because we had to draw them out, and be able to use equations to graph them. This meant I needed to be able to explain and understand how each variable in the parabola equations, created a parabola correctly.
I also rarely used this habit, but the few times I did use it was for kinematics problems, because I struggle most with these, and it makes them slightly easier to explain and draw out in diagrams what these problems are asking. I also used this habit when we were working with parabolas because we had to draw them out, and be able to use equations to graph them. This meant I needed to be able to explain and understand how each variable in the parabola equations, created a parabola correctly.
Seek why and prove:
I used this habit most with problems I didn't understand, because thats the only way one can be able to understand something previously un-understandable to them.
I used this habit most with problems I didn't understand, because thats the only way one can be able to understand something previously un-understandable to them.
Be confident, patient, and persistent:
I used this habit a lot, because I struggle with math, which means I need to be patient when trying to solve math problems. I also used this most with kinematics problems, and I also used it with trying to solve economics problems, as well as trying to properly graph parabolas with equations without using the Desmos site.
I used this habit a lot, because I struggle with math, which means I need to be patient when trying to solve math problems. I also used this most with kinematics problems, and I also used it with trying to solve economics problems, as well as trying to properly graph parabolas with equations without using the Desmos site.
Collaborate and listen:
This habit was used when we did group work, and when going over in small groups on how to solve certain problems given. This habit was always used during this time.
This habit was used when we did group work, and when going over in small groups on how to solve certain problems given. This habit was always used during this time.
Generalize:
Lastly, I used this habit to help me memorize how equations worked, and how to solve word problems. I need to work on this habit a lot still, because I have a habit of moving on to other problems the second I solve something correctly, before first understanding why its correct, and how I solved it.
Lastly, I used this habit to help me memorize how equations worked, and how to solve word problems. I need to work on this habit a lot still, because I have a habit of moving on to other problems the second I solve something correctly, before first understanding why its correct, and how I solved it.