I remember that at the very start of this unit, we were given a problem about a Victory Celebration (involving fireworks, of course) and a rocket containing fireworks that was to be launched and activated at the top of its trajectory. We were to use an equation showing the height of the rocket (physics displacement equation) as a function of time to solve for certain variables, such as how long it would take the rocket to reach the top of its trajectory, how high the rocket would go, and how long the rocket will be in the air. This first problem helped me to substitute variables in where they should be, simplify equations, and look for specific things within an equation. After working on this problem for a while, we went into different things, like geometric equations, parabolas, their equations and relationships between a parabola and its equation, other forms of equations, using area diagrams, and economic problems, such as the problem dealing with how one might find maximum revenue. I think that a main goal of the unit was to familiarize ourselves more with quadratic equations and the many forms they appear in, as well as the relationship between them and parabolas.
Vertex Form of a Quadratic Equation:
The format of an equation in vertex form is: y=a(x-h)^2+k.
When I assign values to a, h, and k so that my equation is y=1(x-2)^2+3, I get a graph looking like this:
Vertex Form of a Quadratic Equation:
The format of an equation in vertex form is: y=a(x-h)^2+k.
When I assign values to a, h, and k so that my equation is y=1(x-2)^2+3, I get a graph looking like this:
As you can see, the X and Y values are +1. Looking at the equation we can confirm this. Inside the parenthesis there is a -X, and with our knowledge of x values in standard form we can confirm that the X value is +1. Next we can look for the Y value and we see the +1 after the (X-1)e2. This means that the Y value is 1, and by observing the graph we can confirm that it is.
Various Forms of the Quadratic Equation:
The three variations/forms of equations we learned about during this unit were:
-standard form: y=ax^2+bx+c
Various Forms of the Quadratic Equation:
The three variations/forms of equations we learned about during this unit were:
-standard form: y=ax^2+bx+c
-vertex form: y=a(x-h)^2+k
(see image above)
-factored form: y=(x-r)(x-r)
(see image above)
-factored form: y=(x-r)(x-r)
(on top of this, there is expanded form, which shows you a FOILed version of factored form, in essence.)
When you convert to standard form, you know the y intercepts of the equation. Converting to factored form lets you see the x intercepts, and vertex form gives you the vertex of the parabola.
Converting Between Forms:
You have seen the different forms of quadratic equations, but how do you move between these forms? Fear not, as there are easy to access methods to convert to all of the 3 forms. You would want to convert between these different forms to find, as stated before, the vertex, the x values, and the y values by looking at the equation. We learned how to:
1. Convert from vertex form to standard form
When you convert to standard form, you know the y intercepts of the equation. Converting to factored form lets you see the x intercepts, and vertex form gives you the vertex of the parabola.
Converting Between Forms:
You have seen the different forms of quadratic equations, but how do you move between these forms? Fear not, as there are easy to access methods to convert to all of the 3 forms. You would want to convert between these different forms to find, as stated before, the vertex, the x values, and the y values by looking at the equation. We learned how to:
1. Convert from vertex form to standard form
In the example above, a=3, h=1, and k=2.
In order to convert from vertex to standard, you need to first, FOIL whatever is in the parenthasis, then distribute the value that is a, then combine like terms, and the resulting equation should look like: y=ax^2+bx+c.
2. Convert from standard form to vertex form
In order to convert from vertex to standard, you need to first, FOIL whatever is in the parenthasis, then distribute the value that is a, then combine like terms, and the resulting equation should look like: y=ax^2+bx+c.
2. Convert from standard form to vertex form
In the example above, a=0, b=12, and c=32.
First, you group the x values together and complete the square so that you can write them as a perfect square. What you would do, then, to complete the perfect square, is take your b value, halve it, and then square it, to get 36. Once you add that halved and squared b value to your original 12x, you have a perfect square trinomial. But once you've all of the sudden added this 36 to 12x, you have an imbalance. In order to counteract this balance, you simply subtract 36 back out of the equation, so you'd essentially be adding a value of zero. Then, you group that negative you added on to balance the equation with your c value, which is 32 in this case. Then, the x^2 +12x+36 automatically becomes (x+6)^2, and the resulting equation is y=(x+6)^2-4.
3. Convert from factored form to standard form
First, you group the x values together and complete the square so that you can write them as a perfect square. What you would do, then, to complete the perfect square, is take your b value, halve it, and then square it, to get 36. Once you add that halved and squared b value to your original 12x, you have a perfect square trinomial. But once you've all of the sudden added this 36 to 12x, you have an imbalance. In order to counteract this balance, you simply subtract 36 back out of the equation, so you'd essentially be adding a value of zero. Then, you group that negative you added on to balance the equation with your c value, which is 32 in this case. Then, the x^2 +12x+36 automatically becomes (x+6)^2, and the resulting equation is y=(x+6)^2-4.
3. Convert from factored form to standard form
I thought that converting from factored form to standard was one of the simpler conversions, because you're basically just FOILing and distributing the constant value afterwards. The area diagram (on the right) is also helpful so that you may have a more visual method of simplifying the equation (helpful for when you start out, but not crucial once you have the steps down more mentally).
Solving problems with Quadratic Equations:
Quadratic equations have many applications through various problems, and we worked on some of the different ones throughout this unit.
Kinematic Equations (projectile motion):
A good example of using a kinematic equation was at the start of the unit, when we were trying to figure out certain things based on the projectile motion of a rocket:
Geometry (triangle problems and rectangle area problems:
Another relationship is the relationship between quadratics and geometrics. We used a number of different problems to practice this relationship including 'A Corral Variation', where we worked with different areas using the same perimeter, and found an equation for the value that would maximize the area :
Another relationship is the relationship between quadratics and geometrics. We used a number of different problems to practice this relationship including 'A Corral Variation', where we worked with different areas using the same perimeter, and found an equation for the value that would maximize the area :
Economics (maximizing revenue/profit or minimizing expenses/losses):
I think that using quadratic functions in economics for maximizing revenue is one of the most applicable uses to quadratics, and one that I found more interesting personally. In this problem, we found the maximum amount of revenue a company could get by charging just so much for each product that the sales will be maximized and the price isn't too low. The formula, I later found out, is called the Marketing Director's Formula.
I think that using quadratic functions in economics for maximizing revenue is one of the most applicable uses to quadratics, and one that I found more interesting personally. In this problem, we found the maximum amount of revenue a company could get by charging just so much for each product that the sales will be maximized and the price isn't too low. The formula, I later found out, is called the Marketing Director's Formula.
Another problem involving projectile motion was 'Is it a Homer?'.
Our task was to find out whether a baseball would hit a fence or clear it, given that the fence is 15 feet tall, the ball reaches its vertex at 80 feet, when it's at the peak, it's above a spot 200 feet from the home plate, and the fence is 380 feet from the home plate.
I used vertex form to plug in all of my given variables and find my a value, and then find out how high the ball is at the point where it reaches the fence.
Vertex form:
y=a(x-h)^2+k
y=a(x-200)^2+80
0=a(0-200)^2+80
y=-.0025(x-20)^2+80
y=-.0025(x-20)^2+80=50.2
From this, I now know that the ball will be at 50.2 feet at the point it reaches the fence, and therefore can conclude that the ball does clear the fence.
I used vertex form to plug in all of my given variables and find my a value, and then find out how high the ball is at the point where it reaches the fence.
Vertex form:
y=a(x-h)^2+k
y=a(x-200)^2+80
0=a(0-200)^2+80
y=-.0025(x-20)^2+80
y=-.0025(x-20)^2+80=50.2
From this, I now know that the ball will be at 50.2 feet at the point it reaches the fence, and therefore can conclude that the ball does clear the fence.
Reflection:
I felt sufficiently challenged during this unit, and was able to apply a great deal of the Habits of Mathematicians in every process I went through, whether it was graphing an equation on Desmos, looking for a set of coordinates within an equation, FOILing an equation in order to convert it into a different form, or determining how long a rocket would take to reach the ground.
I feel as though when we needed to convert between forms of equations and look for the useful information in each equation we were the most looking for patterns because this was when we were looking for something in particular, and had to use the patterns we had gone through to look for other patterns. I think that when we converted between forms and used FOIL (First, Outer, Inner, Last) as a method of doing this, this was a way of starting small by working on a select section of the equation at a time. I thought that every time we converted, numerous steps were used, and I took a systematic approach when dealing with the more complicated conversions.
Taking apart and putting back together: When using an area diagram during this unit to convert say, factored form to standard form, we were taking the equation apart and putting it back together by solving for one section of the equation at a time. I think that towards the beginning of the unit was when we were conjecturing and testing the most, namely when we were experimenting on Desmos by trying to find what the variables a, h, and k did within a vertex form equation. Staying organized was so essential during this unit, and I can't say with complete confidence that I was able to follow this habit completely. This unit involved a lot of writing down and showing work, and I wasn't always able to keep my work in an orderly manner, but now know how essential it is when working with similar equations in the future.
Describing and articulating after converting forms, or solving for coordinates was crucial during this unit because it helped to reiterate how we got there. Every time we found the coordinates of a point, we needed to seek why and prove how we got there, and what method/equation we used. Because this was the last unit in Math 2 this year, everyone, including myself, felt anxious and overwhelmed at times. I also found this unit to be one of the more challenging and rigorous units of the others. This called for confidence, patience, and persistence because not only was I constantly doubting my abilities at times, but sometimes I simply lost my motivation. This habit of mathematicians helped me to combat my self-doubt and keep me moving through the rigors.
I felt as though collaborating and listening during this unit was one of the more helpful aspects of the Habits of Mathematicians because every time I struggled with setting up something (an area diagram or equation) correctly, I would simply listen to a peer who had already gone through the steps I was having difficulty with and ask them questions, or collaborate with them to come to a solution together.
I think that "quadratic equation" is a very broad, or general term, and that every form of equation we learned how to convert to can be applied to something.