Quadratic functions. Parabolas. Second Degree Polynomials. There's so many names, you just know this has to be important. For your blog, you'll find one aspect of quadratics to focus on and to develop further. Choose something that's a little outside what we did in class but still includes quadratic functions. My blog, hopefully, will send you to a bunch of different places where you can check out two categories of information about these functions. The first category is the cool mathematics of these things. The second category is the application of quadratics to the "real world". Of course, if you want to expand this and discuss something else relevant, then that's ok, too. If you have questions about whether your idea is ok, then just ask me.
As an intro, here's a site that has a slider (you'll have to scroll down to see what I mean) that shows you how the transformations to the quadratic function are paralleled with an athlete kicking a soccer ball. mathisfun.
Please go to that site now and use the slider. Scroll down. See if there's something else on that page that catches your eye. There's several good ideas for your study there.
How many different mathematical ways can you think about a quadratic function?
1. Geometrically, as a conic section (for GA2 kids, if you don't know what that means, google conic section).
2. Geometrically, as a set of points equidistant from a single point and a line. Here's an example of this. Again, you'll have to scroll down a little to see the circles and line. (Artsy folks: you might like the first image on this page; surely a child of the 1960's came up with this idea.)
3. Using algebra to plot points on a Cartesian plane using a quadratic equation..lo and behold, you get the parabola. (Sorry, no link for this one; you've all done this hundreds of times before.)
4. Use algebra to express the function in a variety of helpful forms: vertex form, descending form, factored form. (Again, we did this in class.)
5. Using creative forms of construction: folding paper or using the circles paper shown in #2 link above. (There's some interesting youtube videos that show paper folding to create other conic sections; that can be an interesting exploration for you.)
So...what about interesting applications?
1. A parabolic reflector. (Someone's trying to make money with this one...)
2. A parabolic dish for a telescope.
3. A football punt. (NFL and NSF got together for this video.) The idea applies, as you'll see in the video, to any projectile.
Find something that strikes your fancy. Choose ONE idea and develop it. Make it more than 8 sentences; it must be a single full idea. Cut and paste. And, of course, submit through canvas.
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DeleteI know from physics class last year that as the ball travels up on its parabolic path gravity gradually slows it down until it reaches its peak where it stops shortly before falling (accelerating) towards the ground.
I started googling math problems dealing with parabolas and soccer balls. Here is an example:
If a soccer ball is kicked straight up from the ground with an initial velocity of 32 feet per second, then its height above the earth in feet is given by s(t) = -16t2 + 32t, where t is time in seconds. What is the maximum height reached by the ball?
As defined by the mathisfun link on Mrs. Mariner’s blog, the maximum height is the vertex and “the vertex (where the parabola makes its sharpest turn).”
In order to find the vertex, we need a point. And a good way to start is finding an x value. As learned today in math class, one way to find the x value of the vertex is to use –b/2a. Therefore, it is -32/ 2(-16) = 1. Now we that the x coordinate of the vertex is x = 1. Now you can plug x = 1 into the (t) value in order to find the value of y. So y = -16(1)2+ 32(1) = 16. Now putting the x and y values together yields a vertex of (1, 16). So the maximum height is 16 feet.
NOW, what if we solved this problem by using another method we learned in class today (completing the square). Here is how we would do it: We start with s(t) = -16t2 + 32t. Now take out the -16 to get t2 to have a coefficient of 1. Now we have -16 (t2 – 2t). Now we take ½ of 2 which is 1 and square it which is 1 and then multiply it by -16 (don’t forget it because we factored it out). Now our equation looks something like this: -16(t-1)2 0 – (-16) = -16(t-1)2+ 16. From this we can tell that the vertex is (1, 16)!
As a soccer player, I realize that I can affect the path the ball takes and how far it goes in three major ways: I can control the velocity, angle, and rotation. If the ball is kicked from my foot at a steep angle, that is when I hit it with my toes pointing more up towards the sky, the ball will have more velocity and thus it travels higher vertically. But if when I kick the ball my toes are pointed straight out or slightly turned to the ground, my ball will have less speed and will follow a more horizontal path (so the parabola of this kick will not be as steep, therefore it won’t go as high).
This link to my blog will show you a picture of my graph:
http://remylink.blogspot.com/2014/04/paraballa-i-know-from-physics-class.html
I have always loved dinosaurs. So as Mrs. Mariner was talking about parabolas, I was thinking in the back of my head, the sail of a spinosaurus is a parabola. So I looked up how high the highest point of a spinosaurus' sail was; about 8.55ft, and decided to derive a function from that. If the vertex of the sail is (0,8.55) , then the x-int. are equal distances away from the origin. Using this website: http://www.walkingwithdinosaurs.com/dinosaurs/detail/spinosaurus/
ReplyDelete(There are pictures and interactive stuff here! It's cool!)
I was able to measure about how long the sail was, which was about 16.74ft. So each x-int was at (-8.37,0) and (8.37,0). Using these values, we can calculate a in y=a(x+h)^2 +k. We plug in 0=y and 8.37=x and h=0 and k=8.55 to solve a. 0-8.55=a(8.37+0)^2. a=-0.12. Now plug a back into the equation; y=-0.12(x)^2 +5. And the function of the sail of a spinosaurus is f(x)=-0.12(x)^2 +5!
When searching the internet for an interesting property of parabolas and quadratic functions, one interesting concept stood out, the property of reflection. If you draw a parabola, and then draw rays parallel to that quadratic function, something very interesting happens. The rays, when reflected off of the parabola, all intersect in one point called the focus. This reflector property of parabolas can then be used in real world applications, such as a satellite dish. There are two types of satellite dishes, transmitting and receiving, but both work on the same reflector principal. In a receiving dish, the transmitted signals act like the rays in the basic geometric diagram and get reflected off the surface of the parabolic bowl into the feed horn, which sends the signal on to other receiving equipment. In transmission, the feed horn transmits a signal which gets reflected off of the parabolic bowl and out towards equipment that will receive the transmission. So in transmission, the principal of the reflector property is working in reverse, originating at the focus and then spreading outward. I never knew how satellite transmission and receiving worked, and its cool to know that this whole process is governed by the reflector principal of a parabola.
ReplyDeleteQuadratic functions are amazing and have seemingly endless functions and meanings. I really like how mathisfun.com showed a parabola in soccer. I found this youtube clip that showed the importance of understanding parabolas in basketball: https://www.youtube.com/watch?v=A1R_TDTv6fg. If a player can mentally estimate “a” the distance from the hoop, and “b” the height of the hoop, which is always 10ft in the NBA. Using the formula for parabola, technically a player could figure the path of the ball to from his hands to the hoop. Basketball coaches and specialists could change the game, teaching players how to use precise shots to make baskets perfectly. As technology progresses and players get better, understanding the math behind movements will become more important and essential to winning. I predict that sports mathematics will become an actual professional field in some form someday.
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