Quadratic Function. Parabola. Second Degree Polynomial. GA2 and TPC
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, googleconic 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.
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, googleconic 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.
I found it interesting that I can find an infinite amount of parabolas in real life. For example, in architecture, rollercoasters, and nature. The McDonald's arches even is made of two parabolas. A dolphin's jump out of the water is a parabola. The Axis of Symmetry becomes very important in each case. This is so we understand the what the other half of the parabola is going to be like. I found this image below to understand the engineering in rollercoasters. I didn't realize it'd be so complicated.
Also, there are parabolas in football. I remember the video we watched in class and found it interesting that they needed a mathematician to help the punter improve. I re-watched the video at home and found it interesting how the trajectory is a parabola. it follows this pathway due to gravity and the decrease of the vertical vector. The last thing I found interesting about the video is the closing statement where it it said "but takes NFL punters like Craig Heinrich, to turn science into a an art form". The distance between points of a parabola mean so much and can be studied to hit perfect spots.
