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Juxtaposition or Parabolic?


“Para” means “alongside” and “bola” comes from the Greek word “to throw“. So, the math word “Parabola” (and “parable”), for example, means to be “thrown alongside“. A parable is a story that has a meaning “alongside”. The parabola is a curve that is “thrown alongside” a fixed point and a fixed line, as we defined in an earlier post (using a canon to do the throwing of course). So, what does this have to do with “juxtaposition”? Juxtaposition is where two things are placed close together with contrasting effect. For example, H3 once entered a photo competition where a person’s face had one half smiling and the other frowning – hence the “contrasting effect” or “juxtaposition” (which was the title of entries for the competition). But, let’s get back to the Mathematics!

H3 were recently resting up in a hotel which looked out at this building.

curved buildingNote the really dynamic curve in the architecture. But, asked H3, “What shape is this curve“? Is there software that could take the picture of the curve and then calculated the equation of its shape, without using Excel or any specialised software? A very good question, you might ask.

But, another approach to this might be for you, dear students, to take the picture of the curve and then, using a parabola app, see if you can make a parabolic curve to fit the shape in the picture. In this way, you will learn more about what changes the shape of a parabola, than just playing with the software on its own – in this case, H3 used http://www.mathopenref.com/quadraticexplorer.html.

graph tool

Fitting Parabolas2So, the parabola is made shallower by changing the coefficient (number) before the x. Changing the x coefficient or the constant will not alter the basic shape of the parabola to fit it to the curve in the photo. Good fun and good Mathematics too!

A parabola is defined at the “locus” or path of moving points that remain the same distance from a fixed point as a fixed line. Think of the locus as the points of blue light in this next photo of the same building – taken at night;


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10 x 9 x 8 + (7 + 6) x 5 x 4 x (3 + 2) x 1 = 2020

NCEA Level 2 Algebra Problem. Using the information given, the shaded area = 9, that is:
y(y-8) = 9 –> y.y – 8y – 9 =0
–> (y-9)(y+1) = 0, therefore y = 9 (can’t have a distance of – 1 for the other solution for y)
Using the top and bottom of the rectangle,
x = (y-8)(y+2) = (9-8)(9+2) = 11
but, the left side = (x-4) = 11-4 = 7, but rhs = y+? = 9+?, which is greater than the value of the opp. side??
[I think that the left had side was a mistake and should have read (x+4)?]

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