﻿ Quadratics from Apollonius to Bezier

# Quadratics from Apollonius to Bezier

An introduction to this week's activities on quadratic relations and functions.
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Last week we were talking about lines and linear relations. This week we want to go up a level to quadratic relations involving second degree curves, and the conic sections of the ancient Greeks. The story goes back to Apollonius, and then into the 17th century with Descartes’ introduction of Cartesian coordinates, allowing us to really concretely get a hold algebraically of quadratic relations. Applications will be coming from physics, prominently– for example, kinetic energy– very important example. Also, the trajectories of various objects, described beautifully by quadratic curves. We’ll also talk about some economic applications, then we’ll be talking about the algebra of quadratic relations involving completing the square, and the connection with Descartes’ factor theorem.
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And finally, we’ll be looking at applications to modern design, introduced by two French car engineers in the ’60s, called De Casteljau bezier curves. So lots of interesting applications of quadratic relations.
Well done everyone in learning a lot about linear relations and lines in our first week! In this week, the course steps into high gear, with a lot of interesting and important mathematics.
We will be introducing quadratic relations, described by a quadratic equation of the form
$\Large{y=ax^2}.$
These are very special cases of the conic sections of the ancient Greeks, which were studied intensively by Apollonius, and given algebraic life by Descartes through more general degree two equations in two variables.
This is definitely a step up from the linear equations of the form $$\normalsize{y=mx+b}$$ that we have been looking at last week, and we will want to enlarge our view to include also quadratic functions like
$\Large{y=ax^2+bx+c}.$
These are examples of polynomials, combining multiples of $$\normalsize{1}$$, $$\normalsize{x}$$ and $$\normalsize{x^2}$$. In this video we give a quick overview of some of the rich theory and variety of applications that we will be looking at this week.