We use cookies to give you a better experience, if that’s ok you can close this message and carry on browsing. For more info read our cookies policy.
3.1

## UNSW Sydney

Skip to 0 minutes and 11 secondsLast 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.

Skip to 1 minute and 0 secondsAnd 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.

# Quadratics from Apollonius to Bezier

Well done everyone in learning a lot about linear relations, graphing and lines in our first two weeks! 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

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 two weeks, and we will want to enlarge our view to include also quadratic functions like

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.

## Course highlights Get a taste of this course before you join:

• ##### Australian bush tucker
article

Adventurer Bear Grylls has claimed that, "Pound for pound, insects contain more protein than beef". Let's investigate!

• ##### The Cartesian plane and the beauty of graph paper
video

The Cartesian plane, modelled by a sheet of graph paper, is fundamental for much of modern mathematics. Norman Wildberger explains the basics.

• ##### Galileo's ball
article

What happens to an object, like a ball, when it falls, or better yet, if you toss it in the air? Galileo was the first to understand the answer.