We described the graphs of f(x-a), f(ax), f(x)+a, af(x), for a\in\mathbb{R}, in terms of the the graph of f(x) (assuming we know everything about the graph of f(x)).

We defined the quadratic function and what a root of a function is. We derived the familiar x=-b\pm\dots equation for the roots of the quadratic, and explained how the b^2-4ac term determines the nature of the roots. We noted the inherent symmetry of the quadratic, and how if we know the graph of f(x)=x^2 we have a rough idea of the shape of a quadratic by looking at the sign of the squared term.

We defined a polynomial function – and noted that the quadratic function is an example of one. We wrote down the Factor Theorem (without proof – a proof is given on this webpage). We wrote down the Fundamental Theorem of Algebra (whose proof is beyond the scope of the course). We defined a rational function and described a method of solving inequalities involving rational functions (although we made an assumption about polynomials – namely that they are continuous – this assumption will be shown to be true in the next few weeks).

We defined the absolute value function and proved some of its properties. Finally we described the sets \{x\in\mathbb{R}:|x|<\delta\} and \{x\in\mathbb{R}:|x-a|<\delta\}.

Exercises

Q.4-6 and 7(i) from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise1.pdf

Q. 1-2 from Problems

From class:

Let x\in\mathbb{R}, x< 0. Prove that

\left|\frac{1}{x}\right|=\frac{1}{|x|}