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

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