2.17 Curve Sketching

Let $y=f(x)$ and investigate the following:

(i) the intercepts of the graph of $f$ with the $y$ axis at $(0,f(0))$, and the $x$ axis at $(a,0)$ for $a$ such that $f(a)=0$;

(ii) what happens to $y$ as $x\to\pm\infty$?

(iii) which values of $x$ make $y\to\pm\infty$?

(iv) symmetry, do we have $f(-x)=f(x)$ ($f$ is even), or $f(-x)=-f(x)$ ($f$ is odd)?

(v) periodicity, as in the sine or sawtooth functions.

(vi) $f$ is strictly increasing if $f(a)<f(b)$ whenever $a<b$, so the graph is going upwards.

(vii) $f$ is strictly decreasing if $f(a)>f(b)$ whenever $a<b$, so the graph of such a function is going downwards;

(viii) asymptotes are straight lines that get close to the graph as $x\rightarrow\pm\infty$ or $y\rightarrow\pm\infty$.