Features of a graph

1. The x- and y-intercepts: If $y = f(x),$ find the $x$-intercept(s) by setting $y = 0$ and solving for $x$; find the $y$-intercept by setting $x = 0$ and solving for $y$.

2. Extrema: Locate the maxima and minima. (When drawing the curve by hand, it is not necesary at this point to identify which extrema relative or absolute, as that will be clear once the entire curve is graphed.

3. Points of inflection: Locate the points of inflection.

4. Behavior near singular points of $f:$$a$ is a singular point of $f$ if $f(a)$ is not defined, but $f(x)$ is defined for (at least some) points arbitrarily close to and on both sides of $a.$ If $a$ is a singular point of $f,$ consider $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ to see how the graph of $f$ behaves as $x$ approaches $a$.

5. Behavior at infinity: Consider $\lim_{x \to -\infty} f(x)$ and $\lim_{x \to \infty} f(x)$ if appropriate, to see how the graph of $f$ behaves far to the left and right: