[Latexpage]
A complete graph of a function must include all of the following if they exist:
- The x- and y-intercepts
- Local maxima and minima (turning points)
- Concavity and points of inflection
- Vertical asymptotes
- Horizontal asymptotes
- The end behavior of the graph (behavior as x goes to infinity or – infinity) – horizontal asymptotes, if they exist, are one form of end behavior.
- The domain of the function should be clear from the graph
Exception: in the case of periodic functions, we often sketch only one period of the function.
Comments on some of these:
The y-intercept is the point where x=0, if 0 is in the domain of the function.
The x-intercepts are the zeroes of the function: set y=0 and solve.
There is a vertical asymptote x=c when c is not in the domain of the function, and at least one of the following is true:
- $\displaystyle \lim_{x\rightarrow c}f(x) = \infty$ or $-\infty$
- $\displaystyle \lim_{x\rightarrow c^{+}f(x) = \infty$or $-\infty$
- $\displaystyle \lim_{x\rightarrow c^{-}f(x) = \infty$or $-\infty$
There is a horizontal asymptote y=c if
$\displaystyle \lim_{x\rightarrow\infty}f(x) = c$
