Science:Math Exam Resources/Courses/MATH104/December 2016/Question 11 (d)/Hint 1

To produce an accurate sketch of a given function $f$, we may consider the following steps:

1. Find the domain of $f$. Generally, we assume that the domain is the entire real line then find restrictions, such as where a denominator is $0$ or where negatives appear under the radical.
2. Find the critical values of $f$.
3. Find the possible points of inflection of $f$.
4. Find the location of any vertical asymptotes of $f$ (usually done in conjunction with item 1 above).
5. Consider the limits $\lim _{x\to -\infty }f(x)$ and $\lim _{x\to \infty }f(x)$ to determine the end behaviour of the function.
6. Create a number line that includes all critical points, possible points of inflection, and locations of vertical asymptotes. For each interval created, determine whether $f$ is increasing or decreasing, concave up or down.
7. Evaluate $f$ at each critical point and possible point of inflection. Plot these points on a set of axes. Connect theses points with curves exhibiting the proper concavity. Sketch asymptotes and $x$ and $y$ intercepts where applicable.