â Guidelines for Sketching a Curve The following checklist is intended as a guide to sketching a curve y f(x) by hand. Not every item is relevant to every function. (For instance, a given curve might not have an asymptote or possess symmetry.) But the guidelines provide all the information you need to make a sketch that displays the most important aspects of the function. A. Domain It's often useful to start by determining the domain D of f, that is, the set of values of x for which f(x) is defined. B. Intercepts The y-intercept is f(0) and this tells us where the curve intersects the y-axis. To find the x-intercepts, we set y0 and solve for x. (You can omit this step if the equation is difficult to solve.) -x C. Symmetry (i) If f(-x) =f(x) for all x in D, that is, the equation of the curve is unchanged when x is replaced by -x, then f is an even function and the curve is symmetric about the y-axis. This means that our work is cut in half. If we know what the curve looks like for x0, then we need only reflect about the y-axis to obtain the com- plete curve [see Figure 3(a)]. Here are some examples: y-x, y x", y-lal, and y cos x. metry (ii) If f(-x) =-f(x) for all t in D, then f is an odd function and the curve is symmetric about the origin. Again we can obtain the complete curve if we know what it looks like for x0. [Rotate 180° about the origin; see Figure 3(b).] Some simple examples of odd functions are y-x,y-x, y = x, and y-sin x. nmetry
