By Robert Miller
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Extra resources for Calc I
Both and are inflection points. The ends f(100) and f(-100) are positive. Both ends go to plus infinity. The sketch in two stages is as follows: Example 26— x intercept: y = 0 (3,0). Note x = -3 is not in the domain. y intercept: x = 0 (0,9). Possible max, min: y' = 0 x = 0. We get the point (0,9). (0,9) is a maximum since y" is always negative. No inflection point since y" is never equal to 0. Since the domain is finite, we must get values for the left and right ends of x = -1, y = 8 (-1,8). x = 4, y = -7 (4,7).
F(x) = -f(-x). Symmetrical about the origin. 3. g(y) = g(-y). Symmetrical about the x axis. This is only found in graphing curves that are not functions of x. Example 31— y axis symmetry Example 32— Symmetric origin Example 33— x axis symmetry For completeness, we will sketch a curve that is not a function of x or y. Example 34— Intercept is (0,0). It turns out to be an isolated point since for all x values between -1 and 1, except x = 0, is negative, making y imaginary. Vertical asymptotes: x = 1, x =-1.
There is no y intercept. Warnings: 1. If you get the sign of the y intercept wrong, you will never, never sketch the curve properly. 2. Functions have one y intercept at most (one or none). 3. If we have the intercept (0,0), it is one of the x intercepts, maybe the only one, but the only y intercept. We do not have to waste time trying to find another one! Vertical Asymptotes A rational function has a vertical asymptote whenever the bottom of the fraction is equal to 0. Example 6— Asymptotes are vertical lines x = 0, x = 4, and x = -3.