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Graphing by Hand

Here are tips for graphing.

  • First, you need to find x-intercepts and y-intercept, if possible.  For x-intercepts, setting y=0 to find x-values and similarly, setting x=0 to find y-intercept.
  • Then, investigate if there are any asymptotes for fractional functions. For the horizontal asymptotes, we need to take the limit at infinity.  If you get the limit, then y= the limit value should be the equation of horizontal asymptote.
  • For the vertical asymptote, we need to set the denominator function be zero to find the x-values and check the numerator function if it is not zero when it takes that x-value. Once you checked, x = that x-value should be the equation of vertical asymptote.
  • Then, of course, find the behavior  of increasing or decreasing, and local maxima or minima of the function if any.
  • Lastly, find out the concavity as well as the inflection points.   Those points guide you to graph easily.