ࡱ> .0-  bjbjT~T~ 4H66 +++++????4s?$<^uuuu!!!|$~$~$~$~$~$~$$9&($+!\!!!$++uu% $$$$!F+u+u|$$!|$$$$up[3_?<"$h$$0$$q)"q)$q)+$L!!$!!!!!$$#!!!$!!!!q)!!!!!!!!! : To Graph a Function Completely CharacteristicHow to examineA. DomainDomain is usually restricted by denominators that equal 0 and by radicals that contain negative values. Sometimes there may be an explicit domain.B. Interceptsy-intercepts are found by computing f(0). x-intercepts are found by solving f(x) = 0C. SymmetryEven: f(-x)=f(x) will have y-axis symmetry Odd: f(-x)=-f(x) will have origin symmetry Periodic: trig functions will repeat over a certain period of x-valuesD. AsymptotesHorizontal:  EMBED Equation.3 , then y=a is an asymptote to the right  EMBED Equation.3 , then y=a is an asymptote to the left Vertical:  EMBED Equation.3  EMBED Equation.3 , then x = a is an asymptote from the left  EMBED Equation.3 , then x = a is an asymptote from the right Slant: Perform long division, and ignore the remainderE. Increasing/Decreasingf(x) is increasing whenever f (x) is positive f(x) is decreasing whenever f (x) is negativeF. Extremaf(a) is a maximum if f (a) is 0 or undefined, AND f(x) is changing from increasing to decreasing (this would show up as a sign change on f (x) from positive to negative, and also as a negative value for f (a) Also, any closed endpoints must also be examined for potential maximums. f(a) is a minimum if f (a) is 0 or undefined, AND f(x) is changing from decreasing to increasing (this would show up as a sign change on f (x) from negative to positive, and also as a positive value for f (a) Also, any closed endpoints must also be examined for potential minimums.G. Concavityf(x) is concave up whenever f (x) is positive. (this would also show up as f (x) increasing.) f(x) is concave down whenever f (x) is negative. (this would also show up as f (x) decreasing.)H. Inflection Points(a, f(a) ) is an inflection point if f(x) is changing concavity at a. This would show up as a sign change on f (x), in either direction.I. SKETCH CAREFULLYIf needed, calculate a few y-values by substitution * Verify the sketchCheck the points and/or intervals for each characteristic to make sure your sketch matches your analysis, Be sure you have labeled the scale on each axis, and that your graph and work are legible. 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