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Section 4.2 Families of Curves • We have seen before that knowledge of one function can provide knowledge of many others • For instance, what we know about y = x3 can give us information about y = x3, y = 5x3, y = (x - 10)3, or y = 3(x + 2)3 - 7 • We call these families of functions • This helps us in their use in mathematical modeling • The position of an object moving vertical under the influence of gravity is given by h(t ) 4.9t 2 v0t h0 • Where t is time in seconds and h is the height above the ground • What do you notice about this function? • Curves of the form y = Asin(Bx) – This family is used to model a wave • Curves of the form y e ( x a ) 2 / b – This family is related to the normal density function used in probability and statistics • Curves of the form y a(1 e bx ) – This family of functions represents a quantity that is increasing but leveling off • The temperature of an object placed in a refrigerator will cool off quickly at first, but slow down as it approaches the temperature of the refrigerator Examples ( x a ) 2 / b • Find a function of the form y e for b > 0 with a local maximum at x = 2 and points of inflection at x = 1 and x = 3 – This is number 11 from 4.2 in the book • Number 29 from 4.2 in the book