What model describes the relationship between the amount of bacteria and time, given that the bacteria quadruple every two hours? linear, cubic, quadratic, exponential

This is exponential.  Start with time increments of 1.  If we have 4^x, then the bacteria population triples every hour (x=0 -> 1, x=1 -> 4, x=2 -> 16, etc).  Now, the problem is that is quadruples every two hours.  If you substitute one hour for two, the equation becomes 4^(x/2).  (Now at 2 hours, it is 4, at 4 hours, it is 16, ect).  I am assuming that the population starts at 1, but it doesn't have to.  Let's say the starting population at time 0 is P.  Then, the population at time x would be P*4^(x/2).  You can verify this for any starting population P>=0 and for any time x>=0.

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