The function \( b(t) = 300(1.6)^t \) represents the number of bacteria after \( t \) hours, where \( t \) is the time in hours.
To understand why this function cannot be graphed in quadrant 2, we need to look at the characteristics of the function and the quadrants of the Cartesian coordinate system:
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Quadrant 2: In the second quadrant, the x-coordinates (which represent the time \( t \) in this case) are negative, and the y-coordinates (the number of bacteria \( b(t) \)) are positive.
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Function Behavior: The variable \( t \) in this function represents time in hours and cannot be negative, as negative time does not have a physical interpretation in this context. Therefore, \( t \geq 0 \).
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Number of Bacteria: The function \( b(t) \) produces only positive values for all non-negative values of \( t \). For example, when \( t = 0 \), \( b(0) = 300(1.6)^0 = 300 \); when \( t = 1 \), \( b(1) = 300(1.6)^1 = 480 \), and so on.
Given these points, since the function \( b(t) \) only exists for \( t \geq 0 \) (which corresponds to the right side of the y-axis) and outputs positive values (which corresponds to the upper half of the graph), it cannot be represented in quadrant 2 where \( t < 0 \) (negative x-values) and \( b(t) \) would have to be positive (upper part).
Thus, the function cannot be graphed in quadrant 2 because the variable \( t \) (time) cannot take negative values.