The function \( n(t) = 50(1.2)^t \) describes the growth of bacteria over time, where:
- \( n(t) \) represents the number of bacteria,
- \( t \) represents time in hours,
- \( 50 \) represents the initial quantity of bacteria (at \( t = 0 \)),
- \( 1.2 \) is the growth factor per hour.
When graphing this function, we typically consider the following:
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Quadrant 1 of the Cartesian Plane: Quadrant 1 is defined as the area where both the x-axis (horizontal) and y-axis (vertical) have positive values: \( (x > 0, y > 0) \).
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Positive Values of \( t \): Since time \( t \) is generally non-negative in this context (we don't consider negative time for this scenario), \( t \) will be greater than or equal to 0 (i.e., \( t \geq 0 \)).
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Positive Values of \( n(t) \): The function \( n(t) = 50(1.2)^t \) will also always yield positive results for \( n(t) \):
- At \( t = 0 \): \[ n(0) = 50(1.2)^0 = 50(1) = 50, \]
- As \( t \) increases, \( (1.2)^t \) will also increase, and thus \( n(t) \) will always remain positive.
Given these points, the graph of the function will always lie in Quadrant 1. The values of time will be non-negative (initially 0 and increasing), and the values of the number of bacteria will also be positive, making it suitable to be plotted in the first quadrant.