The growth of a bacterial culture that doubles every hour can be represented mathematically by an exponential growth function. In this case, if the initial amount of bacteria \( N_0 \) is 10, the number of bacteria at time \( t \) (in hours) can be expressed by the equation:
\[ N(t) = N_0 \cdot 2^t \]
Substituting the initial population:
\[ N(t) = 10 \cdot 2^t \]
To visualize this on a graph:
- X-axis: This will represent time in hours (t).
- Y-axis: This will represent the number of bacteria (N).
When you plot this function, the values at key points in time would be:
- At \( t = 0 \), \( N(0) = 10 \cdot 2^0 = 10 \)
- At \( t = 1 \), \( N(1) = 10 \cdot 2^1 = 20 \)
- At \( t = 2 \), \( N(2) = 10 \cdot 2^2 = 40 \)
- At \( t = 3 \), \( N(3) = 10 \cdot 2^3 = 80 \)
- At \( t = 4 \), \( N(4) = 10 \cdot 2^4 = 160 \)
- At \( t = 5 \), \( N(5) = 10 \cdot 2^5 = 320 \)
As the time progresses, you'll observe that the graph will rise steeply, illustrating exponential growth.
The resulting graph will start at the point (0, 10) and rise sharply, curving upwards as time increases. The growth is not linear; instead, it accelerates as time goes on, showing a characteristic shape of exponential growth where the increase becomes more dramatic after each hour.