Question

for the given function, use your intuition or additional research, if necessary, to complete parts (a) through (c). (time, number of bacteria), where time is measured in hours of a day a. describe an appropriate domain and range for the function. b. make a rough sketch of a graph of the function. c. briefly discuss the validity of the graph as a model of the true function. b. choose the correct graph. let t represent time in hours and b represent number of bacteria. a. b. c. d.

Explanation:

To determine the correct graph for the function relating time (in hours) to the number of bacteria, we analyze the axes:

• The independent variable (time, \( t \)) should be on the x - axis, and the dependent variable (number of bacteria, \( b \)) on the y - axis.
• Bacterial growth is typically exponential, so the graph of the number of bacteria against time should show an increasing curve, often exponential in nature (starting with a slower growth and then accelerating, or a consistent exponential rise).
• Option A: Has time on the y - axis and bacteria on the x - axis, which is incorrect as time is the independent variable and should be on the x - axis.
• Option B: Has time (\( t \)) on the x - axis and number of bacteria (\( b \)) on the y - axis. The curve shows an increasing trend, consistent with bacterial growth (exponential or otherwise increasing over time).
• Option C: Has time on the x - axis but the scale (1900 - 2010) is not relevant for a daily (0 - 24 hours) time scale for bacterial growth.
• Option D: Has time on the y - axis and bacteria on the x - axis, which is incorrect.

Answer:

B. The graph with \( t \) (hours) on the x - axis, \( b \) (bacteria in millions) on the y - axis, and an increasing curve representing bacterial growth over time.