Please answer "PATTERN RECOGNITION". The other pictures are related to the activity.

Mathematical Modeling A mathematical model is a representation of a real-world situation using mathematical objects or symbols in order to study relationships and, if possible, predict outcomes that | play a significant role in the problem. A function is a mathematical model that is very much useful in various fields such as in biology, chemistry, economics, psychology, engineering, physics, and others. In this unit, you have learned the important concepts of functions and you studied special kinds of functions. You will use your learnings to formulate a mathematical model for a widely studied research area in biology: bacterial growth. Vibrio natriegens is a species of bacteria found in mud estuaries. An experiment was - conducted to observe the population growth of this bacteria in a nutrient growth medium whose pH was adjusted to 6.25. The experimental data are shown in the table below. Table 1. Population growth of V. natriegens over time (Lifted from "Math Insight") Time (per 16-minute interval) Population Density 0 0.022 0.036 N 0.060 W 0.101 4 0.169 5 0.266Pattern Recognition Explore the dataset. Look for and write down in the box below patterns that will enable you to create the general formula or function for the problem. (Hint: See first example under the 'Exponential and Logarithmic Functions' section.)Exponential and Logarithmic Functions One of the special kinds of functions is the exponential function. It is a one-to-one function and thus, has an inverse which is also specially named as logarithmic function. These two functions have wide real-world applications. Example: In a small village in Africa, the rate of acquiring the flu virus was studied. It was observed that at the start of the epidemic, only 1 person had the virus. However, for every subsequent week, the number of people who acquired the virus quadrupled. so that a week after the onset of the disease. 4 people already had the virus. On the second week, a total of 16 people acquired the virus. And this pattern continued. Let f(t) be the number of people who acquired the flu virus at week f. During the start of the epidemic, f(0) = 1 since 1 person initially had the virus. One week after (t = 1). the number of people infected quadrupled. Thus, f(1) = 4((0)) = 4(1) = 4 In other words, the number of people infected was four times the initial number. On the second week, the number quadrupled again such that f(2) = 40(1)) Substituting f(1) = 4(f(0)), /(2) = 4(4(f(0))) = 4(4(1)) = 16. We could then expect that on the third week, the number of people infected will be f(3) = 4(f(2)) = 4(4(1(1))) 33and since f(1) = 4(f(0)) f(3) = 4(4(4(f(0)))) = 4(4(4(1))) = 64. Observe that the function is a power of 4. Particularly, the time, which is the number of weeks that passed after the onset of flu, serves as the exponent of 4 that results in the number of people already infected by the virus. Hence, the function that would represent the epidemic is f (1) = 4% Use this formula and verify whether this function definition matches the values we computed manually