Question

a biologist is researching the growth of a certain species of bacteria. the growth rate of the bacteria is shown in the data below, where t is the number of days and b is the number of bacteria. determine whether the data, (t,b), can be modeled by a linear, exponential, or a quadratic function and then select a function rule to model the situation. (0,0.74), (1,1.48), (2,2.96), (3,5.92), (4,11.84) a. $f(x) = 0.74x$ b. $f(x) = 0.74 \cdot 2^x$ c. $f(x) = 0.74x^2$

Explanation:

Step1: Comprobar la función lineal

Para una función lineal \(y = mx + b\) (en este caso \(a. f(x)=0.74x\), \(m = 0.74\), \(b = 0\)).
Si \(x = 1\), \(y=0.74\times1 = 0.74
eq1.48\).

Step2: Comprobar la función cuadrática

Para una función cuadrática \(y = ax^{2}+bx + c\) (en este caso \(c. f(x)=0.74x^{2}\)).
Si \(x = 1\), \(y = 0.74\times1^{2}=0.74
eq1.48\).

Step3: Comprobar la función exponencial

Para una función exponencial \(y = ab^{x}\) (en este caso \(b. f(x)=0.74\cdot2^{x}\)).
Si \(x = 0\), \(y=0.74\cdot2^{0}=0.74\).
Si \(x = 1\), \(y = 0.74\cdot2^{1}=1.48\).
Si \(x = 2\), \(y=0.74\cdot2^{2}=0.74\times4 = 2.96\).
Si \(x = 3\), \(y=0.74\cdot2^{3}=0.74\times8 = 5.92\).
Si \(x = 4\), \(y=0.74\cdot2^{4}=0.74\times16 = 11.84\).

Answer:

B. \(f(x)=0.74\cdot2^{x}\)