Question

1. the table below shows bacteria growth over time.

which regression model best fits the data?
a) linear
b) quadratic
c) exponential

Explanation:

Step1: Analyze linear model

A linear model has a constant rate of change. The differences between consecutive population values: \(20 - 10=10\), \(40 - 20 = 20\), \(80 - 40=40\). The differences are not constant.

Step2: Analyze quadratic model

A quadratic model has a second - order constant rate of change. Let \(y = ax^{2}+bx + c\). For \(x = 0,y = 10\) (so \(c = 10\)). For \(x=1,y=20\): \(a + b+10 = 20\) (i.e., \(a + b=10\)). For \(x = 2,y = 40\): \(4a+2b + 10=40\) (i.e., \(4a + 2b=30\)). Solving \(

$$\begin{cases}a + b=10\\4a + 2b=30\end{cases}$$

\), from the first equation \(b = 10 - a\), substitute into the second: \(4a+2(10 - a)=30\), \(4a + 20-2a=30\), \(2a=10\), \(a = 5\), \(b = 5\). Check for \(x = 3\): \(y=5\times3^{2}+5\times3 + 10=45 + 15+10=70
eq80\).

Step3: Analyze exponential model

An exponential model is of the form \(y = ab^{x}\). When \(x = 0,y = 10\), so \(a = 10\). When \(x = 1,y = 20\), \(10b^{1}=20\), \(b = 2\). The model is \(y = 10\times2^{x}\). Check for \(x = 2\): \(y=10\times2^{2}=40\). Check for \(x = 3\): \(y=10\times2^{3}=80\).

Answer:

C. Exponential