Question

the data in the table represents the predicted price of a gallon of milk, \\(y\\), for number of years, \\(x\\).

price of milk
\

$$\begin{array}{|c|c|} \\hline \\text{years} & \\text{price} \\\\ \\hline 0 & \\$3.00 \\\\ \\hline 1 & \\$3.12 \\\\ \\hline 2 & \\$3.24 \\\\ \\hline 3 & \\$3.37 \\\\ \\hline 4 & \\$3.51 \\\\ \\hline \\end{array}$$

which form of an equation was used to model the price of a gallon of milk?

\\(y = ax + b\\)
\\(y = ax^2 + b\\)
\\(y = ab^x\\)
\\(y = a + bx\\)

Explanation:

Analyze first differences of the data

Using the Finite Differences knowledge point

$$
LATEXBLOCK0
$$

Analyze ratios of consecutive terms

Using the Exponential Modeling knowledge point

$$
LATEXBLOCK1
$$

Determine the best-fitting model

Using the Polynomial Model Selection knowledge point
The constant ratio of approximately \(1.04\) (a \(4\%\) annual growth rate) indicates that the price of milk grows exponentially over time. Thus, the exponential function model \(y = ab^x\) is the most appropriate equation form.

Answer:

• \(y = ax + b\)
• \(y = ax^2 + b\)
• \(y = ab^x\) (Correct answer)
• \(y = a + bx\)