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A company launches two new products. The market price, in dollars, of the two products after a different number of years, [tex]\(x\)[/tex], is shown in the following table:

[tex]\[
\begin{tabular}{|l|l|l|l|l|}
\hline
Product & Function & Year 1 (dollars) & Year 2 (dollars) & Year 3 (dollars) \\
\hline
Product 1 & \(g(x) = x^2 + 11\) & 12 & 15 & 20 \\
\hline
Product 2 & \(h(x) = 4x\) & 4 & 16 & 64 \\
\hline
\end{tabular}
\][/tex]

Based on the data in the table, for which product does the price eventually exceed all others, and why?

A. Product 1, because it has a greater start value
B. Product 1, because the function is exponential
C. Product 2, because it has a higher Year 3 value
D. Product 2, because the function is exponential



Answer :

GinnyAnswer
Let's examine the given information carefully to determine which product's price eventually exceeds all others and why.

The table provided shows the prices of two products across three years:

[tex]\[ \begin{tabular}{|l|l|l|l|l|} \hline \text{Product} & \text{Function} & \begin{tabular}{c} \text{Year 1} \\ \text{(dollars)} \end{tabular} & \begin{tabular}{c} \text{Year 2} \\ \text{(dollars)} \end{tabular} & \begin{tabular}{c} \text{Year 3} \\ \text{(dollars)} \end{tabular} \\ \hline \text{Product 1} & \( g(x) = x^2 + 11 \) & 12 & 15 & 20 \\ \hline \text{Product 2} & \( h(x) = 4x \) & 4 & 16 & 64 \\ \hline \end{tabular} \][/tex]

Step-by-Step Analysis:

1. Identify Functions and Yearly Values:

- Product 1: [tex]\( g(x) = x^2 + 11 \)[/tex]
- Product 2: [tex]\( h(x) = 4x \)[/tex]

- Yearly values for Product 1 are:
- Year 1: 12 dollars
- Year 2: 15 dollars
- Year 3: 20 dollars

- Yearly values for Product 2 are:
- Year 1: 4 dollars
- Year 2: 16 dollars
- Year 3: 64 dollars

2. Compare Yearly Values Directly:

- Year 1:
- Product 1: 12 dollars
- Product 2: 4 dollars
- Product 1 has the higher value.

- Year 2:
- Product 1: 15 dollars
- Product 2: 16 dollars
- Product 2 has the higher value.

- Year 3:
- Product 1: 20 dollars
- Product 2: 64 dollars
- Product 2 has the higher value.

3. Analyze the Growth Functions:

- The function [tex]\( g(x) = x^2 + 11 \)[/tex] for Product 1 is quadratic, meaning it grows at a polynomial rate.
- The function [tex]\( h(x) = 4x \)[/tex] for Product 2 is linear, meaning it grows at a constant rate.

4. Determine which product's price eventually exceeds all others:

Based on the values from the table:
- In Year 3, Product 2 has a value of 64 dollars, which is significantly higher than Product 1's value of 20 dollars.

Therefore, the analysis shows that:

Product 2 eventually exceeds Product 1 in price due to its higher value in the later years.

Conclusion:

Product 2, because it has a higher Year 3 value.

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