Question
A company launches two new products. The market price, in dollars, of the two products after a different number of years, x, is shown in the following table:
Product Function Year 1
(dollars) Year 2
(dollars) Year 3
(dollars)
Product 1 g(x) = 2x 2 4 8
Product 2 h(x) = x2 + 12 13 16 21
Based on the data in the table, for which product does the price eventually exceed all others, and why?
Product 1, because it has a lower start value
Product 2, because it has a greater Year 3 value
Product 1, because the function is exponential
Product 2, because the function is exponential
Product Function Year 1
(dollars) Year 2
(dollars) Year 3
(dollars)
Product 1 g(x) = 2x 2 4 8
Product 2 h(x) = x2 + 12 13 16 21
Based on the data in the table, for which product does the price eventually exceed all others, and why?
Product 1, because it has a lower start value
Product 2, because it has a greater Year 3 value
Product 1, because the function is exponential
Product 2, because the function is exponential
Answers
GPT-4o mini
To analyze which product's price will eventually exceed all others, we can look at the given functions and their growth rates.
1. **Product 1**: \( g(x) = 2x \)
- Year 1: \( g(1) = 2 \cdot 1 = 2 \)
- Year 2: \( g(2) = 2 \cdot 2 = 4 \)
- Year 3: \( g(3) = 2 \cdot 3 = 6 \)
The price follows a linear growth pattern.
2. **Product 2**: \( h(x) = x^2 + 12 \)
- Year 1: \( h(1) = 1^2 + 12 = 13 \)
- Year 2: \( h(2) = 2^2 + 12 = 16 \)
- Year 3: \( h(3) = 3^2 + 12 = 21 \)
The price follows a quadratic growth pattern.
### Analysis:
- Product 1 has linear growth, which means it will continue to increase steadily, but the rate of increase is constant.
- Product 2 has quadratic growth, which means that the price will increase at a faster rate as \( x \) increases.
### Conclusion:
As \( x \) (the number of years) increases, the values of \( h(x) \) for Product 2 will eventually exceed the values of \( g(x) \) for Product 1 due to the quadratic nature of Product 2's growth compared to the linear growth of Product 1.
Therefore, the correct answer is:
**Product 2, because the function is quadratic (and grows faster than the linear function of Product 1).**
From the answer choices provided, there is no option that specifically states that Product 2's growth is quadratic; however, we know that the exponential mention in the options seems to be a confusion because Product 1 grows linearly and Product 2 grows quadratically. The closest appropriate reasoning would still lean towards recognizing that Product 2 eventually exceeds the growth of Product 1.
1. **Product 1**: \( g(x) = 2x \)
- Year 1: \( g(1) = 2 \cdot 1 = 2 \)
- Year 2: \( g(2) = 2 \cdot 2 = 4 \)
- Year 3: \( g(3) = 2 \cdot 3 = 6 \)
The price follows a linear growth pattern.
2. **Product 2**: \( h(x) = x^2 + 12 \)
- Year 1: \( h(1) = 1^2 + 12 = 13 \)
- Year 2: \( h(2) = 2^2 + 12 = 16 \)
- Year 3: \( h(3) = 3^2 + 12 = 21 \)
The price follows a quadratic growth pattern.
### Analysis:
- Product 1 has linear growth, which means it will continue to increase steadily, but the rate of increase is constant.
- Product 2 has quadratic growth, which means that the price will increase at a faster rate as \( x \) increases.
### Conclusion:
As \( x \) (the number of years) increases, the values of \( h(x) \) for Product 2 will eventually exceed the values of \( g(x) \) for Product 1 due to the quadratic nature of Product 2's growth compared to the linear growth of Product 1.
Therefore, the correct answer is:
**Product 2, because the function is quadratic (and grows faster than the linear function of Product 1).**
From the answer choices provided, there is no option that specifically states that Product 2's growth is quadratic; however, we know that the exponential mention in the options seems to be a confusion because Product 1 grows linearly and Product 2 grows quadratically. The closest appropriate reasoning would still lean towards recognizing that Product 2 eventually exceeds the growth of Product 1.