Question

Could a quadratic function model the data in the table below? Justify your answer.

Answer 1

A quadratic function could model the data if there is a consistent second difference in the y-values of the table.

Step-by-step explanation:

To determine if a quadratic function could model the data, we need to check for a consistent second difference in the y-values. If the second difference is constant, it indicates that the data may be modeled by a quadratic function. To find the second difference, we calculate the differences between consecutive first differences. If these second differences are constant, it suggests a quadratic relationship.

Consider the table below:

x y

1 2

2 5

3 10

4 17

5 26

​

The first differences in y are 3, 5, 7, and 9. The second differences are 2, 2, and 2, which are constant. This constant second difference indicates a quadratic relationship. Therefore, a quadratic function could indeed model the data in this table.

In conclusion, examining the second differences in the y-values is a straightforward method to determine if a quadratic function can model a given set of data. If the second differences are constant, it suggests a quadratic relationship, providing a mathematical justification for the modeling choice.

Answer 2

Fianl Answer:

No, a quadratic function cannot accurately model the data.

Step-by-step explanation:

The given table represents a set of data points where x values are 1, 2, 3, and 4, and the corresponding y values are 3, 7, 12, and 18. To check if a quadratic function can model this data, we need to analyze if the differences between consecutive y-values are constant. For a quadratic function (y = ax² + bx + c), the second differences between consecutive y-values should be constant. However, upon calculating the first and second differences for the given data, the second differences are not constant:

First differences: 4, 5, 6

Second differences: 1, 1

Since the second differences are not constant, the data cannot be accurately modeled by a quadratic function. The pattern in the given data doesn’t align with the behavior of a quadratic equation, implying that the relationship between x and y may not follow a quadratic pattern. Therefore, another type of function or model might better represent this dataset, as it doesn't satisfy the conditions required for a quadratic function.

Question:

Could a quadratic function model the data in the table below? Justify your answer.

| x | y |

|---|---|

| 1 | 3 |

| 2 | 7 |

| 3 | 12 |

| 4 | 18 |