Question

Which type of function best models the data in the table?
x y
2 0.5
5 -1.75
8 -4
(1 point)
linear
quadratic
exponential
cubic

Answer 1

To determine the type of function that best models the data provided in the table, we can analyze the relationship between the \( x \) and \( y \) values.

The given data points are:
- \((2, 0.5)\)
- \((5, -1.75)\)
- \((8, -4)\)

1. **Linear Function**: A linear function has a constant rate of change. We can check if the differences in \( y \) divided by the differences in \( x \) remain constant.

- From \( (2, 0.5) \) to \( (5, -1.75) \):
\[
\frac{-1.75 - 0.5}{5 - 2} = \frac{-2.25}{3} = -0.75
\]

- From \( (5, -1.75) \) to \( (8, -4) \):
\[
\frac{-4 - (-1.75)}{8 - 5} = \frac{-4 + 1.75}{3} = \frac{-2.25}{3} = -0.75
\]

The rate of change is consistent; however, we should consider higher-order models as well since the nature of the changes could suggest curvature.

2. **Quadratic Function**: A quadratic function has the form \( y = ax^2 + bx + c \). We can check for a constant second difference:

- The \( y \) values are \( 0.5, -1.75, -4 \). We calculate the first differences:
- From \( 0.5 \) to \( -1.75 \): \( -1.75 - 0.5 = -2.25 \)
- From \( -1.75 \) to \( -4 \): \( -4 - (-1.75) = -2.25 \)
- First differences: \(-2.25, -2.25\) (constant)

Next, we calculate the second differences:
- The second difference is the same: \( -2.25 - (-2.25) = 0 \) (constant)

Since the second difference is constant, this suggests the data could be modeled well by a quadratic function.

3. **Exponential and Cubic Functions**: Generally refer to non-linear relationships that tend to either demonstrate exponential growth/decay or more complex curvature with a maximum or minimum point. The provided data does not suggest such behavior.

### Conclusion

Based on the differences observed:

- The first differences are consistent, and the second differences are constant, indicating that the data is best modeled by a **quadratic function**.