Question

Which type of function best models the data in the table?

x y
2 0.5
5 -1.75
8 -4
(1 point)
Responses

linear
linear

quadratic
quadratic

exponential
exponential

cubic

Answer 1

To determine which type of function best models the given data, we can analyze the relationship between the x and y values provided in the table:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
2 & 0.5 \\
5 & -1.75 \\
8 & -4 \\
\hline
\end{array}
\]

### Step 1: Look for a Linear Relationship
A linear function has a constant rate of change. We can calculate the slopes between the points:

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

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

Since the slopes between the points are consistent, this suggests a linear relationship.

### Step 2: Check for Quadratic Relationship
A quadratic function would show a second difference that is constant. Let's find the differences:

1. First differences:
- Between (2, 0.5) and (5, -1.75): \(-1.75 - 0.5 = -2.25\)
- Between (5, -1.75) and (8, -4): \(-4 + 1.75 = -2.25\)

The first differences are the same, which is consistent with a linear function rather than quadratic or cubic.

### Step 3: Check for Exponential or Cubic Relationships
Exponential functions would present a multiplicative growth or decay, while cubic functions would be dependent on third differences remaining constant. Since we already discerned a likely linear relationship, it's not necessary to test these options.

### Conclusion
Based on the analysis, the best function that models the data appears to be a linear function.

**Final Answer:** linear