Question

Which type of function best models the data in the table?
Type of Function
x y
1 26
4 86
7 146
A: quadratic B: exponential C: linear

Answer 1

To determine which type of function best models the given data, we need to analyze the relationship between the x and y values from the table:

```
x | y
-----------
1 | 26
4 | 86
7 | 146
```

1. **Linear Function**: A linear function has a constant rate of change, meaning that the difference in y-values divided by the difference in x-values (the slope) should remain constant. Let's calculate the differences:

From (1, 26) to (4, 86):
\[
\Delta y = 86 - 26 = 60 \quad \text{and} \quad \Delta x = 4 - 1 = 3
\]
Thus, the slope is \( \frac{60}{3} = 20 \).

From (4, 86) to (7, 146):
\[
\Delta y = 146 - 86 = 60 \quad \text{and} \quad \Delta x = 7 - 4 = 3
\]
The slope is still \( \frac{60}{3} = 20 \).

Since the slope is constant, the data could potentially be linear, but we need to check further.

2. **Quadratic Function**: A quadratic function would show a varying rate of change. Let's check the second differences:

First differences:
- From 26 to 86: \( 60 \)
- From 86 to 146: \( 60 \)

The first differences are the same, which suggests it could be linear. For a quadratic function, we would expect non-constant first differences.

3. **Exponential Function**: An exponential function shows a multiplicative rate of change. We should check the ratios of successive y-values:
\[
\frac{y(4)}{y(1)} = \frac{86}{26} \approx 3.3077
\]
\[
\frac{y(7)}{y(4)} = \frac{146}{86} \approx 1.6988
\]
The ratios are not constant, which suggests it is not exponential.

### Conclusion:
Based on the analysis:

- The differences between the y-values suggest a linear relationship.
- The constant slope means that the best fit for this data is a linear function.

Thus, the answer is **C: linear**.