Question

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

Type of Function

This table presents the x and y values to identify the function that most accurately models the given dataset.
x y
1 10
2 100
3 1000
4 10000

Answer 1

To determine the type of function that best models the given data in the table, let's analyze the values provided:

```
x | y
---------
1 | 10
2 | 100
3 | 1000
4 | 10000
```

We can observe that as \( x \) increases, \( y \) also increases significantly. Let's look at the ratio of consecutive \( y \) values:

- From \( x = 1 \) to \( x = 2 \): \( \frac{100}{10} = 10 \)
- From \( x = 2 \) to \( x = 3 \): \( \frac{1000}{100} = 10 \)
- From \( x = 3 \) to \( x = 4 \): \( \frac{10000}{1000} = 10 \)

This indicates that the \( y \) values increase by a factor of 10 for each increase in \( x \) by 1. This suggests an exponential relationship.

We can express \( y \) in terms of \( x \):

\[
y = 10^x
\]

To confirm this fits the data:

- For \( x = 1 \): \( y = 10^1 = 10 \)
- For \( x = 2 \): \( y = 10^2 = 100 \)
- For \( x = 3 \): \( y = 10^3 = 1000 \)
- For \( x = 4 \): \( y = 10^4 = 10000 \)

All values match the dataset perfectly.

Thus, the type of function that best models the data is an **exponential function** given by

\[
y = 10^x.
\]