To determine what kind of function is depicted in the provided table, let's analyze the values:
- When \( x = -1 \), \( f(x) = 10 \)
- When \( x = 0 \), \( f(x) = 14 \)
- When \( x = 1 \), \( f(x) = 26 \)
- When \( x = 2 \), \( f(x) = 46 \)
- When \( x = 3 \), \( f(x) = 74 \)
Next, let's look at the changes in \( f(x) \):
- From \( x = -1 \) to \( x = 0 \): \( 14 - 10 = 4 \)
- From \( x = 0 \) to \( x = 1 \): \( 26 - 14 = 12 \)
- From \( x = 1 \) to \( x = 2 \): \( 46 - 26 = 20 \)
- From \( x = 2 \) to \( x = 3 \): \( 74 - 46 = 28 \)
Now, let's analyze the differences of the differences:
- From \( 4 \) to \( 12 \): \( 12 - 4 = 8 \)
- From \( 12 \) to \( 20 \): \( 20 - 12 = 8 \)
- From \( 20 \) to \( 28 \): \( 28 - 20 = 8 \)
Since the second differences are constant (equal to 8), this indicates that the function is quadratic.
Therefore, we can conclude:
This function is quadratic because the second differences are constant.