Question

Question
Classify the function represented by the input-output table.

Function table with values (-1, 1), (0, -1), (1, 1), and (2, 7)

Responses

Exponential

Quadratic

Absolute Value

Linear

Answer 1

To classify the function represented by the input-output table with values (-1, 1), (0, -1), (1, 1), and (2, 7), we can analyze the behavior of the outputs based on the inputs.

1. **Linear Function**: A linear function has the form \( y = mx + b \). Check if this table forms a straight line:
- The slope between (-1, 1) and (0, -1) is:
\[
\frac{-1 - 1}{0 - (-1)} = \frac{-2}{1} = -2
\]
- The slope between (0, -1) and (1, 1) is:
\[
\frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2
\]
- The slope between (1, 1) and (2, 7) is:
\[
\frac{7 - 1}{2 - 1} = \frac{6}{1} = 6
\]

Since the slopes are not constant, this cannot be a linear function.

2. **Quadratic Function**: A quadratic function has the form \( y = ax^2 + bx + c \). The outputs seem to vary at an increasing rate, suggesting a possibility of a quadratic function. Let's examine the second differences:
- First differences:
- From (-1, 1) to (0, -1): -1 - 1 = -2
- From (0, -1) to (1, 1): 1 - (-1) = 2
- From (1, 1) to (2, 7): 7 - 1 = 6

First differences: -2, 2, 6

- Second differences:
- From -2 to 2: 2 - (-2) = 4
- From 2 to 6: 6 - 2 = 4

Second differences are constant (4), so this indicates a quadratic relationship.

3. **Exponential Function**: Exponential functions have a form where the rate of change is not constant (e.g., \( y = ab^x \)). The outputs change by non-linear increments, but they don't match the typical form of an exponential growth.

4. **Absolute Value Function**: Absolute functions tend to create V-shaped graphs but the increase in values doesn’t reflect that.

Based on the analysis, the function represented by the input-output table is classified as **Quadratic**.