Question

Use differences or ratios to tell whether the table of values represents a linear function, and exponential function, or a quadratic function.

x -2 -1 0 1 2
y -6 -6 -4 0 6

First differences:

second differences:

Answer 1

To determine the type of function represented by the given table of values, we will analyze the first and second differences of the \( y \) values.

### Given Data
\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & -6 \\
-1 & -6 \\
0 & -4 \\
1 & 0 \\
2 & 6 \\
\hline
\end{array}
\]

### First Differences
We will calculate the first differences, which are found by subtracting each \( y \) value from the next one.

1. \( -6 - (-6) = 0 \)
2. \( -6 - (-4) = -2 \)
3. \( -4 - 0 = -4 \)
4. \( 0 - 6 = -6 \)

Thus, the first differences are:
\[
0, -2, -4, -6
\]

### Second Differences
Now, we calculate the second differences by subtracting each first difference from the next one.

1. \( -2 - 0 = -2 \)
2. \( -4 - (-2) = -2 \)
3. \( -6 - (-4) = -2 \)

Thus, the second differences are:
\[
-2, -2, -2
\]

### Conclusion
- The **first differences** are not constant, which indicates this is not a linear function.
- The **second differences** are constant, which indicates the function is a quadratic function.

Therefore, the table of values represents a **quadratic function**.