To determine whether the given table represents a linear function, a quadratic function, or an exponential function, we need to analyze the differences between the [tex]\( y \)[/tex]-values over the equally spaced [tex]\( x \)[/tex]-values.
Here’s the table provided:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & -9 \\
\hline
1 & -4 \\
\hline
2 & 5 \\
\hline
3 & 18 \\
\hline
4 & 35 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Calculate the First Differences:
The first differences are obtained by subtracting each [tex]\( y \)[/tex]-value from the next [tex]\( y \)[/tex]-value.
[tex]\[
\begin{align*}
y_1 - y_0 &= -4 - (-9) = 5 \\
y_2 - y_1 &= 5 - (-4) = 9 \\
y_3 - y_2 &= 18 - 5 = 13 \\
y_4 - y_3 &= 35 - 18 = 17 \\
\end{align*}
\][/tex]
So, the first differences are: 5, 9, 13, and 17.
2. Calculate the Second Differences:
The second differences are calculated by subtracting each first difference from the next first difference.
[tex]\[
\begin{align*}
9 - 5 &= 4 \\
13 - 9 &= 4 \\
17 - 13 &= 4 \\
\end{align*}
\][/tex]
So, the second differences are 4, 4, and 4, which are constant.
### Conclusion:
- Since the second differences are constant, this indicates that the function is quadratic.
- Linear functions have constant first differences.
- Exponential functions have constant ratios between consecutive [tex]\( y \)[/tex]-values, which is not the case here.
Therefore, the function represented by the given table is a quadratic function.
