To determine whether the given table of values represents a linear or exponential function, we need to analyze the relationship between the [tex]\(x\)[/tex]-values and the [tex]\(y\)[/tex]-values. A linear function will have a constant rate of change, meaning the difference between consecutive [tex]\(y\)[/tex]-values should be consistent. In contrast, an exponential function will exhibit multiplication by a common ratio between consecutive [tex]\(y\)[/tex]-values.
Let's proceed step-by-step:
1. Record the [tex]\(x\)[/tex]-values and [tex]\(y\)[/tex]-values:
[tex]\[
\begin{align*}
x & : -2, -1, 0, 1, 2 \\
y & : 7, 4, 1, -2, -5 \\
\end{align*}
\][/tex]
2. Compute the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[
\begin{align*}
y_{-1} - y_{-2} & = 4 - 7 = -3 \\
y_{0} - y_{-1} & = 1 - 4 = -3 \\
y_{1} - y_{0} & = -2 - 1 = -3 \\
y_{2} - y_{1} & = -5 - (-2) = -3 \\
\end{align*}
\][/tex]
3. Determine if the differences are constant:
From the calculations above, the differences between consecutive [tex]\(y\)[/tex]-values are:
[tex]\[
-3, -3, -3, -3
\][/tex]
Since all these differences are equal, the rate of change is consistent.
4. Conclusion:
The constant differences indicate that the [tex]\(y\)[/tex]-values change by a fixed amount as [tex]\(x\)[/tex] increases. This characteristic is typical of a linear function, not an exponential function.
Therefore, the table of values represents a linear function.
