Answered

In Exercises 5 and 6, tell whether the table of values represents a linear or an exponential function.

5.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -2 & -1 & 0 & 1 & 2 \\
\hline
[tex]$y$[/tex] & 7 & 4 & 1 & -2 & -5 \\
\hline
\end{tabular}



Answer :

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.