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Decide whether the data in the table represent a linear function or an exponential function. Explain how you know.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 4 \\
\hline
2 & -5 \\
\hline
3 & -14 \\
\hline
4 & -23 \\
\hline
5 & -32 \\
\hline
\end{tabular}

A. The data represent an exponential function because there is a common ratio of 4.

B. The data represent a linear function because there is a common difference of -9.

C. The data represent an exponential function because there is a common ratio of [tex]$-\frac{5}{4}$[/tex].

D. The data represent a linear function because there is a common difference of 9.

Sagot :

GinnyAnswer
To determine whether the data represents a linear function or an exponential function, we need to analyze the Y-values corresponding to the given X-values in the table.

We are given the following pairs:
[tex]\[ (1, 4), (2, -5), (3, -14), (4, -23), (5, -32) \][/tex]

### Step 1: Calculate the Differences Between Successive Y-values

We will find the difference between each successive Y-value:

[tex]\[ \Delta y_1 = y_2 - y_1 = -5 - 4 = -9 \][/tex]
[tex]\[ \Delta y_2 = y_3 - y_2 = -14 - (-5) = -14 + 5 = -9 \][/tex]
[tex]\[ \Delta y_3 = y_4 - y_3 = -23 - (-14) = -23 + 14 = -9 \][/tex]
[tex]\[ \Delta y_4 = y_5 - y_4 = -32 - (-23) = -32 + 23 = -9 \][/tex]

### Step 2: Determine If the Differences Are Constant

If the differences between successive Y-values are constant, then the data represents a linear function. From the calculations above, we can see that:

[tex]\[ \Delta y_1 = -9 \][/tex]
[tex]\[ \Delta y_2 = -9 \][/tex]
[tex]\[ \Delta y_3 = -9 \][/tex]
[tex]\[ \Delta y_4 = -9 \][/tex]

The differences are constant and equal to [tex]\(-9\)[/tex].

### Step 3: Conclusion

A common difference of [tex]\(-9\)[/tex] indicates that the function is linear. Therefore, the correct answer is:

B. The data represent a linear function because there is a common difference of -9.
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