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.
