Determine whether the following table represents a linear or an exponential function. Explain why or why not.

Question

x
0
1
2
3
y
7
9
11
13
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Part 1
Does the table represent a linear or an exponential​ function? Why or why​ not?
A.
​Exponential; all of the​ x-values have a common difference and all of the​ y-values have a common ratio.
B.
​Linear; all of the​ x-values have a common difference and all of the​ y-values have a common difference.
C.
​Exponential; all of the​ x-values have a common ratio and all of the​ y-values do not have a common difference.
D.
​Linear; all of the​ x-values have a common difference and all of the​ y-values have a common ratio.

Answer 1

To determine whether the given table represents a linear or an exponential function, we need to analyze the differences in the x-values and y-values.

Table:

x:  0, 1, 2, 3
y:  7, 9, 11, 13

1. Identify the common differences:

   • The x-values increase by a constant amount of 1 (common difference of 1).
   • For the y-values:
     • From 7 to 9: increase of 2
     • From 9 to 11: increase of 2
     • From 11 to 13: increase of 2
   • The difference between consecutive y-values is consistent (also a common difference of 2).

2. Determine the function type:

   • A function is linear if the differences between the y-values are constant for equal increments in x (which has been observed here).
   • A function is exponential if the ratios of the y-values for equal increments in x are constant, but that is not the case here.

From the analysis above, we see that:

• All of the x-values have a common difference (1).
• All of the y-values have a common difference (2).

Therefore, the correct choice is:

B. Linear; all of the x-values have a common difference and all of the y-values have a common difference.