Question

determine whether the following table represents a linear or an exponential function. explain why or why not.

x	y
1	9
2	13
3	17

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. linear; all of the x - values have a common difference and all of the y - values have a common ratio
d. exponential; all of the x - values have a common ratio and all of the y - values do not have a common difference

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Explanation:

Step1: Check x - values' difference

The x - values are 0, 1, 2, 3. The difference between consecutive x - values: \(1 - 0=1\), \(2 - 1 = 1\), \(3 - 2=1\). So the x - values have a common difference of 1.

Step2: Check y - values' difference

The y - values are 5, 9, 13, 17. The difference between consecutive y - values: \(9 - 5 = 4\), \(13 - 9=4\), \(17 - 13 = 4\). So the y - values have a common difference of 4.

Step3: Determine function type

For a linear function, the x - values have a common difference and the y - values have a common difference. For an exponential function, the x - values have a common difference and the y - values have a common ratio. Since x - values have common difference (1) and y - values have common difference (4), it is a linear function. Now check the options:

• Option A: Says exponential with x having common difference and y having common ratio. But y has common difference, so A is wrong.
• Option B: Says linear, x has common difference and y has common difference. This matches our calculation.
• Option C: Says linear with x having common difference and y having common ratio. But y has common difference, so C is wrong.
• Option D: Says exponential with x having common ratio (x has common difference, not ratio) and y having no common difference (y has common difference). So D is wrong.

Answer:

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