Question

which table represents a linear function?

x	y
2	7
3	11
4	15

x	y
2	8
3	15
4	21

x	y
2	9
3	3
4	9

x	y
2	9
3	27

Explanation:

Step1: Recall linear function property

A linear function has a constant rate of change (slope), meaning the difference in \( y \)-values (\( \Delta y \)) for equal differences in \( x \)-values (\( \Delta x = 1 \) here) is constant.

Step2: Analyze first table

For \( x = 1,2,3,4 \) and \( y = 3,7,11,15 \):

• \( \Delta y \) between \( x=1 \) and \( x=2 \): \( 7 - 3 = 4 \)
• \( \Delta y \) between \( x=2 \) and \( x=3 \): \( 11 - 7 = 4 \)
• \( \Delta y \) between \( x=3 \) and \( x=4 \): \( 15 - 11 = 4 \)

Constant \( \Delta y = 4 \), so slope is constant.

Step3: Analyze second table

For \( x = 1,2,3,4 \) and \( y = 3,8,15,21 \):

• \( \Delta y \) (2 - 1): \( 8 - 3 = 5 \)
• \( \Delta y \) (3 - 2): \( 15 - 8 = 7 \) (not constant)

Step4: Analyze third table

For \( x = 1,2,3,4 \) and \( y = 3,9,3,9 \):

• \( \Delta y \) (2 - 1): \( 9 - 3 = 6 \)
• \( \Delta y \) (3 - 2): \( 3 - 9 = -6 \) (not constant)

Step5: Analyze fourth table (partial)

For \( x = 1,2,3 \) and \( y = 3,9,27 \):

• \( \Delta y \) (2 - 1): \( 9 - 3 = 6 \)
• \( \Delta y \) (3 - 2): \( 27 - 9 = 18 \) (not constant, exponential growth)

Answer:

The table with \( x: 1,2,3,4 \) and \( y: 3,7,11,15 \) (the first table) represents a linear function.