Question

which table represents a linear function?

x	y
2	6
3	12
4	24

x	y
2	5
3	9
4	14

x	y
2	-5
3	-7
4	-9

x	y
2	-4
3	-2
4	0

Explanation:

To determine which table represents a linear function, we check the rate of change (slope) between consecutive points. For a linear function, the slope should be constant. The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Since the \( x \)-values increase by 1 each time (\( x_2 - x_1 = 1 \)), we can simply check the difference in \( y \)-values (\( y_2 - y_1 \)) for constancy.

Step 1: Analyze the first table

• From \( x = 1 \) to \( x = 2 \): \( \Delta y = 6 - 3 = 3 \)
• From \( x = 2 \) to \( x = 3 \): \( \Delta y = 12 - 6 = 6 \)
• From \( x = 3 \) to \( x = 4 \): \( \Delta y = 24 - 12 = 12 \)

The differences are not constant (3, 6, 12), so this is not a linear function.

Step 2: Analyze the second table

• From \( x = 1 \) to \( x = 2 \): \( \Delta y = 5 - 2 = 3 \)
• From \( x = 2 \) to \( x = 3 \): \( \Delta y = 9 - 5 = 4 \)
• From \( x = 3 \) to \( x = 4 \): \( \Delta y = 14 - 9 = 5 \)

The differences are not constant (3, 4, 5), so this is not a linear function.

Step 3: Analyze the third table

• From \( x = 1 \) to \( x = 2 \): \( \Delta y = -5 - (-3) = -2 \)
• From \( x = 2 \) to \( x = 3 \): \( \Delta y = -7 - (-5) = -2 \)
• From \( x = 3 \) to \( x = 4 \): \( \Delta y = -9 - (-7) = -2 \)

The differences are constant (\(-2\) for each interval), so this is a linear function.

Step 4: Analyze the fourth table (for confirmation)

• From \( x = 1 \) to \( x = 2 \): \( \Delta y = -4 - (-2) = -2 \)
• From \( x = 2 \) to \( x = 3 \): \( \Delta y = -2 - (-4) = 2 \)
• From \( x = 3 \) to \( x = 4 \): \( \Delta y = 0 - (-2) = 2 \)

The differences are not constant (\(-2, 2, 2\)), so this is not a linear function.

Answer:

The third table (with \( x \) values 1, 2, 3, 4 and \( y \) values -3, -5, -7, -9) represents a linear function.