Question

which table represents a linear function?

x	y
1	5
2	9
3	5
4	9

x	y
1	-5
2	10
3	-15
4	20

x	y
1	5
2	10
3	20
4	40

x	y
1	-5
2	0
3	5

Explanation:

To determine which table represents a linear function, we check the rate of change (slope) between consecutive points. A linear function has a constant slope.

Step 1: Analyze the first table

For \( x = 1 \) to \( x = 2 \): \( \Delta y = 9 - 5 = 4 \), \( \Delta x = 2 - 1 = 1 \), slope \( m = \frac{4}{1} = 4 \)
For \( x = 2 \) to \( x = 3 \): \( \Delta y = 5 - 9 = -4 \), \( \Delta x = 3 - 2 = 1 \), slope \( m = \frac{-4}{1} = -4 \)
Slopes are not constant, so not linear.

Step 2: Analyze the second table

For \( x = 1 \) to \( x = 2 \): \( \Delta y = 10 - (-5) = 15 \), \( \Delta x = 2 - 1 = 1 \), slope \( m = \frac{15}{1} = 15 \)
For \( x = 2 \) to \( x = 3 \): \( \Delta y = -15 - 10 = -25 \), \( \Delta x = 3 - 2 = 1 \), slope \( m = \frac{-25}{1} = -25 \)
Slopes are not constant, so not linear.

Step 3: Analyze the third table

For \( x = 1 \) to \( x = 2 \): \( \Delta y = 10 - 5 = 5 \), \( \Delta x = 2 - 1 = 1 \), slope \( m = \frac{5}{1} = 5 \)
For \( x = 2 \) to \( x = 3 \): \( \Delta y = 20 - 10 = 10 \), \( \Delta x = 3 - 2 = 1 \), slope \( m = \frac{10}{1} = 10 \)
Slopes are not constant, so not linear.

Step 4: Analyze the fourth table (assuming the last table is: \( x: 1, 2, 3; y: -5, 0, 5 \))

For \( x = 1 \) to \( x = 2 \): \( \Delta y = 0 - (-5) = 5 \), \( \Delta x = 2 - 1 = 1 \), slope \( m = \frac{5}{1} = 5 \)
For \( x = 2 \) to \( x = 3 \): \( \Delta y = 5 - 0 = 5 \), \( \Delta x = 3 - 2 = 1 \), slope \( m = \frac{5}{1} = 5 \)
Slope is constant (5), so this is linear.

Answer:

The table with \( x = 1, 2, 3 \) and \( y = -5, 0, 5 \) (the last table shown, though partially, but following the pattern) represents a linear function. Assuming the full table is:

\( x \)	\( y \)
2	0
3	5
4	10	(extending the pattern)

This table has a constant slope of 5, so it represents a linear function.