Question

svlc algebra 1a - standard (15260)
introduction to linear functions
which table represents a linear function?

first table:

x	y
2	9
3	13
4	21

second table:

x	y
2	6
3	16
4	30

third table:

x	y
2	1
3	\\(1\frac{1}{2}\\)
4	2

fourth table:

x	y
2	\\(\frac{1}{2}\\)
3	\\(\frac{1}{3}\\)
4	\\(\frac{1}{4}\\)

Explanation:

Step1: Recall linear function property

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

Step2: Analyze Table 1 (Top - Left)

\( x = 1, y = 7 \); \( x = 2, y = 9 \): \( \Delta y = 9 - 7 = 2 \)
\( x = 2, y = 9 \); \( x = 3, y = 13 \): \( \Delta y = 13 - 9 = 4 \)
\( \Delta y \) is not constant. Not linear.

Step3: Analyze Table 2 (Top - Right)

\( x = 1, y = 0 \); \( x = 2, y = 6 \): \( \Delta y = 6 - 0 = 6 \)
\( x = 2, y = 6 \); \( x = 3, y = 16 \): \( \Delta y = 16 - 6 = 10 \)
\( \Delta y \) is not constant. Not linear.

Step4: Analyze Table 3 (Bottom - Left)

\( x = 1, y = \frac{1}{2} \); \( x = 2, y = 1 \): \( \Delta y = 1 - \frac{1}{2} = \frac{1}{2} \)
\( x = 2, y = 1 \); \( x = 3, y = 1\frac{1}{2} \): \( \Delta y = 1\frac{1}{2} - 1 = \frac{1}{2} \)
\( x = 3, y = 1\frac{1}{2} \); \( x = 4, y = 2 \): \( \Delta y = 2 - 1\frac{1}{2} = \frac{1}{2} \)
\( \Delta y \) is constant (\( \frac{1}{2} \)). Linear.

Step5: Analyze Table 4 (Bottom - Right)

\( x = 1, y = 1 \); \( x = 2, y = \frac{1}{2} \): \( \Delta y = \frac{1}{2} - 1 = -\frac{1}{2} \)
\( x = 2, y = \frac{1}{2} \); \( x = 3, y = \frac{1}{3} \): \( \Delta y = \frac{1}{3} - \frac{1}{2} = -\frac{1}{6} \)
\( \Delta y \) is not constant. Not linear.

Answer:

The table with \( x \) values 1, 2, 3, 4 and \( y \) values \( \frac{1}{2} \), 1, \( 1\frac{1}{2} \), 2 (bottom - left table) represents a linear function.