Which table represents a linear Function?

Let’s check each table for a constant rate of change (slope).

First table

(1,1),(2,1),(3,1.5),(4,2)(1, 1), (2, 1), (3, 1.5), (4, 2)

• Rate of change:
  • 1−12−1=0\frac{1 – 1}{2 – 1} = 0
  • 1.5−13−2=0.5\frac{1.5 – 1}{3 – 2} = 0.5
  • 2−1.54−3=0.5\frac{2 – 1.5}{4 – 3} = 0.5
• Not constant → Not linear

Second table

(1,1),(2,12),(3,13),(4,14)(1,1), (2, \frac{1}{2}), (3, \frac{1}{3}), (4, \frac{1}{4})

• The y-values are fractions decreasing differently → Not linear

Third table

(1,7),(2,9),(3,13),(4,21)(1, 7), (2, 9), (3, 13), (4, 21)

• Rate of change:
  • 9−72−1=2\frac{9 – 7}{2 – 1} = 2
  • 13−93−2=4\frac{13 – 9}{3 – 2} = 4
  • 21−134−3=8\frac{21 – 13}{4 – 3} = 8
• Not constant → Not linear

Fourth table

(1,0),(2,6),(3,16),(4,30)(1, 0), (2, 6), (3, 16), (4, 30)

• Rate of change:
  • 6−02−1=6\frac{6 – 0}{2 – 1} = 6
  • 16−63−2=10\frac{16 – 6}{3 – 2} = 10
  • 30−164−3=14\frac{30 – 16}{4 – 3} = 14
• Not constant → Not linear

Conclusion

None of the tables are linear.