Which of these tables represents a linear​ function? Use pencil and paper to explain your reasoning.

Question

Table I
Input
Output
0
1
1
0
2
1
3
4
4
9
Table II
Input
Output
0
28
1
23
2
18
3
13
4
8
Question content area bottom
Part 1
Choose the correct answer below.
A.
Both Table I and Table II
B.
Neither Table I nor Table II
C.
Only Table II
D.
Only Table

Answer 1

To determine which table represents a linear function, we need to check for consistent changes between inputs and outputs.

A linear function has a constant rate of change, meaning that the difference in output values divided by the difference in input values remains the same between any two points.

Analysis of Table I:

Input (x) | Output (y)
-----------------------
   0     |    1  
   1     |    0  
   2     |    1  
   3     |    4  
   4     |    9  

Calculating the differences in output for each pair of inputs:

• From \(x = 0\) to \(x = 1\): \(y\) goes from \(1\) to \(0\) → change is \(-1\)
• From \(x = 1\) to \(x = 2\): \(y\) goes from \(0\) to \(1\) → change is \(1\)
• From \(x = 2\) to \(x = 3\): \(y\) goes from \(1\) to \(4\) → change is \(3\)
• From \(x = 3\) to \(x = 4\): \(y\) goes from \(4\) to \(9\) → change is \(5\)

The output does not change by a constant amount; therefore, Table I is not a linear function.

Analysis of Table II:

Input (x) | Output (y)
-----------------------
   0     |   28  
   1     |   23  
   2     |   18  
   3     |   13  
   4     |   8  

Calculating the differences in output for each pair of inputs:

• From \(x = 0\) to \(x = 1\): \(y\) goes from \(28\) to \(23\) → change is \(-5\)
• From \(x = 1\) to \(x = 2\): \(y\) goes from \(23\) to \(18\) → change is \(-5\)
• From \(x = 2\) to \(x = 3\): \(y\) goes from \(18\) to \(13\) → change is \(-5\)
• From \(x = 3\) to \(x = 4\): \(y\) goes from \(13\) to \(8\) → change is \(-5\)

The outputs change by a constant amount of \(-5\). Therefore, Table II represents a linear function.

Conclusion:

The correct answer is: D. Only Table II