Question

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