Question

Which table of values represents a linear function?
A
xx yy
00 33
33 44
66 66
99 77
B
xx yy
minus, 3−3 99
minus, 1−1 66
11 44
55 minus, 2−2
C
xx yy
minus, 4−4 minus, 6−6
minus, 2−2 minus, 3−3
11 00
55 66
D
xx yy
minus, 9−9 minus, 4−4
minus, 4−4 minus, 2−2
11 00
66 22
Answer
Multiple Choice Answers
AA
BB
CC
DD

Answer 1

To determine which table of values represents a linear function, we can check if the change in \( y \) (the output) is proportional to the change in \( x \) (the input). In other words, if for every equal increment in \( x \), \( y \) changes by a constant amount, then the relationship is linear.

Let's analyze each table:

**Table A:**
```
x y
0 3
3 4
6 6
9 7
```
Calculating the differences:
- From \( (0, 3) \) to \( (3, 4) \): change in \( x = 3 \), change in \( y = 4 - 3 = 1 \)
- From \( (3, 4) \) to \( (6, 6) \): change in \( x = 3 \), change in \( y = 6 - 4 = 2 \)
- From \( (6, 6) \) to \( (9, 7) \): change in \( x = 3 \), change in \( y = 7 - 6 = 1 \)

Since the changes in \( y \) are not constant, this table does not represent a linear function.

**Table B:**
```
x y
-3 9
-1 6
1 4
5 -2
```
Calculating the differences:
- From \( (-3, 9) \) to \( (-1, 6) \): change in \( x = 2 \), change in \( y = 6 - 9 = -3 \)
- From \( (-1, 6) \) to \( (1, 4) \): change in \( x = 2 \), change in \( y = 4 - 6 = -2 \)
- From \( (1, 4) \) to \( (5, -2) \): change in \( x = 4 \), change in \( y = -2 - 4 = -6 \)

The changes in \( y \) are also not constant, so this table does not represent a linear function.

**Table C:**
```
x y
-4 -6
-2 -3
1 0
5 6
```
Calculating the differences:
- From \( (-4, -6) \) to \( (-2, -3) \): change in \( x = 2 \), change in \( y = -3 - (-6) = 3 \)
- From \( (-2, -3) \) to \( (1, 0) \): change in \( x = 3 \), change in \( y = 0 - (-3) = 3 \)
- From \( (1, 0) \) to \( (5, 6) \): change in \( x = 4 \), change in \( y = 6 - 0 = 6 \)

The changes in \( y \) are not consistent, so this table does not represent a linear function.

**Table D:**
```
x y
-9 -4
-4 -2
1 0
6 2
```
Calculating the differences:
- From \( (-9, -4) \) to \( (-4, -2) \): change in \( x = 5 \), change in \( y = -2 - (-4) = 2 \)
- From \( (-4, -2) \) to \( (1, 0) \): change in \( x = 5 \), change in \( y = 0 - (-2) = 2 \)
- From \( (1, 0) \) to \( (6, 2) \): change in \( x = 5 \), change in \( y = 2 - 0 = 2 \)

Since the changes in \( y \) are constant (2 for an increment of 5 in \( x \)), this table **does represent a linear function.**

**Answer:** D