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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

1 answer

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