Which table of values represents a linear function?

Question

A
xx yy
minus, 2−2 66
22 44
55 22
88 00
B
xx yy
minus, 9−9 minus, 8−8
minus, 3−3 minus, 4−4
33 minus, 1−1
99 22
C
xx yy
minus, 5−5 minus, 7−7
minus, 2−2 minus, 6−6
11 minus, 4−4
44 minus, 2−2
D
xx yy
minus, 5−5 44
minus, 2−2 11
11 minus, 2−2
44 minus, 5−5

Answer 1

To determine which table of values represents a linear function, we need to check if the change in \(y\) (the dependent variable) is consistent for a consistent change in \(x\) (the independent variable). This means examining whether the slopes between consecutive points are the same.

Let's analyze each table individually:

Table A

x   |  y
---------
-2  |  6
 2  |  4
 5  |  2
 8  |  0

Calculating the differences in \(y\) for each pair of \(x\):

• From \(-2\) to \(2\): \(4 - 6 = -2\)
• From \(2\) to \(5\): \(2 - 4 = -2\)
• From \(5\) to \(8\): \(0 - 2 = -2\)

The slope is constant \(-2\). Table A represents a linear function.

Table B

x   |  y
---------
-9  | -8
-3  | -4
 3  | -1
 9  |  2

Calculating the differences in \(y\):

• From \(-9\) to \(-3\): \(-4 - (-8) = 4\)
• From \(-3\) to \(3\): \(-1 - (-4) = 3\)
• From \(3\) to \(9\): \(2 - (-1) = 3\)

The changes are \(4\), \(3\), and \(3\). The slope is not constant. Table B does not represent a linear function.

Table C

x   |  y
---------
-5  | -7
-2  | -6
 1  | -4
 4  | -2

Calculating the differences in \(y\):

• From \(-5\) to \(-2\): \(-6 - (-7) = 1\)
• From \(-2\) to \(1\): \(-4 - (-6) = 2\)
• From \(1\) to \(4\): \(-2 - (-4) = 2\)

The changes are \(1\), \(2\), \(2\). The slope is not constant. Table C does not represent a linear function.

Table D

x   |  y
---------
-5  |  4
-2  |  1
 1  | -2
 4  | -5

Calculating the differences in \(y\):

• From \(-5\) to \(-2\): \(1 - 4 = -3\)
• From \(-2\) to \(1\): \(-2 - 1 = -3\)
• From \(1\) to \(4\): \(-5 - (-2) = -3\)

The slope is constant \(-3\). Table D represents a linear function.

Summary

Only Tables A and D represent linear functions.