Table A

Question

x y
negative 2
one-sixteenth

negative 1 one-fourth
0 1
1 4
2 16

,begin bold,Table B,end bold,
x y
negative 2 8
negative 1 4
0 0
1 4
2 8

Table C
x y
negative 2 negative 8
negative 1 negative 1
0 0
1 1
2 8

Table D
x y
negative 2 negative 4
negative 1 negative 1
0 0
1 1
2 4

Table E
x y
negative 2 5
negative 1 2
0 1
1 2
2 5

Question
Which table represents a quadratic relationship between x and y ?

Answer options with 5 options
A.
Table A

B.
Table B

C.
Table C

D.
Table D

E.
Table E

Answer 1

To determine which table represents a quadratic relationship between \(x\) and \(y\), we can look for a relationship where the second differences of \(y\) values (as \(x\) values change) are constant. A quadratic relationship generally follows the form \(y = ax^2 + bx + c\).

Let's analyze each table:

Table A:

x      y
-2     ?
-1     1/4
0      1
1      4
2      16

We cannot compute differences because not all \(y\) values are given.

Table B:

x      y
-2     8
-1     4
0      0
1      4
2      8

First differences:

• From -2 to -1: \(4 - 8 = -4\)
• From -1 to 0: \(0 - 4 = -4\)
• From 0 to 1: \(4 - 0 = 4\)
• From 1 to 2: \(8 - 4 = 4\)

Second differences:

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

Not constant.

Table C:

x       y
-2     -8
-1     -1
0      0
1      1
2      8

First differences:

• From -2 to -1: \(-1 - (-8) = 7\)
• From -1 to 0: \(0 - (-1) = 1\)
• From 0 to 1: \(1 - 0 = 1\)
• From 1 to 2: \(8 - 1 = 7\)

Second differences:

• From 7 to 1: \(1 - 7 = -6\)
• From 1 to 1: \(1 - 1 = 0\)
• From 1 to 7: \(7 - 1 = 6\)

Not constant.

Table D:

x       y
-2     -4
-1     -1
0      0
1      1
2      4

First differences:

• From -2 to -1: \(-1 - (-4) = 3\)
• From -1 to 0: \(0 - (-1) = 1\)
• From 0 to 1: \(1 - 0 = 1\)
• From 1 to 2: \(4 - 1 = 3\)

Second differences:

• From 3 to 1: \(1 - 3 = -2\)
• From 1 to 1: \(1 - 1 = 0\)
• From 1 to 3: \(3 - 1 = 2\)

Not constant.

Table E:

x       y
-2     5
-1     2
0      1
1      2
2      5

First differences:

• From -2 to -1: \(2 - 5 = -3\)
• From -1 to 0: \(1 - 2 = -1\)
• From 0 to 1: \(2 - 1 = 1\)
• From 1 to 2: \(5 - 2 = 3\)

Second differences:

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

The second differences are constant here.

Thus, the answer is:

E. Table E