To determine which table represents a quadratic relationship between x and y, we can calculate the second differences of the y-values in each table. If the second differences are constant, then the relationship is quadratic.
Let's analyze each table:
Table A:
x y
-2 1/16
-1 1/4
0 1
1 4
2 16
First differences:
- (1/4 - 1/16) = 3/16
- (1 - 1/4) = 3/4
- (4 - 1) = 3
- (16 - 4) = 12
Second differences:
- (3/4 - 3/16) = 9/16
- (3 - 3/4) = 9/4
- (12 - 4) = 8
Not constant, not quadratic.
Table B:
x y
-2 8
-1 4
0 0
1 4
2 8
First differences:
- (4 - 8) = -4
- (0 - 4) = -4
- (4 - 0) = 4
- (8 - 4) = 4
Second differences:
- (-4 + 4) = 0
- (4 - (-4)) = 8
Not constant, not quadratic.
Table C:
x y
-2 -8
-1 -1
0 0
1 1
2 8
First differences:
- (-1 - (-8)) = 7
- (0 - (-1)) = 1
- (1 - 0) = 1
- (8 - 1) = 7
Second differences:
- (1 - 7) = -6
- (1 - 1) = 0
- (7 - 1) = 6
Not constant, not quadratic.
Table D:
x y
-2 -4
-1 -1
0 0
1 1
2 4
First differences:
- (-1 - (-4)) = 3
- (0 - (-1)) = 1
- (1 - 0) = 1
- (4 - 1) = 3
Second differences:
- (1 - 3) = -2
- (1 - 1) = 0
- (3 - 1) = 2
Not constant, not quadratic.
Table E:
x y
-2 5
-1 2
0 1
1 2
2 5
First differences:
- (2 - 5) = -3
- (1 - 2) = -1
- (2 - 1) = 1
- (5 - 2) = 3
Second differences:
- (-1 - (-3)) = 2
- (1 - (-1)) = 2
- (3 - 1) = 2
Constant second differences of 2 indicate a quadratic relationship.
Since only Table E has constant second differences, the answer is:
E. Table E