Question

Use the tables to answer the question.

Information
Table A
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 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**