Question
Which dataset represents a quadratic function?
Dataset A
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
25
2
32
3
41
4
52
Dataset A This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 25 2 32 3 41 4 52
Dataset B
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
0.5
2
0.25
3
0.125
4
0.0625
Dataset B This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 0 point 5 2 0 point 2 5 3 0 point 1 2 5 4 0 point 0 6 2 5
Dataset C
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
−23
2
−26
3
−29
4
−32
Dataset C This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 negative 23 2 negative 26 3 negative 29 4 negative 32
Dataset D
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
4
2
16
3
64
4
256
Dataset D This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 4 2 16 3 64 4 256
Dataset A
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
25
2
32
3
41
4
52
Dataset A This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 25 2 32 3 41 4 52
Dataset B
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
0.5
2
0.25
3
0.125
4
0.0625
Dataset B This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 0 point 5 2 0 point 2 5 3 0 point 1 2 5 4 0 point 0 6 2 5
Dataset C
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
−23
2
−26
3
−29
4
−32
Dataset C This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 negative 23 2 negative 26 3 negative 29 4 negative 32
Dataset D
This table displays a set of ordered pairs (x, y), showing the relationship between x and y values.
x y
1
4
2
16
3
64
4
256
Dataset D This table displays a set of ordered pairs (x, y), showing the relationship between x and y values. x y 1 4 2 16 3 64 4 256
Answers
GPT-4o mini
To determine which dataset represents a quadratic function, we can analyze the second differences of the y-values for each dataset. A quadratic function has a constant second difference.
Let's calculate the first and second differences for each dataset:
### Dataset A
- **y values**: 25, 32, 41, 52
- **First differences**:
- 32 - 25 = 7
- 41 - 32 = 9
- 52 - 41 = 11
- **First differences**: 7, 9, 11
- **Second differences**:
- 9 - 7 = 2
- 11 - 9 = 2
- **Second differences**: 2, 2 (Constant)
### Dataset B
- **y values**: 0.5, 0.25, 0.125, 0.0625
- **First differences**:
- 0.25 - 0.5 = -0.25
- 0.125 - 0.25 = -0.125
- 0.0625 - 0.125 = -0.0625
- **First differences**: -0.25, -0.125, -0.0625
- **Second differences**:
- -0.125 - (-0.25) = 0.125
- -0.0625 - (-0.125) = 0.0625
- **Second differences**: 0.125, 0.0625 (Not constant)
### Dataset C
- **y values**: -23, -26, -29, -32
- **First differences**:
- -26 - (-23) = -3
- -29 - (-26) = -3
- -32 - (-29) = -3
- **First differences**: -3, -3, -3
- **Second differences**:
- -3 - (-3) = 0
- -3 - (-3) = 0
- **Second differences**: 0, 0 (Constant)
### Dataset D
- **y values**: 4, 16, 64, 256
- **First differences**:
- 16 - 4 = 12
- 64 - 16 = 48
- 256 - 64 = 192
- **First differences**: 12, 48, 192
- **Second differences**:
- 48 - 12 = 36
- 192 - 48 = 144
- **Second differences**: 36, 144 (Not constant)
### Conclusion:
- **Dataset A** has constant second differences of 2, indicating a quadratic relationship.
- **Dataset B** does not have constant second differences.
- **Dataset C** has constant second differences of 0, indicating it is linear (not quadratic).
- **Dataset D** does not have constant second differences.
Therefore, **Dataset A** represents a quadratic function.
Let's calculate the first and second differences for each dataset:
### Dataset A
- **y values**: 25, 32, 41, 52
- **First differences**:
- 32 - 25 = 7
- 41 - 32 = 9
- 52 - 41 = 11
- **First differences**: 7, 9, 11
- **Second differences**:
- 9 - 7 = 2
- 11 - 9 = 2
- **Second differences**: 2, 2 (Constant)
### Dataset B
- **y values**: 0.5, 0.25, 0.125, 0.0625
- **First differences**:
- 0.25 - 0.5 = -0.25
- 0.125 - 0.25 = -0.125
- 0.0625 - 0.125 = -0.0625
- **First differences**: -0.25, -0.125, -0.0625
- **Second differences**:
- -0.125 - (-0.25) = 0.125
- -0.0625 - (-0.125) = 0.0625
- **Second differences**: 0.125, 0.0625 (Not constant)
### Dataset C
- **y values**: -23, -26, -29, -32
- **First differences**:
- -26 - (-23) = -3
- -29 - (-26) = -3
- -32 - (-29) = -3
- **First differences**: -3, -3, -3
- **Second differences**:
- -3 - (-3) = 0
- -3 - (-3) = 0
- **Second differences**: 0, 0 (Constant)
### Dataset D
- **y values**: 4, 16, 64, 256
- **First differences**:
- 16 - 4 = 12
- 64 - 16 = 48
- 256 - 64 = 192
- **First differences**: 12, 48, 192
- **Second differences**:
- 48 - 12 = 36
- 192 - 48 = 144
- **Second differences**: 36, 144 (Not constant)
### Conclusion:
- **Dataset A** has constant second differences of 2, indicating a quadratic relationship.
- **Dataset B** does not have constant second differences.
- **Dataset C** has constant second differences of 0, indicating it is linear (not quadratic).
- **Dataset D** does not have constant second differences.
Therefore, **Dataset A** represents a quadratic function.