Question

Which dataset represents a quadratic function?(1 point)
Responses

Dataset A

x y
1
25
2
32
3
41
4
52
Dataset A x y 1 25 2 32 3 41 4 52

Dataset B

x y
1
0.5
2
0.25
3
0.125
4
0.0625
Dataset B 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

x y
1
−23
2
−26
3
−29
4
−32
Dataset C x y 1 negative 23 2 negative 26 3 negative 29 4 negative 32

Dataset D

x y
1
4
2
16
3
64
4
256

Answer 1

To determine which dataset represents a quadratic function, we need to analyze how the values in each dataset change as the value of \( x \) increases.

### Dataset A
- \( x = 1 \), \( y = 25 \)
- \( x = 2 \), \( y = 32 \)
- \( x = 3 \), \( y = 41 \)
- \( x = 4 \), \( y = 52 \)

The differences between consecutive \( y \) values:
- \( 32 - 25 = 7 \)
- \( 41 - 32 = 9 \)
- \( 52 - 41 = 11 \)

The second differences:
- \( 9 - 7 = 2 \)
- \( 11 - 9 = 2 \)

The second differences are constant, indicating a quadratic function.

### Dataset B
- \( x = 1 \), \( y = 0.5 \)
- \( x = 2 \), \( y = 0.25 \)
- \( x = 3 \), \( y = 0.125 \)
- \( x = 4 \), \( y = 0.0625 \)

This dataset appears to be exponential decay. The differences are not consistent, implying this is not quadratic.

### Dataset C
- \( x = 1 \), \( y = -23 \)
- \( x = 2 \), \( y = -26 \)
- \( x = 3 \), \( y = -29 \)
- \( x = 4 \), \( y = -32 \)

The differences between consecutive \( y \) values:
- \( -26 + 23 = -3 \)
- \( -29 + 26 = -3 \)
- \( -32 + 29 = -3 \)

The differences are constant, but the second differences are zero, indicating a linear function rather than a quadratic function.

### Dataset D
- \( x = 1 \), \( y = 4 \)
- \( x = 2 \), \( y = 16 \)
- \( x = 3 \), \( y = 64 \)
- \( x = 4 \), \( y = 256 \)

Here, it appears to be exponential growth:
- \( 16 / 4 = 4 \)
- \( 64 / 16 = 4 \)
- \( 256 / 64 = 4 \)

The ratios suggest an exponential function, not quadratic.

### Conclusion
Only **Dataset A** demonstrates constant second differences, indicating that it represents a quadratic function.