Question
Which function best models the dataset?
x
2 4 6 8 10 12
y
7.2 16.2 36.45 82.01 184.53 415.19
(1 point)
Responses
y=−1.5x2+26.6x−52.4
y is equal to negative 1 point 5 x squared plus 26 point 6 x minus 52 point 4
y=7.6⋅1.2x
y is equal to 7 point 6 times 1 point 2 to the x th power
y=6.4x2−52.8x+104.1
y is equal to 6 point 4 x squared minus 52 point 8 x plus 104 point 1
y=3.2⋅1.5x
x
2 4 6 8 10 12
y
7.2 16.2 36.45 82.01 184.53 415.19
(1 point)
Responses
y=−1.5x2+26.6x−52.4
y is equal to negative 1 point 5 x squared plus 26 point 6 x minus 52 point 4
y=7.6⋅1.2x
y is equal to 7 point 6 times 1 point 2 to the x th power
y=6.4x2−52.8x+104.1
y is equal to 6 point 4 x squared minus 52 point 8 x plus 104 point 1
y=3.2⋅1.5x
Answers
GPT-4o mini
To determine which function best models the given dataset of \(x\) and \(y\) values, we need to analyze the nature of the data points.
Given data:
- \(x\): 2, 4, 6, 8, 10, 12
- \(y\): 7.2, 16.2, 36.45, 82.01, 184.53, 415.19
From the values, we can observe that as \(x\) increases, \(y\) increases significantly, suggesting an exponential or polynomial growth rather than a linear relationship.
Let’s briefly analyze the provided function options:
1. **Quadratic Polynomial**: \(y = -1.5x^2 + 26.6x - 52.4\) and \(y = 6.4x^2 - 52.8x + 104.1\) - This would imply that \(y\) decreases at larger \(x\) values since quadratic functions either have maximum or minimum points, which likely wouldn't fit the rapidly increasing \(y\) values.
2. **Exponential Functions**:
- \(y = 7.6 \cdot 1.2^x\)
- \(y = 3.2 \cdot 1.5^x\)
Exponential functions can fit datasets where \(y\) rapidly grows with increasing \(x\).
To determine the best fit, we can evaluate the growth rates:
- \(y = 7.6 \cdot 1.2^x\) might grow more slowly compared to the other option.
- \(y = 3.2 \cdot 1.5^x\) grows faster, which might correlate better with the rapid increases in \(y\).
Based on the overall trend of the dataset showing exponential growth and rapid increase in values of \(y\), it is likely that either of the exponential functions would be the best model.
Among these two exponential functions, \(y = 3.2 \cdot 1.5^x\) will likely yield values that align better with the given \(y\) values due to a higher growth factor (\(1.5 > 1.2\)).
Therefore, the function that likely best models the dataset is:
**y = 3.2 * 1.5^x**.
Given data:
- \(x\): 2, 4, 6, 8, 10, 12
- \(y\): 7.2, 16.2, 36.45, 82.01, 184.53, 415.19
From the values, we can observe that as \(x\) increases, \(y\) increases significantly, suggesting an exponential or polynomial growth rather than a linear relationship.
Let’s briefly analyze the provided function options:
1. **Quadratic Polynomial**: \(y = -1.5x^2 + 26.6x - 52.4\) and \(y = 6.4x^2 - 52.8x + 104.1\) - This would imply that \(y\) decreases at larger \(x\) values since quadratic functions either have maximum or minimum points, which likely wouldn't fit the rapidly increasing \(y\) values.
2. **Exponential Functions**:
- \(y = 7.6 \cdot 1.2^x\)
- \(y = 3.2 \cdot 1.5^x\)
Exponential functions can fit datasets where \(y\) rapidly grows with increasing \(x\).
To determine the best fit, we can evaluate the growth rates:
- \(y = 7.6 \cdot 1.2^x\) might grow more slowly compared to the other option.
- \(y = 3.2 \cdot 1.5^x\) grows faster, which might correlate better with the rapid increases in \(y\).
Based on the overall trend of the dataset showing exponential growth and rapid increase in values of \(y\), it is likely that either of the exponential functions would be the best model.
Among these two exponential functions, \(y = 3.2 \cdot 1.5^x\) will likely yield values that align better with the given \(y\) values due to a higher growth factor (\(1.5 > 1.2\)).
Therefore, the function that likely best models the dataset is:
**y = 3.2 * 1.5^x**.