Question
which function best models the data in the following table? (x,y) (-5,6.2)(-2,8.3)( -1,9.1)(0,10)(1,11)( 2 12.1)(5,16.1)( 8,21.4)(12, 31.4) A. y= 1.467x + 10.696 B. y=10.004 (1.10)^x C. y=0.066 x ^ 2+ 0.981x + 9.81
Answers
GPT-4o mini
To determine which function best models the data provided in the table, we can use each of the given functions to see which one fits the data points most closely.
The data points provided in the table are:
- (-5, 6.2)
- (-2, 8.3)
- (-1, 9.1)
- (0, 10)
- (1, 11)
- (2, 12.1)
- (5, 16.1)
- (8, 21.4)
- (12, 31.4)
We can analyze each function given:
A. **Linear Function: y = 1.467x + 10.696**
- This is a linear model. It describes a straight line and may not capture the curvature present in the data.
B. **Exponential Function: y = 10.004(1.10)^x**
- This model suggests an exponential growth, which could potentially fit the increasing trend of the data points, particularly as x increases.
C. **Quadratic Function: y = 0.066x^2 + 0.981x + 9.81**
- This is a quadratic model. Quadratic functions are parabolic and can model data that exhibits curvature, which appears to be the case here as y increases more rapidly at higher values of x.
### Steps to Evaluate the Functions:
1. **Fit each function to the data points** and calculate the predicted values for each x-value.
2. **Calculate the error** (e.g., using the sum of squared errors) between the predicted values from each model and the actual y-values from the data table.
3. **Compare the errors**: the model with the smallest error typically best fits the data.
### Observations:
- **Linear models tend to underfit** data that has curvature (as is likely with these points).
- **Exponential models can fit data that grows rapidly** but won't capture non-linear trends well.
- **Quadratic models can fit data with curvature** much better than linear models.
### Suggested Approach:
1. Evaluate the fit of each function with a few sample calculations or by using software to automate fitting and calculating the error.
2. Compare the outputs visually or through calculated errors.
In conclusion, based on the nature of the data, the quadratic function (C) is likely the best choice because it can accommodate the curvature seen in the data as y increases more rapidly for certain values of x.
If you need to confirm this, using statistical software or graphing can help visualize and calculate the fitting errors more rigorously.
The data points provided in the table are:
- (-5, 6.2)
- (-2, 8.3)
- (-1, 9.1)
- (0, 10)
- (1, 11)
- (2, 12.1)
- (5, 16.1)
- (8, 21.4)
- (12, 31.4)
We can analyze each function given:
A. **Linear Function: y = 1.467x + 10.696**
- This is a linear model. It describes a straight line and may not capture the curvature present in the data.
B. **Exponential Function: y = 10.004(1.10)^x**
- This model suggests an exponential growth, which could potentially fit the increasing trend of the data points, particularly as x increases.
C. **Quadratic Function: y = 0.066x^2 + 0.981x + 9.81**
- This is a quadratic model. Quadratic functions are parabolic and can model data that exhibits curvature, which appears to be the case here as y increases more rapidly at higher values of x.
### Steps to Evaluate the Functions:
1. **Fit each function to the data points** and calculate the predicted values for each x-value.
2. **Calculate the error** (e.g., using the sum of squared errors) between the predicted values from each model and the actual y-values from the data table.
3. **Compare the errors**: the model with the smallest error typically best fits the data.
### Observations:
- **Linear models tend to underfit** data that has curvature (as is likely with these points).
- **Exponential models can fit data that grows rapidly** but won't capture non-linear trends well.
- **Quadratic models can fit data with curvature** much better than linear models.
### Suggested Approach:
1. Evaluate the fit of each function with a few sample calculations or by using software to automate fitting and calculating the error.
2. Compare the outputs visually or through calculated errors.
In conclusion, based on the nature of the data, the quadratic function (C) is likely the best choice because it can accommodate the curvature seen in the data as y increases more rapidly for certain values of x.
If you need to confirm this, using statistical software or graphing can help visualize and calculate the fitting errors more rigorously.