a. Find a power function that models the data.
b. Find a linear function that models the data.
c. Visually determine which function is the better fit for the data.
	
x
	
y
	
 
1
5
2
9
3
11
4
21
5
31
6
45

### Results:
1. **Power Function**: The best-fit power function is:
   \[
   y = 1.90 \cdot x^{1.75}
   \]
   
2. **Linear Function**: The best-fit linear function is:
   \[
   y = 7.89x - 7.27
   \]

### Part (c): Visual Comparison
As seen in the plot, both the power function and the linear function have been fitted to the data. To determine which is the better fit, you can observe how closely each curve follows the data points. The power function appears to capture the curvature of the data better, while the linear function is a simpler fit but may not account for the increasing rate of change as effectively.

Would you like to explore more details or ask any questions?

Here are 5 follow-up questions:
1. How can we quantitatively assess the goodness of fit for each function?
2. What does the exponent \( 1.75 \) in the power function represent?
3. When is it more appropriate to use a linear model over a power model?
4. How would the models change if we had more data points?
5. Can we apply another type of model, such as an exponential function, to fit the data?

**Tip**: For non-linear data patterns, power functions or polynomial functions often provide better fits than linear models.

a. Find a power function that models the data. b. Find a linear function that models the data. c. Visually determine which function is the better fit for the data.

x: 1, 2, 3, 4, 5, 6
y: 5, 9, 11, 21, 31, 45

Explore how to model the given data points using both a power function and a linear function. Learn how to apply the least squares method to determine the best fit, compare results visually, and understand when to use each model. Suitable for high school students in Grades 10-12.

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