Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Use the regression capabilities of a graphing utility or a spreadsheet to find linear and quadratic models for the data. State which model best fits the data. $$ (1,10.3),(2,14.2),(3,18.9),(4,23.7),(5,29.1),(6,35) $$

Step-by-Step Solution

Answer

Based on above steps, both a linear model and a quadratic model can be found. Their goodness of fit (usually quantified by R^2 value) need comparison. The model with higher value fits the data better. Precise answer requires numerical computation with specific software.

Step 1: Input the data

First, place the given data ((1,10.3),(2,14.2),(3,18.9),(4,23.7),(5,29.1),(6,35)) into a spreadsheet or graphing utility. Usually, the x-values go into one column and the y-values into another one.

Step 2: Plotting the data

The software should have a plotting feature. Utilize this tool to generate a scatter plot. This visualization helps in understanding the data behavior in a more intuitive way.

Step 3: Fit a linear model

Use the tool, usually 'linear regression' or 'linear trendline' in your software. The utility automatically calculates the best fitting linear trend line to the data points. Document this line's equation of the form \(y = ax + b\), where \(a\) is the slope and \(b\) is the y-intercept.

Step 4: Fit a quadratic model

In the similar way, fit a quadratic model to the data through the 'polynomial regression' tool, usually with 2nd order (since it is quadratic). This will output a quadratic equation, typically in the form \(y = ax^2 + bx + c\), where \(a,b,c\) are the coefficients.

Step 5: Evaluate fit

Most graphing utilities or spreadsheets provide a goodness-of-fit measure when doing regression. Often this is the R-squared value. Compare the R-squared value for both models. The one with the higher value is the better fit.