Question

Which equation is a quadratic model for the data set? Hint: Use a graphing calculator or a spreadsheet program.

Data table

Responses

y=20.9330x−93.8918
y equals 20.9330 x minus 93.8918

y=1.2246x2−5.0511x+24.2880
y equals 1.2246 x squared minus 5.0511 x plus 24.2880

y=11.6018⋅1.2207x
y equals 11.6018 times 1.2207 to the power of x

y=1.1556x1.9375

Answer 1

The correct quadratic model for the data set is determined by analyzing the given options with a graphing calculator or spreadsheet software's regression function, where the equation in the form of y = ax^2 + bx + c with values computed to best fit the data is the quadratic model.

Step-by-step explanation:

To determine which equation is the correct quadratic model for the given data set, a graphing approach must be taken. Since quadratic functions are of the form y = ax2 + bx + c, only the options with an x2 term can be considered as quadratic models. The correct equation should resemble a parabola when graphed. The options provided contain two potential quadratic equations and two that are not quadratic (one linear and one exponential). With the use of a graphing calculator or spreadsheet software, the quadratic equation that best fits the data can be determined by performing a regression analysis.

To perform this analysis, the data is inputted into the calculator or software to obtain a graph. If using a calculator, the STAT and REGRESSION functions are typically utilized to calculate the least-squares regression line for the data. This process fits the best possible curve to the data points, minimized by the sum of the squares of the vertical differences (residuals) between the data points and the curve. Quadratic regression will output an equation of the form y = ax2 + bx + c, where a, b, and c are coefficients that define the specific curve.

The correct quadratic model would be the one that produces the lowest sum of squared residuals, often indicated by the coefficient of determination, R2. It is a measure of how well the regression curve fits the data. An R2 value closer to 1 indicates a better fit. Comparing the provided quadratic options against the regression results will identify the correct equation.

If we apply these steps to the given options, we can determine that the option y = 1.2246x2 - 5.0511x + 24.2880 is likely the correct quadratic model, being the only given quadratic equation in the form of ax2 + bx + c.