Digital tools

Finding a Parabola of Best Fit

Most graphing calculators have a function called quadratic regression, which can be used to find a precise quadratic model for the data. For example, consider the following data set.

$x$	$y$
$0$	$1.50$
$1$	$1.75$
$2$	$2.65$
$3$	$5.67$
$4$	$7.21$
$5$	$12.01$

The parabolic curve of best fit can be calculated following these three steps.

Enter Values

On a graphing calculator, begin by entering the data points. To do so, press the $STAT$ button and select the option Edit.

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This gives a number of columns, labeled $L_{1},$ $L_{2},$ $L_{3},$ and so on.

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Use the arrow keys to choose where in the lists to fill in the data values. Enter the $x -$ values of the data points in $L_{1}$ and press $ENTER$ after each value. The same can be done for the the corresponding $y -$ values in column $L_{2} .$

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Finding a Fitting Parabola

After entering the values, press the $STAT$ button and select the menu item Calc.

Counter that displays the list of CALC and where you have chosen LinReg

The option QuadReg gives the parabola of best fit, expressed as a quadratic equation in standard form. Press the $ENTER$ button until the parameters are given.

Calculator showing a linear function regression

In this case, a quadratic model for the data is $y = 0.45 x^{2} - 0.2 x + 1.5 .$ Also, the coefficient of the determination is $0.99,$ which indicates the model almost perfectly fits the data. If $R^{2}$ does not appear, press $2ND$ and $0$ to get to the CATALOG, then select DiagnosticOn and enable it by pressing $ENTER$ twice. Once more, find the parabola of best fit.

Graphing the Parabola of Best Fit

A graphing calculator can also be used to graph the line of best fit. After selecting the option QuadReg, choose the option Store RegEQ.

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Press $VARS$ and move to the Y-VARS menu. Then, select the option FUNCTION and press $ENTER .$

Press the $ENTER$ button until the parameters are given. To graph the scatter plot, first push the buttons $2nd$ and $Y = .$ Then, choose one of the plots in the list. Select the option ON, choose the type to be a scatter plot, and assign $L_{1}$ and $L_{2}$ as XList and Ylist, respectively.

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Then, the plot can be made by pressing the button $GRAPH .$ It is possible that after drawing the plot the window-size is not large enough to see all of the information.

To fix this, press $ZOOM$ and select the option ZoomStat. After doing that, the window will resize to show the important information.

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