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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.

1

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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2

Finding a Fitting Parabola

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

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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.

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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.

3

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 $ENTER .$

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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.

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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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