Which quadratic function best fits this data?

Options:

y=−11.41x^2+154.42x−143.9

y=11.41x^2+154.42x+143.9

y=−11.41x^2+154.42x+143.9

y=11.41x2^+154.42x−143.9

Thanks again. Happy valentines day! :)

PhoenixForever Feb 14, 2019

#1**+1 **

Of course, we are gonna' need an app for this

I like this one : https://keisan.casio.com/exec/system/14059932254941

The correct function is : y=−11.41x^2+154.42x−143.9

CPhill Feb 14, 2019

#2**+1 **

For completeness I'll show you how this is solved in general. It's not trivial

\(\text{first we construct the matrix}\\ X = \begin{pmatrix}1 &x_1 &x_1^2\\1 &x_2 &x_2^2\\\vdots\\1 &x_n &x_n^2\end{pmatrix}\)

\(\text{In this problem this is}\\ X=\left( \begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \\ 1 & 4 & 16 \\ 1 & 5 & 25 \\ 1 & 6 & 36 \\ \end{array} \right)\)

\(\text{We then form the matrix}\\ Y=\begin{pmatrix}y_1\\y_2\\\vdots \\ y_n\end{pmatrix}\\ \text{In this problem this is}\\ (32, 78, 178, 326, 390, 337)^T\)

\(\text{The equation for the coefficients of the least squares quadratic fit is given by}\\ c = (X^TX)^{-1}X^TY\)

\(\text{Grinding this through we get}\\ c =(-143.9,154.418,-11.4107)^T \text{ i.e. that the best fit quadratic is}\\ y(t) = -11.41t^2 + 154.42x - 143.9\)

Rom Feb 14, 2019