Look for an R^2 equal to 1 using the QuadReg, CubicReg, and QuartReg information screens.
y=- 0.0404x^4+0.2205x^3+4.026x^2-14.1468x+8.5119
Practice makes perfect
To find a polynomial function whose graph passes through the given points, we should first plot the points in a coordinate plane and try to rule out some obvious regressions. Push the STAT button, choose Edit, and enter your values.
Once the values have been entered, we can plot them by pushing 2nd, Y=, and then choosing one of the plots in the list. Make sure you turn the plot ON before you choose the scatter plot type. Then, match the lists for the x- and y-values accordingly.
By pushing GRAPH, the calculator will plot the data. Note that we will need to change the viewing window so that we can see all of the points.
Immediately, we can see that a linear function will not work because the points do not lie on a straight line. Therefore, let's test the data using the QuadReg, CubicReg, and QuartReg features. If one of them fits the data, the R^2 value will equal 1.
Since R^2 is 0.9930008605 ≈ 1 for the quartic regression, it's almost a perfect match! A quartic function will pass very close to all the given points. The values of a, b, c, d, and e are the coefficients of the function, we will round them to four decimal places.
y= ax^4+ bx^3+ cx^2+ dx+ e
⇓
y= - 0.0404x^4 + 0.2205x^3 + 4.026x^2 - 14.1468x + 8.5119
Let's plot the quartic regression on top of the points just to double check our findings.
