Polynomial Regression

Purposes - Finding a polynomial of best fit for data points; Using the Intermediate Value Theorem

Why use polynomial regression?

Source: arcgis website Source: KS Geological survey

Fitting a surface to the sample points when the surface varies slowly from region to region over the area of interest (e.g., pollution over an industrial area).
Examining and/or removing the effects of long-range or global trends. In such circumstances the technique is often referred to as trend surface analysis.

Warmup

Find the 3rd degree polynomial that fits these data points. Then plot the polynomial from x = 0 to 8:
TI Calculator Notes:
STATE | EDIT enter data points; STAT | CALC Cubic Regression
To graph from x = 0 to 8: Y1 = (your polynomial) ((x>0) and (x<8)) Check your parentheses!
Inequalities → 2nd TEST; and → 2nd TEST LOGIC

x	y
4	6
7	8
2	1
5	9

Check ans: y = -0.267 x³+ 3.1 x²- 8.633 x+ 8

Part I - Fitting a polynomial to the elevation profile of the Leadville, Colorado 100 Ultramarathon Mountain Bike Race

Here is an altitude profile of the racecourse:

Using your grapher, find the Quartic (4th degree polynomial) best fitting the following pairs of points:
Include the endpoints. For the coefficients use three or more decimal places.

Group 1 - Points A through B p(x) =

Group 2 - Points B through C p(x) =

Group 3 - Points C through D p(x) =

Group 4 - Points D through E p(x) =

Group 5 - Points E through F p(x) =

Sketch the graph of your polynomial piece here with scales and units:

TI Calculator Notes:
STATE | EDIT enter data points; STAT | CALC Quartic Regression
WINDOW Note your Domain and Range! The standard window of -10 to 10 won't work!
To graph your piece: Y1 = (your polynomial) ((x>a) and (x<b)) Check your parentheses!
Inequalities → 2nd TEST; and → 2nd TEST LOGIC

Part II - Death Valley and the Intermediate Value Theorem

Death Valley holds the record for the highest reliably reported temperature in the Western Hemisphere, 134 °F and has the lowest elevation in North America, 282 ft below sea level.

Altitude profile of Death Valley altitude in feet vs. a cross section in miles .

You can see the altitude profile of Death Valley is a continuous function (no gaps). Assuming it could be modeled by a
polynomial (see part 1), let's apply the Intermediate Value Theorem to this model.
The Intermediate Value Theorem:
Let P(x) be a polynomial function with real coefficients. If a is not equal to b , and P(a) and P(b) have opposite signs, then the function has a real zero between a and b.

From the profile find these values approximately (include units):
Find P(20 miles) =
Find P(50 miles) =

What does the IVT say about the altitude function between 20 and 50?

Estimate this value between a and b                            What is the altitude at this value?