Equation to look of graphs:

• Even exponent: Both ends point the same direction
• Odd exponent: Each end points a different direction
• Positive coefficient: as x??, p(x) ??
• Negative coefficient: as x??, p(x) ?-?
• Turns = degree – 1
• Branches = zeros =      degree
• A zero of multiplicity bigger than 2 flattens the graph out
• Greater the exponent the more it flattens out
• Multiplicity of root=1 will go through x-axis
• Multiplicity of root = 2 will touch x-axis
• Find the y intercept for a more accurate graph

To show the different details we learned I’ll use an example.

f(x) = -(x + 2)(2x – 1)³(x – 3)²(x +6)

degree = 7 (greatest exponent or for multiplying factors- how many factors there are)

multiplicity: -(x + 2)(2x – 1)(2x – 1)(2x – 1)(x – 3)(x – 3)(x + 6)

zeros: -2, ½, ½, ½, 3, 3, -6 (what makes each factor = 0)

y-intercept: 108 (found by putting 0 in for each x and then solving)

Local behavior- How a function behaves when the graph is zoomed in enough to see the x and y intercepts and the discontinuities/vertical asymptote

Global behavior- How the function behaves when the graph is zoomed way out, and horizontal asymptote

How do we know if we have a good model?

1.      Interpolation: Put points in your model and check how well they match the true point

2.      Extrapolate: Go outside the data set and check how well it fits the model

3.      See how close the “r” value is to 1 (the closer to 1, the better)

4.      Look at it, does the model fit a majority of the points

5.      Look at endpoint behavior

6.      Look at the residual plot and if the points are random or in a pattern (random is good, meaning that the model accounts for all sources of error)

How to do the math using your calculator:

For reach point above, you can input it into your calculator under the stat plot lists as follows—

L₃ = deviations² = (average (L₂) – L₂)²

L₄ = residuals² = (y₁(L₁) – L₂)²

r² = SL₃ – SL₄ (S represents the sum of that list)

SL₃

Now I’ll do the bluegill problem off of an assessment we did.

(Enter this data into your calculator to produce a graph.  Then using the graph and calculations you can analyze the model.  My analysis of the model is below)

As seen on the graph, the logistics function is the best fitting model for the data. Although ln could also be a good model, logistics is better for a number of reasons.  First off, logistics has a r value of .9997, while ln has an r value of .9966; also, the other regressions r values were worse than both of these (power-.9848 and linear-.9775).  Another reason to rule out power and linear is that the model misses a majority of the points while the other two hit a majority of the points; the data is more curved then the power and linear functions.  So after that step I decided to further look into only logistics and ln. When looking at the graph of the residuals both regressions are random and have no pattern, showing that both models account for all sources of error.  For interpolation, I randomly chose the point (68,95). Logistics gave me the point (68, 95.5), and ln gave me (68, 98.1); so logistics is better for interpolation.  Then I look at extrapolation and can see that logistics converges to 177.9, while ln continues to grow.  Converging makes since because bluegills don’t grow forever, so although converging is still growing a tiny amount, that is better then ln which continues to grow more rapidly for the whole graph.  Also, ln starts off with a negative total length for any initial length less then 21.83, and at about 21.84 initial length, there is no growth total growth.  Where as if you look at the logistics graph, there is no negative lengths, and it starts off at 0 initial length with a total growth after one year of 24.86; which makes since fish start off as an egg and grow.  After making these evaluations, I can say logistics the best model for this set of data.