aim: graph of best fit course: alg. 2 & trig. aim: how do we model real-world data with...
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Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Aim: How do we model real-world data with polynomial and other functions?
Do Now: 6 pt. Regents QuestionThe 1999 win-loss statistics for the American League East baseball teams on a particular date is shown in the accompanying chart.
W L
New York 52 34
Boston 49 39
Toronto 47 43
Tampa Bay 39 49
Baltimore 36 51
Find the mean for the number of wins, , and the mean for the number of losses, , and determine if the point ( is a point on the line of best fit. Justify your answer.
WL
,W )L
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Model Problem
The average daily amount of waste generated by each person in the United States is given below. This includes all wastes such as industrial wastes, demolition wastes, and sewage.
Yr. 1980 1985 1990 1991 1992 1993 1994 1995 1996
lbs. of waste per
person per day
3.7 3.8 4.5 4.4 4.5 4.5 4.5 4.4 4.3
Create a scatter plot and determine the regression line. Round to nearest
hundredth
y = .05x – 98.69
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Correlation
Positive Correlation•y tends to increase as x
increases•slope is positive
No Correlation
Negative Correlation•y tends to decrease as x
increases•slope is negative
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Correlation Co-efficient
Data that are linear in nature will have varying degrees of goodness of fit to the lines of fit.
The correlation coefficient r describes the nature of data.
The closer the fit of the data to the line, the closer r gets to + 1 or -1
0 < r < 0.5 positive/weak
0.75 < r < 1 strongly positive
-0.5 < r < 0 moderately
negative
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Real World Data & Poly Function Shapes
linear y = ax + b
4
2
4
2
5
quadratic y = ax2 + bx + c
4
2
cubic y = ax3+bx2+ cx+d
-2
-4
quartic y = ax4 + bx3 + cx2 + dx + e
No Direction Change
1 Direction Change
2 Direction Changes
3 Direction Changes
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Which Function is Best Fit?
2
-2
-4
Determine the type of polynomial function that could be used to represent the data in each scatter plot.
2
-2
-4
Two direction change: cubic function would be
best fit
One direction change: quadratic function would
be best fit
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Functions Modeling Data
Write a polynomial function that models the set of data.
x -1 -.5 0 0.5 1 1.5 2 2.5 3 3.5 4
f(x) -10 -6.4 -5 -5.1 -6 -6.9 -7 -5.6 -2 4.6 15
enter x into L1
enter f(x) into L2
View Stat Plot and determine which function best models the data
f(x) = x3 – 3x2 + x – 5
Determine Cubic Regression Equation and round coefficients to nearest integer STAT 6 ENTER
cubic
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Waste Problem
The average daily amount of waste generated by each person in the United States is given below. This includes all wastes such as industrial wastes, demolition wastes, and sewage.
Yr 1980 1985 1990 1991 1992 1993 1994 1995 1996
lbs. of waste per
person per day
3.7 3.8 4.5 4.4 4.5 4.5 4.5 4.4 4.3
Is a linear function the best fit for this data?
6
5
4
3
6
5
4
3
f x = 0.05x+3.7
6
5
4
3
quadratic1 direction change
STAT 5 ENTER
Quadratic Regression
y = ax2 + bx + c a = -.004209b = .119480c = 3.592550R2 = .819795
y = -.004x2 + .119x + 3.593
6
5
4
3
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Waste Problem
y = -.004x2 + .119x + 3.5936
5
4
3
6
5
4
3
a. Use the model to predict the amount of waste produced per day in 2010.
Since 2010 is 30 years later than 1980, find f(30).
f(30) = -.004x2 + .119x + 3.593 = 3.563 lb.
b. Use the model to predict when waste will drop to 3 pounds per day.
f(x) = -.004x2 + .119x + 3.593 = 3
x ≈ -4 or 34 1976 or 2014
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
6
4
2
Growth6
4
2
Decay
2
-2
-4
5
Growth2
-2
-5
Decay
Exponential Functions y = abx
Logarithmic Functions y = a + b ln x
Growth & Decay
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Model Problem
The table below gives the population of the world in billions for selected years during the 1900’s.
YR
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
0 10 20 30 40 50 60 70 80 90 100
P 1.65 1.75 1.86 2.07 2.3 2.52 3.02 3.7 4.44 5.27 6.06
Determine an equation that models the data.
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
6
5
4
3
2
1
-1
20 40 60 80 100 120
Growth & Decay
0 10 20 30 40 50 60 70 80 90 100
P 1.65 1.75 1.86 2.07 2.3 2.52 3.02 3.7 4.44 5.27 6.06
y = ax + b a = .0435b = .97409r2 = .9008r = .9491y = .04x + 1
6
4
2
Growth
y = a · bx a = 1.4457b = 1.0136r2 = .9700r = .9849y = 1.44 ·1.01x
f x = 1.445691.01369x
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
4 pt. Regents Question
A biologist finds that a colony of bacteria grows exponentially and collects the following data on its size.
On a grid, make a scatter plot of this data. Write an exponential regression equation, expressing the regression coefficients to the nearest tenth.
Time(days)
Population (100s of liters per hour)
0 100
1 310
2 470
3 715
4 1150
5 1650
6 2500
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Natural Log Growth Model Problem
The data in the table gives the yield y (in milligrams) of a chemical reaction after x minutes.x 1 2 3 4 5 6 7 8
y 1.5 7.4 10.2 13.4 15.8 16.3 18.2 18.3
Find a logarithmic model for the datay = 1.538+8.373 lnx
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Oil Tanker Problem
An oil tanker collides with another ship and starts leaking oil. the Coast Guard measure the rate of flow of oil from the tanker and obtains the data shown in the table. Write a polynomial function to model the set of data.
Time(hours)
Flow Rate (100s of liters per hour)
1 18.0
2 20.5
3 21.3
4 21.1
5 19.9
6 17.8
7 15.9
8 11.3
9 7.8
10 3.7
f(x) = -0.4x2 + 2.8x + 16.3
Aim: Graph of Best Fit Course: Alg. 2 & Trig.
Model Problem
Write a polynomial function to model the set of data.
x f(x)
-2.0 -22.0
-1.5 -7.9
-1.0 0.0
-0.5 3.5
0.0 4.1
0.5 2.9
1.0 2.1
1.5 2.5
2.0 5.8
2.5 14.1
3.0 28.0
f(x) = 2x3 – 3x2 – x + 4