1. 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 Question The 1999 win-loss statistics for the American League East baseball teams on a particular date is shown in the accompanying chart. WL New York5234 Boston4939 Toronto4743 Tampa Bay3949 Baltimore3651 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.

2. 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. 198019851990199119921993199419951996 lbs. of waste per person per day 3.73.84.54.44.5 4.44.3 Create a scatter plot and determine the regression line. Round to nearest hundredth y =.05x – 98.69

3. 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

4. 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

5. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Real World Data & Poly Function Shapes linear y = ax + b quadratic y = ax 2 + bx + c cubic y = ax 3 +bx 2 + cx+d quartic y = ax 4 + bx 3 + cx 2 + dx + e No Direction Change 1 Direction Change 2 Direction Changes 3 Direction Changes

6. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Which Function is Best Fit? Determine the type of polynomial function that could be used to represent the data in each scatter plot. Two direction change: cubic function would be best fit One direction change: quadratic function would be best fit

7. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Functions Modeling Data Write a polynomial function that models the set of data. x-.500.511.522.533.54 f(x)f(x)-10-6.4-5-5.1-6-6.9-7-5.6-24.615 enter x into L 1 enter f(x) into L 2 View Stat Plot and determine which function best models the data f(x) = x 3 – 3x 2 + x – 5 Determine Cubic Regression Equation and round coefficients to nearest integer STAT 6 ENTER  cubic

8. 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 198019851990199119921993199419951996 lbs. of waste per person per day 3.73.84.54.44.5 4.44.3 Is a linear function the best fit for this data? quadratic 1 direction change STAT 5 ENTER  Quadratic Regression y = ax 2 + bx + c a = -.004209 b =.119480 c = 3.592550 R 2 =.819795 y = -.004x 2 +.119x + 3.593

9. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Waste Problem y = -.004x 2 +.119x + 3.593 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) = -.004x 2 +.119x + 3.593 = 3.563 lb. b. Use the model to predict when waste will drop to 3 pounds per day. f(x) = -.004x 2 +.119x + 3.593 = 3 x ≈ -4 or 34 1976 or 2014

10. Aim: Graph of Best Fit Course: Alg. 2 & Trig. GrowthDecay GrowthDecay Exponential Functions y = ab x Logarithmic Functions y = a + b ln x Growth & Decay

11. 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. YRYR 19001910192019301940195019601970198019902000 0102030405060708090100 P1.651.751.862.072.32.523.023.74.445.276.06 Determine an equation that models the data.

12. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Growth & Decay 0102030405060708090100 P1.651.751.862.072.32.523.023.74.445.276.06 y = ax + b a =.0435 b =.97409 r 2 =.9008 r =.9491 y =.04x + 1 Growth y = a · b x a = 1.4457 b = 1.0136 r 2 =.9700 r =.9849 y = 1.44 ·1.01 x

13. 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) 0100 1310 2470 3715 41150 51650 62500

14. 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. x12345678 y1.57.410.213.415.816.318.218.3 Find a logarithmic model for the data y = 1.538+8.373 lnx

15. 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) 118.0 220.5 321.3 421.1 519.9 617.8 715.9 811.3 97.8 103.7 f(x) = -0.4x 2 + 2.8x + 16.3

16. Aim: Graph of Best Fit Course: Alg. 2 & Trig. Model Problem Write a polynomial function to model the set of data. xf(x)f(x) -2.0-22.0 -1.5-7.9 0.0 -0.53.5 0.04.1 0.52.9 1.02.1 1.52.5 2.05.8 2.514.1 3.028.0 f(x) = 2x 3 – 3x 2 – x + 4