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Linear Functions and Models Lesson 2.1 Problems with Data l Real data recorded l Experiment results l Periodic transactions l Problems l Data not always recorded accurately l Actual data may not exactly fit theoretical relationships l In any case … l Possible to use linear (and other) functions to analyze and model the data Fitting Functions Viscosity to Data Temperature 160 (lbs*sec/in2) 28 170 26 180 24 l Consider the data 190 21 200 16 given by this example 210 13 220 11 230 9 l Note the plot of the data points l Close to being in a straight line Finding a Line to Approximate the Data l Draw a line “by eye” l Note slope, y-intercept l Statistical process (least squares method) l Use a computer program such as Excel l Use your TI calculator Graphs of Linear Functions l For the moment, consider the first option Given the graph with tic marks = 1 l Determine l Slope l Y-intercept l A formula for the function l X-intercept (zero of the function) Graphs of Linear Functions l Slope – use difference quotient l Y-intercept – observe l Write in form l Zero (x-intercept) – what value of x gives a value of 0 for y? Modeling with Linear Functions l Linear functions will model data when l Physical phenomena and data changes at a constant rate l The constant rate is the slope of the function l Or the m in y = mx + b l The initial value for the data/phenomena is the y-intercept l Or the b in y = mx + b Modeling with Linear Functions l Ms Snarfblat's SS class is very popular. It started with 7 students and now, 18 months later has grown to 80 students. Assuming constant monthly growth rate, what is a modeling function? l Determine the slope of the function l Determine the y-intercept l Write in the form of y = mx + b Creating a Function from a Table l Determine slope by using x y 3 7 Answer: 4 8.5 5 10 6 11.5 Creating a Function from a Table l Now we know slope m = 3/2 l Use this and one of x y the points to determine 3 7 y-intercept, b 4 8.5 l Substitute an ordered 5 10 pair into 6 11.5 y = (3/2)x + b Creating a Function from a Table l Double check results l Substitute a different ordered pair into the formula l Should give a true statement x y 3 7 4 8.5 5 10 6 11.5 Piecewise Function l Function has different behavior for different portions of the domain Greatest Integer Function l = the greatest integer less than or equal to x l Examples l Calculator – use the floor( ) function Assignment l Lesson 2.1A l Page 88 l Exercises 1 – 65 EOO Finding a Line to Approximate the Data l Draw a line “by eye” l Note slope, y-intercept l Statistical process (least squares method) l Use a computer program such as Excel l Use your TI calculator 15 You Try It Weight Calories 100 2.7 l Consider table of ordered pairs 120 3.2 showing calories per minute 150 4.0 170 4.6 as a function of body weight 200 5.4 l Enter data into data matrix of 220 5.9 calculator l APPS, Date/Matrix Editor, New, 16 Using Regression On Calculator l Choose F5 for Calculations l Choose calculation type (LinReg for this) l Specify columns where x and y values will come from 17 Using Regression On Calculator l It is possible to store the Regression EQuation to one of the Y= functions 18 Using Regression On Calculator l When all options are set, press ENTER and the calculator comes up with an equation approximates your data Note both the original x-y values and the function which approximates the data 19 Using the Function l Resulting function: l Use function to find Calories Weight Calories 100 2.7 for 195 lbs. 120 3.2 l C(195) = 5.24 150 4.0 170 4.6 This is called extrapolation 200 5.4 220 5.9 l Note: It is dangerous to extrapolate beyond the existing data l Consider C(1500) or C(-100) in the context of the problem l The function gives a value but it is not valid 20 Interpolation l Use given data Weight Calories l Determine 100 2.7 proportional 120 3.2 “distances” 150 4.0 170 4.6 25 x 30 195 ?? 0.8 200 5.4 220 5.9 Note : This answer is different from the extrapolation results 21 Interpolation vs. Extrapolation l Which is right? l Interpolation l Between values with ratios l Extrapolation l Uses modeling functions l Remember do NOT go beyond limits of known data 22 Correlation Coefficient l A statistical measure of how well a modeling function fits the data l -1 ≤ corr ≤ +1 l |corr| close to 1 ó high correlation l |corr| close to 0 ó low correlation l Note: high correlation does NOT imply cause and effect relationship 23 Assignment l Lesson 2.1B l Page 94 l Exercises 85 – 93 odd