Session 4 2:40 3:30. If neither the first nor second differences repeat, we need to try another

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1 Linear Quadratics & Exponentials using Tables We can classify a table of values as belonging to a particular family of functions based on the math operations found on any calculator. First differences are found by subtracting successive y values. If the first differences are the same for the entire list, then the table represents a linear function. The function definition can also be identified from the table. Let s look at a pattern in table form: What is the change in the x value? What is the slope for this function? What is the y intercept for this data? for What is a function representation for this table? Find the first differences for our next example: What do you notice about the first differences? Teaching Algebra, Functions, and Data Analysis Page 59

2 Second differences are found by subtracting the successive values in the list of first differences. If the second differences are the same for the entire list, then the table represents a quadratic function, and the leading coefficient will be half the second difference (when the x values change by 1). Find the second differences for the data. What type of function does the table represent? What is the coefficient of the leading variable? What is a function representation for this table? Even if the numbers get large, you can still use this method to identify the type of function and arrive at a function definition from the data tables. Find the first and second differences for this data. What type of function does the table represent? If neither the first nor second differences repeat, we need to try another approach. Successive quotients are found by dividing successive pairs of y values. If the quotients repeat for the entire list, then the table represents an exponential function. The initial value can be identified from the table, and the common ratio is the repeated successive quotient value. Find the successive quotients for the data. What is the common ratio? What is the initial value? What is a function definition for this table of data? Teaching Algebra, Functions, and Data Analysis Page 60

3 Adapted from Teaching Algebra, Functions, and Data Analysis Page 61

4 Calculating Regressions that Model Data using a TI Graphing Calculator WHY USE A GRAPHING CALCULATOR? A graphing calculator can help determine the type of function that models data in ways you wouldn t want to do by hand. Sometimes the numbers get big and difficult to work with and there are also times in real life that the sequence does no t have perfect finite differences. You must determine the type of function accurately if you are planning to use your regression to predict future data and the calculator is a great tool to help with that. Also a graphing calculator allows students to visualize what they are finding and make connections about what they are learning. HOW TO USE YOUR CALCULATOR TO GET REGRESSION EQUATION: One way to calculate your data s regression equation is to put the data into the Statistics feature on a TI series calculator. (Before starting, press 2 ND zero and select DIAGNOSTICS ON. This will show you your correlation coefficient r and r 2. These will help you determine which equation best fits your data.) To enter the data into the calculator, press STAT and the EDIT. Put all the x s in List1, and the y s in List2. Then press STAT, CALC, and predict what kind of function best fits your data. You can choose Linear (LinReg), Quadratic (QuadReg), or exponential (ExpReg). There are other options in your calculator which will be explored in other classes. The closer r 2 is to 1 the better the equation fits the given data. You can try calculating different equations from your data and see which one fits best. Just make sure you re a value isn t zero or it isn t the best equation for your data. HOW TO GRAPH THE DATA AND REGRESSION EQUATION: To see the graph of your data press 2 nd Y= and turn your Stat Plot1 on. It should be a scatterplot using L1 and L2 for your data. Then press Zoom and 9 for ZoomStat. To graph the regression equation you have just calculated press Y=, VARS, go down to 5: Statistics, press the right arrow to go over to EQ (for equations) and you want 1: RegEQ for your regression equation. Then when you press graph you can see how your equation fits your data. HOW TO USE THE TABLE FEATURE TO PREDICT FUTURE DATA: To use your regression equation to predict future data you must put your equation into Y1 and then press 2 nd GRAPH to see the TABLE of values. Scroll down to the x value you want. To set the x to start at the value you want, press 2 nd WINDOW and put the x value you want into TblStart=. Then when you press 2 nd GRAPH the Table should start at that x value. Here is a real world example contributed by Texas Instruments found at Exploring Data on the TI 84 The average height in inches of United States children from ages 7 to 15 is given: Teaching Algebra, Functions, and Data Analysis Page 62

5 Age (yrs.) Height (in.) to enter the data STAT, 1:Edit, enter data into L 1 and L 2 2. to set up the plot of the data 2 nd,stat PLOT, 1:PLOT1, ENTER On Type: scatter Xlist: L 1 Ylist: L 2 Mark (any) 3. to graph the scatter plot ZOOM, 9: ZoomStat 4. to turn on correlation coefficient CATALOG DiagnosticON ENTER 5. to find the linear regression STAT, CALC, 4:LinReg(ax+b) L 1, L 2, y 1 and paste it into the y 1 (found under VARS,Y VARS,1:Function, 1: Y 1 ) equation on the home screen EXIT SLIP Linear Quadratic Exponential Sequence A Sequence B Sequence C Sequence D Teaching Algebra, Functions, and Data Analysis Page 63