Graphing. Enter the function in the input line. Resize the graphics window, if needed, to get a good view of the function.

Transcription

1 Example 5: The path of a small rocket is modeled by the function ht ( ) 16t 18t 1 where initial velocity is 18 feet per second and initial height is 1 feet. The model gives the height of the rocket in feet, t seconds after launch. Find the height of the rocket: A. seconds after launch. B. 4 seconds after launch. C. 5 seconds after launch. D. 8 seconds after launch. Enter the function in the input line. Evaluate the function at each t value given above. Command for A: Command for B: Command for C: Command for D: Graphing Example 6: Use GGB to graph the function f ( x) x 3. Enter the function in the input line. Resize the graphics window, if needed, to get a good view of the function. TO MOVE GRID: Make sure that is selected then put the cursor anywhere on the grid, and move it. TO VIEW STANDARD VIEW: Ctrl+M or right-click on the grid and select Standard View. TO ADJUST X- Y- AXES: Put the cursor on the respective axis and move up, down, right or left. Lesson An Introduction to GGB 8

2 Example 7: Graph the function 3 hx ( ) 3x 3x 96x 180 and find an appropriate viewing window. Lesson An Introduction to GGB 9

3 Finding Some Features of a Graphed Function You can find the zeros (also called roots or x intercepts) of a function using GGB. Example 8: Suppose 3 gx ( ) x x 9x 18. Find the zeros of the polynomial function. A. Enter the function in the input line. Graph and resize if needed. B. Find the zeros of the polynomial function. (When you begin to enter the command, a list will appear.) Lesson An Introduction to GGB 10

4 Example 9: Suppose f( x) x 3x 5. Find the zeros of the function. x 3 A. Enter the function in the input line. Graph and resize if needed. B. Find the zeros of the function. The command in the input line will be the same as in Part B of the previous example; however, this function is not a polynomial so we ll need to choose the command: Roots[<function>, <Start value>, <End value>] Make sure to choose the command ROOTS NOT ROOT. Lesson An Introduction to GGB 11

5 The relative extrema of a function are the high points and low points of the graph of a function, when compared to other points that are close to the relative extremum. A relative maximum will be higher than the points near it, and a relative minimum will be lower than the points near it. GGB will help you find these points. Example 10: Suppose 3 gx ( ) x 5x x 3. Find any relative extrema. A. Enter the function in the input line. Graph and resize, if needed. B. Find the relative extrema. Relative Max: Relative Min: Lesson An Introduction to GGB 1

6 Example 11: Suppose 1 3 hx ( ) x 3x. Find any relative extrema. A. Enter the function in the input line. Graph and resize, if needed. B. Find the relative extrema. Since the function is not a polynomial, the command is: Extremum[<Function>,<Start x-value>,<end x-value>] Relative Max: Relative Min: Lesson An Introduction to GGB 13

7 Intersection of Two Functions Example 1: Find any points where f( x) 1.45x 7.x 1.6 and gx ( ).84x 1.9 intersect. A. Enter the functions one by one in the input line. Graph and resize if necessary to view all points of intersection. B. Find intersection points. Lesson An Introduction to GGB 14

8 6 x 1 Example 13: Find any points where f( x) x and gx ( ) 4e xintersect. x A. Enter the functions one by one in the input line. Graph and resize if necessary to view all points of intersection. B. Find intersection points. Since the function is not a polynomial, the command is: Intersect[<Function>,<Function>,<Start x-value>,<end x-value>] Lesson An Introduction to GGB 15

9 Lesson 3: Regressions Using GeoGebra In this course, you will frequently be given raw data about a quantity, rather than a function. If this is the case, you will need to have a method for finding a function that fits the data that is given that is, a function that passes through or passes close to many or most of the points of data that are given. These equations are called regression equations. You ll be able to use GGB to find these. The process involves several steps. Example 1: Suppose you are given the data shown in the table below. x y A. Create a table of values in the spreadsheet view of GGB. Recall: To view the spreadsheet, go to the Veiw menu and select Spreadsheet. B. Create a list of ordered pairs in Column C of the spreadsheet. Result: Lesson 3 Regressions 1

10 C. Select the ordered pairs and create a list using the list icon in spreadsheet mode. D. Find a linear regression model. E. Find a cubic regression model. F. an exponential regression model. Lesson 3 Regressions

11 You will also be asked to find a value for regression models called r or R. These values are measures of the goodness of fit for a regression model, and will be between 0 and 1. The closer the value is to 1, the better the linear regression model predicts the trend of the given data. The closer it is to 0, the less useful it will be in predicting future values. There are differences between the two values, in terms of how they are computed. For our purposes in this class, they will give us a piece of information for determining how well a regression equation fits the underlying data. GGB will compute this value for you if you specify the list of points to use and the name of the regression model. The command is RSquare[<List of points>, <Function>]. Example : Use the data and the cubic and exponential regression models that you found in Example 1 to find values for r or R for each of the two regression models. A. r or R for the cubic regression model: B. r or R for the exponential regression model: Example 3: Suppose that we know the revenues of a company each year since 005. This information is given in the table below: year revenues ( in millions of dollars) Before starting this problem, rescale the data so that the year 005 corresponds to x 0. A. Enter the data into the GGB spreadsheet and draw a scatterplot. Then create a list of the ordered pairs. Lesson 3 Regressions 3

12 B. Find the linear, cubic and quartic regression models. Find the value for r or R for each model. Linear Model Linear Model: r or R : Cubic Model: Cubic Model: r or R : Lesson 3 Regressions 4

13 Quartic Model: Quartic Model: r or R : C. Which model would be the best one to use? Why? Lesson 3 Regressions 5

14 D. Use that model to predict revenues in 01. Note: There is also a power regression model command: fitpow[<list of points>] Lesson 3 Regressions 6