A L C C U G LATORS N I H P A R G IN THE SCIENCE CLASSROOM GLENCOE McGraw-Hill New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois
Technology Consultants Christine A. Lucas Mathematics Teacher Whitefish Bay High School Whitefish Bay, Wisconsin Jill Baumer-Piña Former Program Assistant The Ohio State University Columbus, Ohio Other Titles in the Glencoe Science Professional Series Alternate Assessment in the Science Classroom Cooperative Learning in the Science Classroom Performance Assessment in the Science Classroom Copyright © 1999 by Glencoe/McGraw-Hill. All rights reserved. Printed in the United States of America.
GRAPHING CALCULATORS IN THE SCIENCE CLASSROOM s t n e t n o C f o e l Tab Introduction An Introduction to Graphing Calculators ..............1 Using Graphing Calculators in Cooperative Learning ..........................................................5 Graphing Calculators and the Mathematics Curriculum .................................6 Graphing Calculator Capabilities ..........................7 Getting to Know Your Graphing Calculator ........................................................
Using Programs................................................62 Templates Plotting Points in a Relation ................................63 Solving a System of Linear Equations ..................63 Using the Quadratic Formula..............................64 Using the Law of Cosines....................................65 Using Hero’s Formula .........................................65 Inner and Cross Products of Vectors ....................66 Sums of a Series.................................................
GRAPHING CALCULATORS IN THE SCIENCE CLASSROOM n o i t c u d o r t n I An raphing to G lators u c l a C The graphing calculator offers you much more flexibility in studying graphs than can be obtained by paper and pencil graphing. Within seconds you can graph an equation or a system of equations that could take you several minutes to sketch. The various functions on a graphing calculator can also help you to analyze the graph in a way not possible with traditional graphing techniques.
Casio Calculators Hewlett Packard Calculators There are several different Casio graphing calculators. There is also more than one model available within each number series of calculator. This guide includes keying sequences for the Casio fx-7700GE calculator. It has a variety of capabilities that permit you to perform calculations and statistical analyses. It can graph equations in a rectangular coordinate system, as well as graph inequalities and parametric equations.
Sharp Calculators Texas Instruments Calculators The Sharp EL-9200C and EL-9300C graphing calculators have an equation editor feature that allows you to enter an expression exactly the way it appears in the textbook. Both models include menus from which you can select specialized modes. Matrix operations are included in the calculation mode, while the graph mode lets you graph functions using rectangular or parametric coordinates. A unique feature in the graph mode is the jump key.
GRAPHING CALCULATORS IN THE SCIENCE CLASSROOM g n i h p a r G g n i s U ulators in Calc erative Coop ning Lear Cooperation is part of the real world at all levels, whether it be among nations, with co-workers, with neighbors, with your community, or within your family. Individuals must develop skills and understandings of working in groups to be successful in today’s world.
Graphing Calculators and the Mathematics Curriculum The graphing calculator skills you will develop in the following pages can be used in a variety of courses in the mathematics curriculum. The chart below lists some of those skills and the course(s) for which they may be appropriate. This list is not all inclusive, and many of the skills thought of as second-year algebra topics are now available to students in first-year algebra or middle school courses by the power of the calculator.
Graphing Calculator Capabilities Capability Casio Sharp fx-7700GE HP 38G EL-9200C TI-81 TI-82/83 TI-92 ✓ ✓ ✓ ✓ ✓ Evaluate numerical expressions ✓ ✓ ✓ Perform operations with fractions resulting in fractions ✓ ✓ ✓ Graph linear and quadratic functions ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
GRAPHING CALCULATORS IN THE SCIENCE CLASSROOM o t g n i Gett Your Know ing h p a r G lator u c l a C As with any new acquaintance, you need time to get to know your graphing calculator. Before beginning any specific task, look at the keyboard. Think of something you are used to seeing on your familiar scientific calculator and see if you can find it on your graphing calculator.
In the keystroke sequences in this booklet, these functions are shown using the appropriate access key and the function name rather than the actual key pressed. How do I clear what’s on the screen? Casio: Press å to completely clear the displayed formulas, numeric values, and texts. Use the Cls (ClearScreen) function to clear graphs from the screen. Press S∞e. TI: Press Ç to completely clear the displayed formulas, numeric values, or text. Clear graphs by pressing YÇé for each line that has an equation.
How do I know what’s on each menu? The TI calculators and Casio fx-7700GE have many functions that are accessed by using menus (lists of options) and making selections. Sometimes you may know what function you want to use, but you have no idea where to find it. In the Appendix on pages 80–81, you will find many of the menus that are available on each calculator, how to access them, and what each item on the menu means.
Evaluating Mathematical Expressions ● Casio fx-7700GE ● TI-81 ● TI-82/83 All graphing calculators observe the order of operations when evaluating a mathematical expression. The calculators also have parentheses that are used in the same way as in writing to group terms in an expression or to clarify the meaning of the expression. When you enter an expression in a graphing calculator, the entire expression appears as you enter it.
Mode and Range ● ● Casio fx-7700GE MODE TI-81 ● TI-82/83 Each of the calculators has a 5 key. However, the type of mode and the way in which you select the mode is slightly different for each calculator. In future activities in this booklet, we will indicate when you need to change modes and what mode(s) to use. Casio fx-7000GE: Pressing 5 once accesses the first menu screen shown below. Pressing 5 again accesses the second screen.
Normal Sci Eng Float 0123456789 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul FullScreen Split Type of notation for display Number of decimal places displayed Type of angle measure Type of graph Whether to connect plotted points How to plot selected functions How to display results Type of screen display Normal Sci Eng Float 0123456789 Radian Degree Func Par Pol Seq Connected Dot Sequential Simul Real a+b i re^ i Full Horiz G-T TI-82 RANGE or WINDOW TI-83 Use the range feature to set t
Functions ● Casio fx-7700GE ● TI-81 ● TI-82/83 A graphing calculator is a powerful tool for studying functions. Any of the graphing calculators will graph functions, but the procedure for graphing is slightly different for each one. On any of the calculators, you must set an appropriate range before you can graph a function. A viewing window of [ 10, 10] by [ 10, 10] with a scale factor of 1 on both axes denotes the domain values 10 x 10 and the range values 10 y 10.
2 Graph y 16x 2 11 so that a complete graph is shown. First graph on a standard viewing window. Casio fx-7700GE: TI-81: g m 16 ? › + 11 e Y m 16 | › + 11 G TI-82/83: Y m 16 [ › + 11 G The standard viewing window does not show a complete graph. Try the range [ 3, 3] by [ 5, 15] with a scale factor of 1 for both axes. When you finish changing the range values, you do not need to reenter the equation. Press 6 to exit the range menu and then press e to graph on a Casio calculator.
Trace and Zoom ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use a graphing calculator to find approximations of the coordinates of points on the graphs of functions. These coordinates can be found with great accuracy by using the trace and zoom features. TRACE 1 The trace feature moves the cursor along the graph of a function and displays approximations of the coordinates of points on the graph. Graph y x 3 6 in the standard viewing window.
Box Zoom Another method of zooming in that is available on the Casio fx-7700GE and the TI calculators is box zoom. With box zoom, the calculator prompts you to use the arrow keys to position the cursor at two opposite corners of a box to make a new viewing window. When you press the e or é key, the calculator redraws the graph of the function in the range that you specified with the box.
Zoom In and Zoom Out by Factors A third method of zooming in and out, which uses factors, is available on each graphing calculator. With zooming by factors, the calculator will enlarge or reduce the graph around a chosen center point by a factor or factors that you choose. On the Casio fx-7700GE and the TI calculators, you can choose different factors for the two axes. For example, if you start with a standard viewing window and choose 1.
Systems of Equations ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use a graphing calculator to graph and solve systems of equations since several equations can be graphed on the screen at one time. Graph the related functions for the system. Then use the trace and zoom features to approximate the solutions. 1 Graph the system of equations y 3.4x 2.1 and y 5.1x 8.3 in the standard viewing window. Then determine the coordinates of the intersection point(s).
2 Graph the system of equations y –x3 3x 2 9x and y –x2 10x 25. Determine the coordinates of the intersection point(s). Try graphing in the standard viewing window. Casio fx-7700GE: TI-81: gm?^33? › + 9? ⁄:gm? › + 10 ? - 25 e Ym|^3-3|›+9| é m | › + 10 | - 25 G TI-82/83: Ym[^3-3[›+9[ é m [ › + 10 [ - 25 G The standard viewing window does not show a complete graph. You must zoom out to view a complete graph so that you can determine how many solutions exist.
Inequalities ● Casio fx-7700GE ● T1-81 ● T1-82/83 The Casio fx-7700GE and TI calculators allow you to graph inequalities on the graphics screen. The procedures for graphing inequalities on a Casio fx-7700GE are similar to the procedures for graphing functions. Graphing inequalities on a TI requires using the “Shade(“ command from the draw menu and entering a function for a lower boundary of the inequality and a function for the upper boundary.
2 Graph y x 3 6x 2 8. Be sure to clear the graphics screen before graphing. Press S 3 e on the Casio fx-7700GE. Press ‹ Î 1 é on the TI since the draw menu was used instead of the Y list to create the graph. Try the standard viewing window. Casio fx-7700GE: g™?x3-6 ?›+8e TI-81: Since the inequality symbol is , the equation of the related function should be entered as the upper boundary in the “Shade“ command. The Ymin value will be entered as the lower bound.
Systems of Inequalities ● Casio fx-7700GE ● T1-81 ● T1-82/83 You can graph and solve a system of inequalities on a graphing calculator. When graphing on a Casio fx-7700GE, you will use the colon to separate the inequalities to be graphed. When graphing on a TI, you will use the shade command with the two inequalities entered as the upper and lower boundaries instead of a maximum or minimum value.
The graphing calculator can be used to solve inequalities by shading the area on the graph that makes the inequality true. To use this method, let each side of the inequality represent a separate inequality and graph the system of inequalities. 3 Solve 5 x 3 3x 8 by graphing a system of inequalities. x 3 y or y Use the inequalities 5 system of inequalities to be graphed.
Rational Functions ● Casio fx-7700GE ● TI-81 ● TI-82/83 A rational function is one in which one polynomial is divided by another, p(x) q(x) or in mathematical terms, f(x) where q(x) 0. Rational functions usually have features that polynomial functions do not have, and the graphing calculator is a good tool to explore these graphs. Some graphs of rational functions are discontinuous. The breaks in continuity can appear as asymptotes or as point discontinuities.
TI-81: Y(2|+3)/(|-1)G TI-82/83: Y(2[+3)/([-1) G First find the vertical asymptote by tracing along the graph and observing the x-values near the discontinuity. As y increases, x approaches 1. On the TI calculators, use the integer function on the Zoom menu to adjust your viewing window. Press Ω 8. Then use the arrow keys to move the cursor to the origin and press é. You can trace to a point where x 1 and the calculator gives no y-value. This shows that the equation of the vertical asymptote is x 1.
Radical Functions ● Casio fx-7700GE ● ● TI-81 TI-82/83 You can use a graphing calculator to graph various kinds of radical functions quickly and easily. To graph a radical function on the Casio graphing calculator, you can use the R, ã, or ß keys. The TI calculators do not have a ß key, although the TI-82/83 calculators do have ß on the MATH menu that you access by pressing the µ key.
Quadratic Relations ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use a graphing calculator to graph relations; however you must do some algebra before you graph. A graphing calculator will only graph functions, so quadratic relations that are not functions must first be written as a combination of functions. For example, the equation of the parabola x y 2 cannot be entered directly into the graphing calculator. We must solve the equation for y. In this case, you obtain y x.
2 Graph the circle whose equation is x 2 y 2 16. Solve the equation for y. x 2 y 2 16 y 2 16 x 2 y 16 x 2 Subtract x 2 from each side. Take the square root of each side. Use the standard viewing window. Casio fx-7700GE: g S R ( 16 - ? › ) ⁄:gmSR ( 16 - ? › ) e The Y-vars menu on the TI calculators allows you to use the name of a function as a variable in other expressions.
Trigonometric Functions and Their Inverses ● Casio fx-7700GE ● TI-81 ● TI-82/83 A graphing calculator is a good tool for exploring the graphs of trigonometric functions and their inverses. Since a graphing calculator is capable of graphing trigonometric functions in degrees or radians, be sure that your calculator is in the correct mode for the function you wish to graph. 1 Graph y cos x for 360° x 360°.
Graphing calculators can also graph the inverses of trigonometric functions. 3 Graph y Arccos x. Casio: Arcsin, Arccos, and Arctan are also built-in functions on Casio calculators, so you do not need to set the range. Press g S c e. TI: The range will need to be set on the TI calculator. A good window to use is [ 1, 1] by [ 20, 180] with a scale factor of 0.5 on the x-axis and 90 on the y-axis. For the TI-81, press Y ‹ C | G. For the TI-82/83, press Y ‹ C [ G.
Exponential and Logarithmic Functions ● ● Casio fx-7700GE TI-81 ● TI-82/83 A graphing calculator can be used to graph exponential and logarithmic functions just as it is used to graph other types of functions. Even functions that would be tedious and time consuming to draw by hand can be graphed quickly. 1 Graph y 92 x in the viewing window [ 7, 1] by [ 1, 9] with a scale factor of 1 on each axis. Be sure that your calculator is in rectangular coordinate mode.
Graphing calculators can be used to graph logarithmic functions with base 10 or base e. If you wish to graph a logarithmic function with another base, log x log a ln x ln a you must use the change of base formula, loga x or loga x . 4 Graph y log4 x. Use the viewing window [ 1, 5] by [ 3, 3] with a scale factor of 1 on each axis. log x log 4 Using the change of base formula, log4 x .
Application: Solving Equations in One Variable ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use the graphs of two linear functions to solve an equation like 3x 6 5x 2. For the first function, let y one side of the equation. For the second function, let y the other side of the equation. Then graph both equations on the same screen. The solution will be the x-coordinate of the point where the two functions intersect. 1 Solve 3x 6 5x 2 by graphing.
Application: Solving Quadratic Equations ● Casio fx-7700GE ● TI-81 ● TI-82/83 A quadratic equation is an equation in the form of Ax2 Bx C 0, where A 0. The solutions for this type of equation can be estimated by graphing the related function, y Ax2 Bx C and locating those points where y 0. The x-coordinates of these points are called the zeros of the function. The zeros of a function are the roots to the associated equation. 1 Solve 2x2 x 15 0. Graph y 2x2 x 15.
3 Solve 10x2 51x 65 0. Graph y 10x2 51x 65 0. The graph appears to intersect the x-axis at a single point between 2 and 3. Zoom in on the point where the graph intersects the x-axis. After several zooms, you can see that the graph actually intersects the axis in two points. The x-coordinates of these points appear to be almost exactly 2.5 and 2.6. If you substitute the values 2.5 and 2.6 in the original equation, you will find that they are in fact exact solutions. 4 Solve 8x2 1 0.
Application: Families of Graphs ● Casio fx-7700GE ● TI-81 ● TI-82/83 A family of graphs is a group of graphs that displays one or more similar characteristics. Many linear functions are related because they have the same slope or the same y-intercept as other functions in the family. All linear functions can be written in the form y mx b, where m represents the slope of the line and b is the y-intercept. You can graph several functions on the same screen and observe if any family traits exist.
Note: On the TI-82/83, families of functions are easily graphed using the bracket keys, { and }, which are second functions of the parentheses keys. The brackets are used to group those numbers that change when all of the other parts of the equations are identical. In Example 1, use the brackets to group the differing coefficients of x. Press Y ‹ { .5 , 1 , 3 , 5 , } [ + 1. Then press G and all of the functions will be graphed. In Example 2, use the brackets to group the differing values of b.
Application: Maxima, Minima, and Zeros of Functions ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use the graphing calculator to estimate the zeros and the relative maximum and minimum points of a polynomial function. First you must graph the function. Then use TRACE and ZOOM to determine the coordinates of the point(s) you desire. 1 Approximate the maximum, minimum, and zeros of y x 2 3x 4. First graph the function.
2 Approximate the maximum, minimum, and zeros of y 2x3 x2 5x 3 to the nearest hundredth. First graph the function. The graph appears to have a relative maximum, a relative minimum, and three zeros. Use TRACE and ZOOM to refine your estimates of the coordinates of each point. The relative maximum occurs at about ( 1.09, 1.05). The relative minimum occurs at about (0.76, 5.34). The three zeros occur at about 1.50, 0.62, and 1.62. 3 Approximate the maxima, minima, and zeros of y x4 5x2 6.
Application: Solving Trigonometric Equations ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use a graphing calculator to solve a trigonometric equation in the same way you have used it to solve linear equations, quadratic equations, and systems of equations. 1 Solve tan x sin x • cos x if 360° x 360°. Make sure your calculator is set for degree measure. Set the RANGE for x-values greater than the given interval, say [ 450°, 450°] with a scale factor of 90°.
Application: Verifying Trigonometric Identities ● ● Casio fx-7700GE TI-81 ● TI-82/83 You can use a graphing calculator to determine whether an equation may be a trigonometric identity by graphing each side of the equation as a separate function on the same screen and then comparing the two graphs. If the graphs appear to coincide, then the equation may be an identity. If the graphs do not match, then the equation is not an identity.
Application: Solving Exponential and Logarithmic Equations ● Casio fx-7700GE ● ● TI-81 TI-82/83 You can use a graphing calculator to solve equations involving logarithms and exponents just as you have used it to solve other types of equations. There are two methods: (1) graph each side of the equation and find the intersection points or (2) rewrite the equation so that it equals zero, graph the related function, and find the x-intercepts of the function.
Entering Matrices ● Casio fx-7700GE ● TI-81 ● TI-82/83 Graphing calculators make many of the tedious paper-and-pencil operations with matrices quick and simple. Each calculator has different matrix capabilities, but they all work through matrix menus. Suppose you wanted to enter the matrix 1 3 –6 . 2 –2 4 Casio fx-7700GE: The Casio fx-7700GE contains memories for five different matrices and a special matrix answer memory. You can enter values into matrices A through E.
If the matrix you chose has the same dimensions as the matrix you wish to enter, press é twice. Element 1, 1 (row 1, column 1) is highlighted. Enter the number you want and press é. Continue the process until all values are entered. If the matrix you chose does not have the same dimensions as the matrix you wish to enter, press the number for the number of rows and é. Then press the number of columns and é. Proceed as shown above to enter each element.
Determinants, Inverses, and Operations ● Casio fx-7700GE ● TI-81 ● TI-82/83 The graphing calculator has simplified the tasks of finding determinants and inverses of matrices to the pressing of only a few keys. 1 Find the determinant of matrix A if A 1 3 . 2 –2 Enter the matrix into your calculator. Casio fx-7700GE: Press É É ™ ¡ ⁄ A e. The determinant appears. Press ¡ ⁄ A ∞ to return to the original matrix screen. TI-81: Press N. Then select 5:det.
Other operations with matrices that can be done with a graphing calculator include scalar multiplication, multiplying two matrices, squaring a matrix, adding and subtracting two matrices, and finding the transpose of a matrix. Casio fx-7700GE: Once you have entered matrices A and B, you can add the two matrices, subtract the two matrices, and multiply the two matrices. A B Press ¡ ⁄ A + ¡ ⁄ B e. The sum appears. A B Press ¡ ⁄ A - ¡ ⁄ B e. The difference appears. A B Press ¡ ⁄ A u ¡ ⁄ B e.
Statistical Computations ● Casio fx-7700GE ● ● TI-81 TI-82/83 In addition to graphing, graphing calculators are capable of doing intricate computations, including finding statistical values for a set of data. The first step on the Casio fx-7700GE is to choose the appropriate statistical mode for statistical computations. This mode is called standard deviation mode. Press … 3. You may choose to store the data for computations or to use the data once and discard it. We will use the data storage mode.
‹_ªé Accesses the data edit option. 51 ééé 40 éé 58 éé and so on 47 é é TI-82/83: Data is entered individually into a list. Before you enter new data, clear data that may have been previously stored in data list L1. Press _ 4 ‹ L é. Use the following procedure to enter the new data. _é Accesses the data edit option. 51 é 40 é 58 é and so on 47 é If you make a mistake entering data into your TI calculator, use the arrow keys to move to the incorrect data and enter the correct value.
2 Find the median of the test scores listed below on a TI calculator. 82 91 74 78 94 68 74 88 64 42 72 82 79 99 98 75 61 78 86 69 Only the Casio fx-7700GE and the TI-82/83 graphing calculators will compute the median of a set of data for you. However, the TI-81 calculator will sort data from least to greatest according to the x- or y-values so that the median can be found easily. Clear the data memory and enter the data into your calculator. Casio fx-7700GE: ™£¡ Clear the data memory.
Histograms ● Casio fx-7700GE ● TI-81 ● TI-82/83 A graphing calculator is capable of drawing statistical graphs as well as graphs of functions. The Casio calculator must be placed in statistical graph draw mode before statistical graphs can be drawn. To do this, press … 3 then S Ñ • • • ¡. Before starting each graph, it is important to clear the statistical memory.
Casio fx-7700GE: 102.5 £ 3 ¡ 117.5 £ 7 ¡ 132.5 £ 15 ¡ 147.5 £ 34 ¡ and so on 252.5 £ 4 ¡ g e Draws the graph. TI-81: ‹ _ Enters the statistical menu. ª é Chooses the data edit option. 102.5 é 3 é 117.5 é 7 é 132.5 é 15 é 147.5 é and so on 252.5 é 4 ‹ _ j éé Chooses histogram from the statistical draw menu and executes. TI-82: The TI-82 requires that the class marks be entered as data list L1 and the frequencies be entered as data list L2. _ é Enters the data edit menu. 102.5 é 117.5 é and so on 252.
Casio fx-7700GE: 84 ¡ 120 ¡ 78 ¡ 89 ¡ and so on 115 ¡ g e TI-81: The calculator also enters 1 as the second value of the data pair automatically, so you can press éto accept this value instead of entering 1 for each y-value. ‹ _ ª 1 Chooses the data edit option. 84 é é 120 é é 78 é é 89 é é and so on 115 ‹ _¶ 1 é Chooses histogram from the statistical draw menu and executes. TI-82/83: _ é Chooses the data edit option. 84 é 120 é 78 é 89 é and so on 115 é ‹ w é Chooses the statplot menu.
Scatter Plots and Lines of Regression ● ● Casio fx-7700GE TI-81 ● TI-82/83 Graphing calculators are capable of drawing scatter plots and lines of regression for data that you enter into the memory. Once the line of regression is plotted, you can write the equation of the line and use the trace function to approximate solutions to other problems.
Casio fx-7700GE: 9 £ 89 ¡ 6 £ 76 ¡ and so on 30 £ 63 ¡ TI-81: ‹ _ ªé Chooses the data edit option. 9 é 89 é 6 é 76 é and so on 30 é 63 é TI-82/83: _ é Chooses the data edit option. 9 é 6 é 2 é and so on 30 é Enters first list into L1. ¶ 89 é 76 é 93 é and so on 63 é Enters second list into L2. Now draw the regression line. Casio: gS¢1e TI-81: ‹_2é Calculates the coefficients of the regression line.
Curve Fitting ● Casio fx-7700GE ● TI-81 ● TI-82/83 Some sets of data do not display a linear pattern when plotted on a scatter plot. A set of data that is fit best by a curve is called curvilinear data. Since most realworld data are positive, most scatter plots are graphed in the first quadrant. If the points of these data sets are connected, the graph might resemble an exponential graph of the form y AeBx or a polynomial graph of the form y AxB.
First set the range parameters. A viewing window of [0, 225] by [0, 275] with scale factors of 25 for both axes is appropriate. Next, set your calculator in the proper statistical mode and clear the statistical memories and the graphics screen. Also, clear any equations from the Y5 list on the TI. Casio fx-7700GE: TI-81: … 4 Sets the regression mode. S Ñ ¡ Sets the rectangular coordinates graph mode. • • • ¡ Sets the statistical draw mode. • £ Sets the exponential regression curve drawing mode.
On the TI calculators, the equation of the regression curve is y 4.721 1.02x, where a A, and b x B. Pressing J V ¶ ¶ 2 é on the TI-81 and J V 5 ¶ ¶ 2 é on the TI-82/83 will give the value of B. The equation then can be written as y 4.721e 0.0208x. On the Casio fx-7700GE, press © º ¡ e ™ e to display the values of A and B. The equation for the regression curve is then y 4.721e 0.0208x.
Now draw the scatter plot on the regression curve or line. Casio: ɺgSH1e TI-81: ‹_¶2é‹_5 éYV¶¶4G‹ _¶2é TI-82/83: _¶⁄Bé Y Ç Clears the exponential equation from the Y list. V5¶¶7G Regression line (TI) As long as the statplot menu has not been changed, the scatterplot will still be displayed. Using a TI calculator, the equation for the regression curve is y 0.0444 x 1.5579, where A a and b B. On the Casio fx-7700GE, find the values of A and B by pressing © and ™ e.
Probability and Combinatorics ● Casio fx-7700GE ● TI-81 ● TI-82/83 You can use a graphing calculator to find permutations and combinations and to generate random numbers. These capabilities are useful in finding probabilities and in modeling situations to study probability. 1 Find 12!. Casio fx-7700GE: 12 S µ Enter the number and access MATH menu. ™ ¡e Access the probability function menu. TI-81: 12 µ Choose x! and execute. Enter the number and access the MATH menu.
3 What is the probability that two cards drawn at random from a standard deck of 52 cards are both hearts? C(13, 2) P(two hearts) C(52, 2) Casio fx-7700GE: TI: ( 13 £ 2 ) / ( 52 £ 2 ) e ( 13 µ ª 3 2 ) / ( 52 µ ª 3 2 ) é The probability that the cards are hearts is 0.06 to the nearest hundredth. Hint: You can display a calculated probability as a fraction on a Casio fx-7700GE by pressing a instead of / in the expression line.
GRAPHING CALCULATORS IN THE SCIENCE CLASSROOM g n i s U ams Progr Like larger computers, a graphing calculator can be used to write and run programs. The programs that follow will run on the TI-82 and the TI-83 calculators, but could be adapted easily for use on another type of graphing calculator or for a computer that runs BASIC or PASCAL. To enter a program on the TI-82/83, press P, highlight NEW, and press é. Type the program name and press é. Enter the program pressing é at the end of each statement.
Plotting Points in a Relation The following program will plot the points in a relation. Name = PLOTPTS :Fn Off Deactivates all current functions. :ClrDraw Clears the graphics screen. :Lbl 1 Sets the marker at position 1. :Disp “X ” Displays the message “X ”. :Input X Accepts an X value. :Disp “Y=” :Input Y :PT-On(X,Y) Plots the point (X, Y). :Pause Waits until the user presses the ENTER key. :Disp “PRESS 0 TO QUIT OR 1 TO PLOT MORE POINTS” :Input A :If A = 0 Tests to see if the entered value is 0.
Using the Quadratic Formula The following program will find the solutions to a quadratic equation using the quadratic formula. The solutions will be given in decimal form.
Using the Law of Cosines This program finds the measure of a side of a triangle using the law of cosines. In the program, A is the measure of the missing side, B and C are the measures of the second and third sides, and is the measure of the angle opposite side A. Name = LAWCOS :Deg :Disp “INPUT B” :Input B :Disp “INPUT C” :Input C :Disp “INPUT “ :Input : (B2 C2 2BC cos ) → A :Disp “A ” :Disp A :End Places the calculator in degree mode. Calculates the value of A.
Inner and Cross Products of Vectors Program 6 can be used to find the inner product of two vectors in a plane or in space. The program will indicate that the vectors are perpendicular if the inner product is 0. Program 7 below calculates the cross product of two vectors in space.
Sum of a Series The following program will find the sum of the first n terms of an arithmetic or geometric series. You will enter the type of series, the first term, the difference for an arithmetic series or the ratio for a geometric series, and the number of terms.
Graphical Iteration This program will perform graphical iteration of a logistic function, which is of the form f (x ) ax(1 x ). Press é for values and to advance the graph. Before you run the program, set the range to [0, 1] by [0, 1] with scale factors of 0.1 on both axes. Note: To put “Fix 3” on the second line of the program, press 5, highlight 3 after FLOAT on the second line, then press é.
Generating Random Numbers The following program will generate random numbers between two values. You input the seed number, which determines the way in which the numbers are generated. Note that the program will not eliminate duplicate numbers, and the factory set seed number is 0. Entering the least and greatest integers of 1 and 6 can simulate rolling one die, or entering 1 and 2 can simulate tossing a coin.
Area Between Two Curves The following program will approximate the area between the graphs of two functions by dividing the region into rectangles. Store the two functions as Y1 and Y2 in the Y list. The program requires you to input the lower and upper bounds and the number of rectangles to use for the calculation.
Mickey Mouse The following program will draw a picture of Mickey Mouse. The program was written by Ryan Kovacik, a sophomore at Strongsville High School in Strongsville, Ohio. It was printed in the April 1991 issue of Mathematics Teacher. Before running the program, set the viewing window to [6.5, 33.5] by [8, 32] with scale factors of 40 on both axes and clear any equation from the Y list.
Casio fx-7700GE CASIO POWER GRAPHIC fx-7700GE • • • • ICON MENU• • • • Trace • Zoom Plot Line Cls Coord F1 F2 F3 F4 F5 F6 A -LOCK QUIT SET UP SHIFT ALPHA EXIT MENU REPLAY G- dx G : d/dx T dx A 10 x B x • DRG 1 • F MEM 0 C 3 sin 1 x 1 N 8 U • DISP 2 • F tan K x L OFF DEL AC /ON R S X W 3 ] Ans EXP T [ Z Defm , INS 6 V • CLR tan 1 O Q • VAR E cos J 9 5 cos 1 ) ( M D sin I 2 P • MATH 4 ex ; M Disp In H G 7 Range
HP 38G HEWLETT PACKARD 38G SETUP PLOT SYMB NUM VIEWS NOTE SKETCH LIB VAR MATH MODES ASIN ACOS ATAN HOME SIN COS TAN A B C ANSWER CHARS ENTER ( a...z LIST A...
Sharp EL-9300C GRAPHICS EL-9300C Scientific calculator SOLVER OFF +– –+ ON JUMP AUTO PLOT EQTN RANGE ZOOM COMMAND OPTION MATH MENU QUIT A-LOCK 2nd F sin–1 A cos–1 sin B Exp i tan–1 L 7 J 9 F ln K RCL ex MAT X/ /T O ( P ( S T U W X Y Z , SPACE ANS –) ENTER 5 2 E log N R V 0 a CL STO 8 4 1 l M Q 10 x x2 ab b BS x –1 D C tan H a DEL SET UP cos G CA INS ALPHA 6 3 ( + – = 75
TI-81 TEXAS INSTRUMENTS Tl-81 Y 2nd RANGE ZOOM TRACE GRAPH INS DEL X|T MODE DRAW C Y-VARS QUIT PRGM VARS A-LOCK ALPHA TEST A STAT B MATH D SIN 1 ABS MATRX 1 I 10 S T X [A] STO ON 76 Z , {y} 0 • RESET [C] 3 2 W V 6 5 Y [B] R 9 U 1 {x} Q 8 4 M ) P 7 LN L ( O H TAN K EE N OFF COS J LOG e E COS 1 F TAN 1 G SIN 2 CLEAR ANS ( ) " ? ENTRY ENTER
TI-82 Tl-82 TEXAS INSTRUMENTS STAT PLOT TblSet Y WINDOW QUIT 2nd LINK ALPHA TEST A MATH EE N , Un-1 LOG S J { X L1 K O Vn-1 1 0 L V W RESET L3 3 , ] 6 2 R 9 Z • M Q [ P n U L6 : ) 5 Y L2 OFF } 8 L5 H TAN ( 4 STO ON COS T L4 CLEAR VARS E COS 1 F TAN 1 G 7 LN C Y-VARS PRGM SIN I RCL STAT MATRX 2 e LIST ANGLE B DRAW 1 10 DEL X,T, D SIN 1 ABS TABLE GRAPH INS MODE A-LOCK CALC TRACE ZOOM ANS ( ) ? " ENT
TI-83 Tl-83 TEXAS INSTRUMENTS STAT PLOT TBLSET WINDOW Y= QUIT 2nd A-LOCK LINK X,T, ,n MATH LIST STAT MATRX D SIN–1 EE N E PRGM ON 78 { L1 0 [ R V L3 3 2 : • Q M ] W MEM " 6 Z i e 9 5 1 L ) U L6 Y L2 CATALOG } P w L5 4 X K 8 T H TAN ( O v L4 LN OFF COS J CLEAR VARS COS–1 F TAN–1 G 7 S STO , u LOG RCL GRAPH DEL SIN ex TABLE TRACE A ANGLE B DRAW C DISTR TEST 10x CALC ZOOM INS MODE ALPHA FINANCE FORMAT ANS ( ) ? ENTRY S
A ON OFF Z CAPS W F8 F4 Q F7 F3 Y= F6 F2 HOME F5 F1 K LOC S 2nd X E D STO RCL C R WINDOW GRAPH F V T TblSet G B Y TABLE TEXAS INSTRUMENTS H N U J \ ; M I i : P ENTER ENTRY L INS DEL K O Tl-92 0 • > 2 1 < CATALOG 5 4 EE 8 MATH 7 d ) } ( COS SIN { COS-1 Y= APPS ESC SIN-1 LN e CLEAR MODE 2nd QUIT 9 ENTER ( ) ENTRY CHAR VAR-LINK ANS 3 CUSTOM 6 MEM -1 ] [ , TAN TAN-1 ENTER TI-92 79
Menus: Casio fx-7700GE There are numerous menus and submenus accessed by the F keys located on the first row of the Casio fx-7700GE calculator. The list below explains some of the abbreviations associated with the F1 through F6 keys in commonly used menus.
Menus: TI-81 The TI-81 has 10 menus from which various functions can be selected. A menu may also contain other menus with additional functions. The lists below contain detailed information on the contents of six of the more commonly-used menus.
Index ALPHA, 9 Arccos, Arcsin, Arctan, 31 asymptote, 25–26, 33, 43 BOX ZOOM, 17, 19 CALC menu, 34, 36, 40 Casio, 4, 7, 11–61, 73, 80 circle, 29, 66 CLEAR, 9, 48, 51, 53–54, 56–58 colon (:), 19, 23, 29 combinations, 60–61 complex numbers, (powers), 64 complete graph, 15, 20, 22, 27, 29 contrast, 8 cooperative groups, 5 correlation value, 55, 58–59 COS, 30, 41–42, 65 critical points of functions, 15, 22–23, 27, 39–40 curriculum, using the graphing calculator, 6 cursor, free–moving, 22, 24 curve fitting, 56–59
