Chapter 5 Graphing Sections 5-1 and 5-2 of this chapter provide basic information you need to know in order to draw a graph. The remaining sections describe more advanced graphing features and functions. Select the icon in the Main Menu that suits the type of graph you want to draw or the type of table you want to generate.
5-1-1 Sample Graphs 5-1 Sample Graphs k How to draw a simple graph (1) Description To draw a graph, simply input the applicable function. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. Execution 2. Input the function you want to graph. Here you would use the V-Window to specify the range and other parameters of the graph. See 5-2-1. 3. Draw the graph.
5-1-2 Sample Graphs ○ ○ ○ ○ ○ Example To graph y = 3x 2 Procedure 1 m GRPH • TBL 2 dvxw 3 5(DRAW) (or w) Result Screen 19990401
5-1-3 Sample Graphs k How to draw a simple graph (2) Description You can store up to 20 functions in memory and then select the one you want for graphing. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. Execution 2. Specify the function type and input the function whose graph you want to draw.
-1-4 Sample Graphs ○ ○ ○ ○ ○ Example Input the functions shown below and draw their graphs Y1 = 2 x 2 – 3, r 2 = 3sin2θ Procedure 1 m GRPH • TBL 2 3(TYPE)b(Y=)cvx-dw 3(TYPE)c(r=)dscvw 3 5(DRAW) Result Screen (Param) (INEQUA) 19990401 (Plot)
5-1-5 Sample Graphs k How to draw a simple graph (3) Description Use the following procedure to graph the function of a parabola, circle, ellipse, or hyperbola. Set Up 1. From the Main Menu, enter the CONICS Mode. Execution 2. Use the cursor fc keys to specify one of the function type as follows.
5-1-6 Sample Graphs ○ ○ ○ ○ ○ Example Graph the circle (X–1)2 + (Y–1) 2 = 22 Procedure 1 m CONICS 2 ccccw 3 bwbwcw 4 6(DRAW) Result Screen (Parabola) (Ellipse) 19990401 (Hyperbola)
5-2-1 Controlling What Appears on a Graph Screen 5-2 Controlling What Appears on a Graph Screen k V-Window (View Window) Settings Use the View Window to specify the range of the x- and y-axes, and to set the spacing between the increments on each axis. You should always set the V-Window parameters you want to use before graphing. u To make V-Window settings 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Press !K(V-Window) to display the V-Window setting screen.
5-2-2 Controlling What Appears on a Graph Screen u V-Window Setting Precautions • Inputting zero for Tθ ptch causes an error. • Any illegal input (out of range value, negative sign without a value, etc.) causes an error. • An error occurs when Xmax is less than Xmin, or Ymax is less than Ymin. When Tθ max is less than Tθ min, Tθ ptch becomes negative. • You can input expressions (such as 2π) as V-Window parameters.
5-2-3 Controlling What Appears on a Graph Screen k Initializing and Standardizing the V-Window u To initialize the V-Window 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Press !K(V-Window). This displays the V-Window setting screen. 3. Press 1(INIT) to initialize the V-Window. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.1, Yscale = 1 Xdot = 0.
5-2-4 Controlling What Appears on a Graph Screen k V-Window Memory You can store up to six sets of V-Window settings in V-Window memory for recall when you need them. u To store V-Window settings 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Press !K(V-Window) to display the V-Window setting screen, and input the values you want. 3. Press 4(STO) to display the pop-up window. 4. Press a number key to specify the V-Window memory where you want to save the settings, and then press w.
5-2-5 Controlling What Appears on a Graph Screen k Specifying the Graph Range Description You can define a range (start point, end point) for a function before graphing it. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Make V-Window settings. Execution 3. Specify the function type and input the function. The following is the syntax for function input. Function ,!+( [ )Start Point , End Point !-( ] ) 4. Draw the graph.
5-2-6 Controlling What Appears on a Graph Screen ○ ○ ○ ○ ○ Example Graph y = x 2 + 3x – 2 within the range – 2 < x < 4 Use the following V-Window settings. Xmin = –3, Xmax = 5, Xscale = 1 Ymin = –10, Ymax = 30, Yscale = 5 Procedure 1 m GRPH • TBL 2 !K(V-Window) -dwfwbwc -bawdawfwi 3 3(TYPE)b(Y=)vx+dv-c, !+( [ )-c,e!-( ] )w 4 5(DRAW) Result Screen # You can specify a range when graphing rectangular expressions, polar expressions, parametric functions, and inequalities.
5-2-7 Controlling What Appears on a Graph Screen k Zoom Description This function lets you enlarge and reduce the graph on the screen. Set Up 1. Draw the graph. Execution 2. Specify the zoom type. 2(ZOOM)b(Box) ... Box zoom Draw a box around a display area, and that area is enlarged to fill the entire screen. c(Factor) d(In)/e(Out) ... Factor zoom The graph is enlarged or reduced in accordance with the factor you specify, centered on the current pointer location. f(Auto) ...
5-2-8 Controlling What Appears on a Graph Screen ○ ○ ○ ○ ○ Example Graph y = (x + 5)(x + 4)(x + 3), and then perform a box zoom. Use the following V-Window settings.
5-2-9 Controlling What Appears on a Graph Screen k Factor Zoom Description With factor zoom, you can zoom in or out, centered on the current cursor position. Set Up 1. Draw the graph. Execution 2. Press 2(ZOOM)c(Factor) to open a pop-up window for specifying the x-axis and y-axis zoom factor. Input the values you want and then press i. 3. Press 2(ZOOM)d(In) to enlarge the graph, or 2(ZOOM)e(Out) to reduce it. The graph is enlarged or reduced centered on the current pointer location. 4.
5-2-10 Controlling What Appears on a Graph Screen ○ ○ ○ ○ ○ Example Enlarge the graphs of the two expressions shown below five times on both the x -and y -axis to see if they are tangent. Y1 = ( x + 4)(x + 1)( x – 3), Y2 = 3x + 22 Use the following V-Window settings.
5-2-11 Controlling What Appears on a Graph Screen k Turning Function Menu Display On and Off Press ua to toggle display of the menu at the bottom of the screen on and off. Turning off the function menu display makes it possible to view part of a graph hidden behind it. When you are using the trace function or other functions during which the function menu is normally not displayed, you can turn on the menu display to execute a menu command.
5-2-12 Controlling What Appears on a Graph Screen k About the Calc Window Pressing u4(CAT/CAL) while a graph or number table is on the display opens the Calc Window. You can use the Calc Window to perform calculations with values obtained from graph analysis, or to change the value assigned to variable A in Y = AX and other expressions and then redraw the graph. Press i to close the Calc Window.
5-3-1 Drawing a Graph 5-3 Drawing a Graph You can store up to 20 functions in memory. Functions in memory can be edited, recalled, and graphed. k Specifying the Graph Type Before you can store a graph function in memory, you must first specify its graph type. 1. While the Graph function list is on the display, press 6(g)3(TYPE) to display the graph type menu, which contains the following items. • {Y=}/{r=}/{Param}/{X=c} ...
5-3-2 Drawing a Graph u To store a parametric function *1 ○ ○ ○ ○ ○ Example To store the following functions in memory areas Xt3 and Yt3 : x = 3 sin T y = 3 cos T 3(TYPE)d(Param) (Specifies parametric expression.) dsvw(Inputs and stores x expression.) dcvw(Inputs and stores y expression.) u To store an X = constant expression *2 ○ ○ ○ ○ ○ Example To store the following expression in memory area X4 : X=3 3(TYPE)e(X = c) (Specifies X = constant expression.) d(Inputs expression.) w(Stores expression.
5-3-3 Drawing a Graph u To create a composite function ○ ○ ○ ○ ○ Example To register the following functions as a composite function: Y1= (X + 1), Y2 = X2 + 3 Assign Y1°Y2 to Y3, and Y2° Y1 to Y4. 2 (Y1° Y2 = ((x2 + 3) +1) = (x2 + 4) Y2°Y1 = ( (X + 1)) + 3 = X + 4 (X ⭌ –1)) 3(TYPE)b(Y=) J4(GRPH)b(Yn)b (1(Yn)c)w 4(GRPH)b(Yn)c (1(Yn)b)w • A composite function can consist of up to five functions.
5-3-4 Drawing a Graph ffffi1(SEL)5(DRAW) The above three screens are produced using the Trace function. See “5-11 Function Analysis” for more information. • If you do not specify a variable name (variable A in the above key operation), the calculator automatically uses one of the default variables listed below. Note that the default variable used depends on the memory area type where you are storing the graph function.
5-3-5 Drawing a Graph k Editing and Deleting Functions u To edit a function in memory ○ ○ ○ ○ ○ Example To change the expression in memory area Y1 from y = 2x2 – 5 to y = 2 x2 – 3 e (Displays cursor.) eeeeDd(Changes contents.) w(Stores new graph function.) u To change the type of a function*1 1. While the Graph function list is on the display, press f or c to move the highlighting to the area that contains the function whose type you want to change. 2. Press 3(TYPE)g(CONV). 3.
5-3-6 Drawing a Graph k Selecting Functions for Graphing u To specify the draw/non-draw status of a graph ○ ○ ○ ○ ○ Example To select the following functions for drawing : Y1 = 2 x2 – 5, r2 = 5 sin3θ Use the following V-Window settings. Xmin = –5, Xmax = 5, Ymin = –5, Ymax = 5, Xscale = 1 Yscale = 1 Tθ min = 0, Tθ max = π, Tθ ptch = 2π / 60 cc (Select a memory area that contains a function for which you want to specify non-draw.) 1(SEL) (Specifies non-draw.) 5(DRAW) or w (Draws the graphs.
5-3-7 Drawing a Graph k Graph Memory Graph memory lets you store up to 20 sets of graph function data and recall it later when you need it. A single save operation saves the following data in graph memory. • All graph functions in the currently displayed Graph function list (up to 20) • Graph types • Draw/non-draw status • View Window settings (1 set) u To store graph functions in graph memory 1. Press 4(GMEM)b(Store) to display the pop-up window. 2.
5-4-1 Storing a Graph in Picture Memory 5-4 Storing a Graph in Picture Memory You can save up to 20 graphic images in picture memory for later recall. You can overdraw the graph on the screen with another graph stored in picture memory. u To store a graph in picture memory 1. After graphing in GRPH • TBL Mode, press 6(g)1(PICT)b(Store) to display the pop-up window. 2. Press a number key to specify the Picture memory where you want to save the picture, and then press w.
5-5-1 Drawing Two Graphs on the Same Screen 5-5 Drawing Two Graphs on the Same Screen k Copying the Graph to the Sub-screen Description Dual Graph lets you split the screen into two parts. Then you can graph two different functions in each for comparison, or draw a normal size graph on one side and its enlarged version on the other side. This makes Dual Graph a powerful graph analysis tool.
5-5-2 Drawing Two Graphs on the Same Screen ○ ○ ○ ○ ○ Example Graph y = x(x + 1)(x – 1) in the main screen and sub-screen. Use the following V-Window settings. (Main Screen) Xmin = –2, Xmax = 2, Xscale = 0.5 Ymin = –2, Ymax = 2, Yscale = 1 Xmin = –4, Xmax = 4, Xscale = 1 Ymin = –3, Ymax = 3, Yscale = 1 (Sub-screen) Procedure 1 m GRPH • TBL 2 u3(SET UP)ccc2(G+G)i 3 !K(V-Window) -cwcwa.
5-5-3 Drawing Two Graphs on the Same Screen k Graphing Two Different Functions Description Use the following procedure to graph different functions in the main screen and sub-screen. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. On the SET UP screen, select G+G for Dual Screen. 3. Make V-Window settings for the main screen. Press 6(RIGHT) to display the sub-graph settings screen. Pressing 6(LEFT) returns to the main screen setting screen. Execution 4.
5-5-4 Drawing Two Graphs on the Same Screen ○ ○ ○ ○ ○ Example Graph y = x(x + 1)(x – 1) in the main screen, and y = 2x2 – 3 in the subscreen. Use the following V-Window settings. (Main Screen) Xmin = –4, Xmax = 4, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 (Sub-screen) Xmin = –2, Xmax = 2, Xscale = 0.5 Ymin = –2, Ymax = 2, Yscale = 1 Procedure 1 m GRPH • TBL 2 u3(SET UP)ccc2(G+G)i 3 !K(V-Window) -ewewbwc -fwfwbw 6(RIGHT)-cwcwa.
5-5-5 Drawing Two Graphs on the Same Screen k Using Zoom to Enlarge the Sub-screen Description Use the following procedure to enlarge the main screen graph and then move it to the subscreen. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. On the SET UP screen, select G+G for Dual Screen. 3. Make V-Window settings for the main screen. Execution 4. Input the function and draw the graph in the main screen. 5. Use Zoom to enlarge the graph, and then move it to the sub-screen.
5-5-6 Drawing Two Graphs on the Same Screen ○ ○ ○ ○ ○ Example Draw the graph y = x( x + 1)(x – 1) in the main screen, and then use Box Zoom to enlarge it. Use the following V-Window settings. (Main Screen) Xmin = –2, Xmax = 2, Xscale = 0.5 Ymin = –2, Ymax = 2, Yscale = 1 Procedure 1 m GRPH • TBL 2 u3(SET UP)ccc2(G+G)i 3 !K(V-Window) -cwcwa.
5-6-1 Manual Graphing 5-6 Manual Graphing k Rectangular Coordinate Graph Description Inputting the Graph command in the RUN • MAT Mode enables drawing of rectangular coordinate graphs. Set Up 1. From the Main Menu, enter the RUN • MAT Mode. 2. Make V-Window settings. Execution 3. Input the commands for drawing the rectangular coordinate graph. 4. Input the function.
5-6-2 Manual Graphing ○ ○ ○ ○ ○ Example Graph y = 2 x 2 + 3 x – 4 Use the following V-Window settings.
5-6-3 Manual Graphing k Integration Graph Description Inputting the Graph command in the RUN • MAT Mode enables graphing of functions produced by an integration calculation. The calculation result is shown in the lower left of the display, and the calculation range is blackened in the graph. Set Up 1. From the Main Menu, enter the RUN • MAT Mode. 2. Make V-Window settings. Execution 3. Input graph commands for the integration graph. 4. Input the function.
5-6-4 Manual Graphing ○ ○ ○ ○ ○ Example Graph the integration ∫ 1 (x + 2)(x – 1)(x – 3) dx. –2 Use the following V-Window settings.
5-6-5 Manual Graphing k Drawing Multiple Graphs on the Same Screen Description Use the following procedure to assign various values to a variable contained in an expression and overwrite the resulting graphs on the screen. Set Up 1. From the Main Menu, Enter GRPH • TBL Mode. 2. Make V-Window settings. Execution 3. Specify the function type and input the function. The following is the syntax for function input. Expression containing one variable ,!+( [ ) variable !.(=) value , value , ...
5-6-6 Manual Graphing ○ ○ ○ ○ ○ Example To graph y = A x 2 – 3 as the value of A changes in the sequence 3, 1, –1. Use the following V-Window settings. Xmin = –5, Xmax = 5, Xscale = 1 Ymin = –10, Ymax = 10, Yscale = 2 Procedure 1 m GRPH • TBL 2 !K(V-Window) -fwfwbwc -bawbawcwi 3 3(TYPE)b(Y=)av(A)vx-d, !+( [ )av(A)!.(=)d,b,-b!-( ] )w 4 5(DRAW) Result Screen # The value of only one of the variables in the expression can change.
5-7-1 Using Tables 5-7 Using Tables k Storing a Function and Generating a Number Table u To store a function ○ ○ ○ ○ ○ Example To store the function y = 3 x2 – 2 in memory area Y1 Use f and c to move the highlighting in the Graph function list to the memory area where you want to store the function. Next, input the function and press w to store it. u Variable Specifications There are two methods you can use to specify value for the variable x when generating a numeric table.
5-7-2 Using Tables u To generate a table using a list 1. While the Graph function list is on the screen, display the SET UP screen. 2. Highlight Variable and then press 2(LIST) to display the pop-up window. 3. Select the list whose values you want to assign for the x-variable. • To select List 6, for example, press gw. This causes the setting of the Variable item of the SET UP screen to change to List 6. 4. After specifying the list you want to use, press i to return to the previous screen.
5-7-3 Using Tables You can use cursor keys to move the highlighting around the table for the following purposes.
5-7-4 Using Tables k Editing and Deleting Functions u To edit a function ○ ○ ○ ○ ○ Example To change the function in memory area Y1 from y = 3x2 – 2 to y = 3 x2 – 5 Use f and c to move the highlighting to the function you want to edit. Use d and e to move the cursor to the location of the change. eeeeeDf w 6(g)5(TABL) • The Function Link Feature automatically reflects any changes you make to functions in the GRPH • TBL Mode list, and DYNA Mode list. u To delete a function 1.
5-7-5 Using Tables k Editing Tables You can use the table menu to perform any of the following operations once you generate a table. • Change the values of variable x • Edit (delete, insert, and append) rows • Delete a table and regenerate table • Draw a connect type graph • Draw a plot type graph While the Table & Graph menu is on the display, press 3(TABL) to display the table menu. • {EDIT} ... {edit value of x-variable} • {DEL·A} ... {delete table} • {Re-T} ...
5-7-6 Using Tables u Row Operations u To delete a row ○ ○ ○ ○ ○ Example To delete Row 2 of the table generated on page 5-7-2 6(g)1(R·DEL) c u To insert a row ○ ○ ○ ○ ○ Example To insert a new row between Rows 1 and 2 in the table generated on page 5-7-2 6(g)2(R·INS) c 19990401
5-7-7 Using Tables u To add a row ○ ○ ○ ○ ○ Example To add a new row below Row 7 in the table generated on page 5-7-2 6(g)3(R·ADD) cccccc u Deleting a Table 1. Display the table and then press 2(DEL·A). 2. Press w(Yes) to delete the table or i(No) to abort the operation without deleting anything.
5-7-8 Using Tables k Copying a Table Column to a List A simple operation lets you copy the contents of a numeric table column into a list. u To copy a table to a list ○ ○ ○ ○ ○ Example To copy the contents of Column x into List 1 K1(LMEM) • You can select any row of the column you want to copy. Input the number of the list you want to copy and then press w.
5-7-9 Using Tables k Drawing a Graph from a Number Table Description Use the following procedure to generate a number table and then draw a graph based on the values in the table. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Make V-Window settings. Execution 3. Store the functions. 4. Specify the table range. 5. Generate the table. 6. Select the graph type and draw it. 4(G • CON) ... line graph*1 5(G • PLT) ...
5-7-10 Using Tables ○ ○ ○ ○ ○ Example Store the two functions below, generate a number table, and then draw a line graph. Specify a range of –3 to 3, and an increment of 1. Y1 = 3 x 2 – 2, Y2 = x 2 Use the following V-Window settings. Xmin = 0, Xmax = 6, Xscale = 1 Ymin = –2, Ymax = 10, Yscale = 2 Procedure 1 m GRPH • TBL 2 !K(V-Window) awgwbwc -cwbawcwi 3 3(TYPE)b(Y=)dvx-cw vxw 4 6(g)2(RANG)-dwdwbwi 5 5(TABL) 6 4(G • CON) Result Screen # You can use Trace, Zoom, or Sketch after drawing a graph.
5-7-11 Using Tables k Specifying a Range for Number Table Generation Description Use the following procedure to specify a number table range when calculating scatter data from a function. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. Execution 2. Store the functions. 3. Specify the table range. 4. Select the functions for which you want to generate a table. The “=” sign of selected functions is highlighted on the screen. 5. Generate the table.
5-7-12 Using Tables ○ ○ ○ ○ ○ Example Store the three functions shown below, and then generate a table for functions Y1 and Y3. Specify a range of –3 to 3, and an increment of 1. Y1 = 3x 2 – 2, Y2 = x + 4, Y3 = x 2 Procedure 1 m GRPH • TBL 2 3(TYPE)b(Y=)dvx-cw v+ew vxw 3 6(g)2(RANG)-dwdwbwi 4 ff1(SEL) 5 5(TABL) Result Screen # You can generate number tables from rectangular coordinate, polar coordinate, and parametric functions.
5-7-13 Using Tables k Simultaneously Displaying a Number Table and Graph Description Specifying T+G for Dual Screen on the SET UP makes it possible to display a number table and graph at the same time. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Make V-Window settings. 3. On the SET UP screen, select T+G for Dual Screen. Execution 4. Input the function. 5. Specify the table range. 6. The number table is displayed in the sub-screen on the right. 7.
5-7-14 Using Tables ○ ○ ○ ○ ○ Example Store the function Y1 = 3x2 – 2 and simultaneously display its number table and line graph. Use a table range of –3 to 3 with an increment of 1. Use the following V-Window settings.
5-7-15 Using Tables k Using Graph-Table Linking Description With Dual Graph, you can use the following procedure to link the graph and table screens so the pointer on the graph screen jumps to the location of the currently selected table value. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. Make the required V-Window settings. Display the SET UP screen, select the Dual Screen item, and change its setting to “T+G”. Execution 3.
5-7-16 Using Tables ○ ○ ○ ○ ○ Example Store the function Y1 = 3logx and simultaneously display its number table and plot-type graph. Use a table range of 2 through 9, with an increment of 1. Use the following V-Window settings.
5-8-1 Dynamic Graphing 5-8 Dynamic Graphing k Using Dynamic Graph Description Dynamic Graph lets you define a range of values for the coefficients in a function, and then observe how a graph is affected by changes in the value of a coefficient. It helps to see how the coefficients and terms that make up a function influence the shape and position of a graph. Set Up 1. From the Main Menu, enter the DYNA Mode. 2. Make V-Window settings. Execution 3. On the SET UP screen, specify the Dynamic Type. 1(Cont) .
5-8-2 Dynamic Graphing ○ ○ ○ ○ ○ Example Use Dynamic Graph to graph y = A (x – 1)2 – 1, in which the value of coefficient A changes from 2 through 5 in increments of 1. The Graph is drawn 10 times. Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.
5-8-3 Dynamic Graphing k Dynamic Graph Application Examples Description You can also use Dynamic Graph to simulate simple physical phenomena. Set Up 1. From the Main Menu, enter the DYNA Mode. 2. Make V-Window settings. Execution 3. On the SET UP screen, specify Stop for Dynamic Type and Deg for Angle. 4. Specify Param (parametric function) as the function type, and input a function that contains a dynamic variable. 5. Specify the dynamic coefficient. 6. Specify the start value, end value, and increment.
5-8-4 Dynamic Graphing ○ ○ ○ ○ ○ Example The path over time T of a ball thrown in the air at initial velocity V and an angle of θ degrees from horizontal can be calculated as follows. X = (Vcos θ ) T, Y = (Vsin θ ) T – (1/2)gT2 (g = 9.8m/s2) Use Dynamic Graph to plot the path of a ball thrown at an initial velocity of 20 meters per second, at horizontal angles of 30, 45, and 60 degrees (Angle: Deg). Use the following V-Window settings.
5-8-5 Dynamic Graphing k Adjusting the Dynamic Graph Speed You can use the following procedure to adjust the Dynamic Graph speed while the draw operation is taking place. 1. While a Dynamic Graph draw operation is being performed, press A to change to the speed adjustment menu. • { } ... {Each step of the Dynamic Graph draw operation is performed each time you press w.} • { }/{ }/{ } ... {slow (1/2 speed)}/{normal (default speed)}/{fast (double speed)} • {STO} ...
5-8-6 Dynamic Graphing k Using Dynamic Graph Memory You can store Dynamic Graph conditions and screen data in Dynamic Graph memory for later recall when you need it. This lets you save time, because you can recall the data and immediately begin a Dynamic Graph draw operation. Note that you can store one set of data in memory at any one time. The following is all of the data that makes up a set.
5-9-1 Graphing a Recursion Formula 5-9 Graphing a Recursion Formula k Generating a Number Table from a Recursion Formula Description You can input up to three of the following types of recursion formulas and generate a number table. • General term of sequence {a n }, composed of a n , n • Linear two-term recursion composed of a n+1, a n, n • Linear three-term recursion composed of a n+2, a n+1, a n , n Set Up 1. From the Main Menu, enter the RECUR Mode. Execution 2. Specify the recursion type.
5-9-2 Graphing a Recursion Formula ○ ○ ○ ○ ○ Example Generate a number table from recursion between three terms as expressed by an +2 = a n+1 + a n, with initial terms of a 1 = 1, a 2 = 1 (Fibonacci sequence), as n changes in value from 1 to 6. Procedure 1 m RECUR 2 3(TYPE)d(a n+2=) 3 4(n. a n ·· )d(a n+1)+2( an )w 4 5(RANG)2( a1 )bwgwbwbwi 5 6(TABL) Result Screen * The first two values correspond to a 1 = 1 and a 2 = 1.
5-9-3 Graphing a Recursion Formula k Graphing a Recursion Formula (1) Description After generating a number table from a recursion formula, you can graph the values on a line graph or plot type graph. Set Up 1. From the Main Menu, enter the RECUR Mode. 2. Make V-Window settings. Execution 3. Specify the recursion formula type and input the formula. 4. Specify the table range, and start and ending values for n. If necessary, specify the initial term value and pointer start point. 5.
5-9-4 Graphing a Recursion Formula ○ ○ ○ ○ ○ Example Generate a number table from recursion between two terms as expressed by a n+1 = 2a n+1, with an initial term of a 1 = 1, as n changes in value from 1 to 6. Use the table values to draw a line graph. Use the following V-Window settings.
5-9-5 Graphing a Recursion Formula k Graphing a Recursion Formula (2) Description The following describes how to generate a number table from a recursion formula and graph the values while Σ Display is On. Set Up 1. From the Main Menu, enter the RECUR Mode. 2. On the SET UP screen, specify On for Σ Display. 3. Make V-Window settings. Execution 4. Specify the recursion formula type and input the recursion formula. 5. Specify the table range, and start and ending values for n.
5-9-6 Graphing a Recursion Formula ○ ○ ○ ○ ○ Example Generate a number table from recursion between two terms as expressed by a n+1 = 2a n+1, with an initial term of a 1 = 1, as n changes in value from 1 to 6. Use the table values to draw a plot line graph with ordinate Σa n , abscissa n. Use the following V-Window settings.
5-9-7 Graphing a Recursion Formula k WEB Graph (Convergence, Divergence) Description y = f(x) is graphed by presuming a n+1 = y, a n = x for linear two-term regression a n+1 = f( a n) composed of a n+1, a n . Next, it can be determined whether the function is convergent or divergent. Set Up 1. From the Main Menu, enter the RECUR Mode. 2. Make V-Window settings. Execution 3. Select 2-term recursion as the recursion formula type, and input the formula. 4.
5-9-8 Graphing a Recursion Formula ○ ○ ○ ○ ○ Example To draw the WEB graph for the recursion formula a n +1 = –3(a n) 2 + 3a n, b n +1 = 3b n + 0.2, and check for divergence or convergence. Use the following table range and V-Window Settings. Table Range Start = 0, End = 6, a0 = 0.01, a n Str = 0.01, b 0 = 0.11, b nStr = 0.
5-10-1 Changing the Appearance of a Graph 5-10 Changing the Appearance of a Graph k Drawing a Line Description The sketch function lets you draw points and lines inside of graphs. Set Up 1. Draw the graph. Execution 2. Select the sketch function you want to use.*1 3(SKTCH) b(Cls) ... Screen clear c(PLOT) {On}/{Off}/{Change}/{Plot} ... Point {On}/{Off}/{Change}/{Plot} d(LINE) {F-Line}/{Line} ... {Freehand line}/{Line} e(Text) ... Text input f(Pen) ... Freehand g(Tangnt) ... Tangent line h(Normal) ...
5-10-2 Changing the Appearance of a Graph ○ ○ ○ ○ ○ Example Draw a line that is tangent to point (2, 0) on the graph for y = x (x + 2)(x – 2). Use the following V-Window settings. Xmin = –5, Xmax = 5, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 Procedure 1 m GRPH • TBL !K(V-Window) -fwfwbwc -fwfwbwi 3(TYPE)b(Y=)v(v+c)(v-c)w 5(DRAW) 2 3(SKTCH)g(Tangnt) 3 e~ew*1 Result Screen * 1 You can draw a tangent line in succession by moving the “ ” pointer and pressing w.
5-10-3 Changing the Appearance of a Graph k Inserting Comments Description You can insert comments anywhere you want in a graph. Set Up 1. Draw the graph. Execution 2. Press 3(SKTCH)e(Text), and a pointer appears in the center of the display. 3. Use the cursor keys to move the pointer to the location where you want the text to be, and input the text. # You can input any of the following characters as comment text: A~Z, r, θ , space, 0~9, .
5-10-4 Changing the Appearance of a Graph ○ ○ ○ ○ ○ Example Insert text into the graph y = x (x + 2)(x – 2). Use the following V-Window settings. Xmin = –5, Xmax = 5, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 Procedure 1 m GRPH • TBL !K(V-Window) -fwfwbwc -fwfwbwi 3(TYPE)b(Y=)v(v+c)(v-c)w 5(DRAW) 2 3(SKTCH)e(Text) 3 f~f d~d a-(Y)!.
5-10-5 Changing the Appearance of a Graph k Freehand Drawing Description You can use the pen option for freehand drawing in a graph. Set Up 1. Draw the graph. Execution 2. Press 3(SKTCH)f(Pen), and a pointer appears in the center of the screen. 3. Use the cursor keys to move the pointer to the point from which you want to start drawing, and then press w. 4. Use the cursor keys to move the pointer. A line is drawn wherever you move the pointer. To stop the line, press w.
5-10-6 Changing the Appearance of a Graph ○ ○ ○ ○ ○ Example Use the pen to draw on the graph y = x (x + 2)(x – 2). Use the following V-Window settings.
5-10-7 Changing the Appearance of a Graph k Changing the Graph Background You can use the set up screen to specify the memory contents of any picture memory area (Pict 1 through Pict 20) as the Background item. When you do, the contents of the corresponding memory area is used as the background of the graph screen. ○ ○ ○ ○ ○ Example 1 With the circle graph X2 + Y2 = 1 as the background, use Dynamic Graph to graph Y = X2 + A as variable A changes value from –1 to 1 in increments of 1.
5-10-8 Changing the Appearance of a Graph Draw the dynamic graph. (Y = X2 – 1) ↓↑ (Y = X2 ) ↓↑ (Y = X2 + 1) • See “5-8-1 Dynamic Graphing” for details on using the Dynamic Graph feature.
5-11-1 Function Analysis 5-11 Function Analysis k Reading Coordinates on a Graph Line Description Trace lets you move a pointer along a graph and read out coordinates on the display. Set Up 1. Draw the graph. Execution 2. Press 1(TRACE), and a pointer appears in the center of the graph.*1 3. Use d and e to move the pointer along the graph to the point at which you want to display the derivative.
5-11-2 Function Analysis ○ ○ ○ ○ ○ Example Read coordinates along the graph of the function shown below. Y1 = x 2 – 3 Use the following V-Window settings. Xmin = –5, Xmax = 5, Xscale = 1 Ymin = –10, Ymax = 10, Yscale = 2 Procedure 1 m GRPH • TBL !K(V-Window) -fwfwbwc -bawbawcwi 3(TYPE)b(Y=)vx-dw 5(DRAW) 2 1(TRACE) 3 d~d 4 -bw Result Screen # The following shows how coordinates are displayed for each function type.
5-11-3 Function Analysis k Displaying the Derivative Description In addition to using Trace to display coordinates, you can also display the derivative at the current pointer location. Set Up 1. On the SET UP screen, specify On for Derivative. 2. Draw the graph. Execution 3. Press 1(TRACE), and the pointer appears at the center of the graph. The current coordinates and the derivative also appear on the display at this time. 4.
5-11-4 Function Analysis ○ ○ ○ ○ ○ Example Read coordinates and derivatives along the graph of the function shown below. Y1 = x 2 – 3 Use the following V-Window settings.
5-11-5 Function Analysis k Graph to Table Description You can use trace to read the coordinates of a graph and store them in a number table. You can also use Dual Graph to simultaneously store the graph and number table, making this an important graph analysis tool. Set Up 1. From the Main Menu, enter the GRPH • TBL Mode. 2. On the SET UP screen, specify GtoT for Dual Screen. 3. Make V-Window settings. Execution 4. Save the function and draw the graph on the active (left) screen. 5. Activate Trace.
5-11-6 Function Analysis ○ ○ ○ ○ ○ Example Save, in a table, the coordinates in the vicinity of the points of intersection at X = 0 for the two graphs shown below, and store the table contents in List 1. Y1 = x2 – 3, Y2 = – x + 2 Use the following V-Window settings.
5-11-7 Function Analysis k Coordinate Rounding Description This function rounds off coordinate values displayed by Trace. Set Up 1. Draw the graph. Execution 2. Press 2(ZOOM)i(Rnd). This causes the V-Window settings to be changed automatically in accordance with the Rnd value. 3. Press 1(TRACE), and then use the cursor keys to move the pointer along the graph. The coordinates that now appear are rounded.
5-11-8 Function Analysis ○ ○ ○ ○ ○ Example Use coordinate rounding and display the coordinates in the vicinity of the points of intersection for the two graphs produced by the functions shown below. Y1 = x 2 – 3, Y2 = – x + 2 Use the following V-Window settings.
5-11-9 Function Analysis k Calculating the Root Description This feature provides a number of different methods for analyzing graphs. Set Up 1. Draw the graphs. Execution 2. Select the analysis function. 4(G-SLV) b(Root) ... Calculation of root c(Max) ... Local maximum value d(Min) ... Local minimum value e(Y-lcpt) ... y-intercept f(Isect) ... Intersection of two graphs g(Y-Cal) ... y-coordinate for given x-coordinate h(X-Cal) ... x-coordinate for given y-coordinate i( ∫dx) ...
5-11-10 Function Analysis ○ ○ ○ ○ ○ Example Draw the graph shown below and calculate the root for Y1. Y1 = x ( x + 2)(x – 2) Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.1, Yscale = 1 (initial defaults) Procedure 1 m GRPH • TBL !K(V-Window) 1(INIT)i 3(TYPE)b(Y=)v(v+c)(v-c)w 5(DRAW) 2 4(G-SLV)b(Root) … 4 e e Result Screen # When analyzing a single graph, results appear as soon as you select an analysis function in step 2, so step 3 is not necessary.
5-11-11 Function Analysis k Calculating the Point of Intersection of Two Graphs Description Use the following procedure to calculate the point of intersection of two graphs. Set Up 1. Draw the graphs. Execution 2. Press 4(G-SLV)5(Isect). When there are three or more graphs, the selection cursor (k) appears at the lowest numbered graph. 3. Use the cursor keys to move the cursor to the graph you want to select. 4. Press w to select the first graph, which changes the shape of the cursor from k to 쏆. 5.
5-11-12 Function Analysis ○ ○ ○ ○ ○ Example Graph the two functions shown below, and determine the point of intersection between Y1 and Y2. Y1 = x + 1, Y2 = x 2 Use the following V-Window settings. Xmin = –5, Xmax = 5, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 Procedure 1 m GRPH • TBL !K(V-Window) -fwfwbwc -fwfwbwi 3(TYPE)b(Y=)v+bw vxw 5(DRAW) 2 4(G-SLV)f(Isect) … 6 e Result Screen # In the case of two graphs, the point of intersection is calculated immediately after you press 4f in step 2.
5-11-13 Function Analysis k Determining the Coordinates for Given Points Description The following procedure describes how to determine the y-coordinate for a given x, and the x-coordinate for a given y. Set Up 1. Draw the graph. Execution 2. Select the function you want to perform. When there are multiple graphs, the selection cursor (k) appears at the lowest numbered graph. 4(G-SLV)g(Y-Cal) ... y-coordinate for given x h(X-Cal) ... x-coordinate for given y 3.
5-11-14 Function Analysis ○ ○ ○ ○ ○ Example Graph the two functions shown below and then determine the ycoordinate for x = 0.5 and the x-coordinate for y = 2.2 on graph Y2. Y1 = x + 1, Y2 = x(x + 2)(x – 2) Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.1, Yscale = 1 (initial defaults) Procedure 1 m GRPH • TBL !K(V-Window) 1(INIT)i 3(TYPE)b(Y=)v+bw v(v+c)(v-c)w 5(DRAW) 2 4(G-SLV)g(Y-Cal) 2 4(G-SLV)h(X-Cal) 3 cw 3 cw 4 a.fw 4 c.
5-11-15 Function Analysis k Calculating the lntegral Value for a Given Range Description Use the following procedure to obtain integration values for a given range. Set Up 1. Draw the graph. Execution 2. Press 4(G-SLV)i(∫dx). When there are multiple graphs, this causes the selection cursor (k) to appear at the lowest numbered graph. 3. Use fc to move the cursor (k) to the graph you want, and then press w to select it. 4. Use d to move the lower limit pointer to the location you want, and then press w.
5-11-16 Function Analysis ○ ○ ○ ○ ○ Example Graph the function shown below, and then determine the integral value at (–2, 0). Y1 = x ( x + 2)(x – 2) Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –4, Ymax = 4, Yscale = 1 Procedure 1 m GRPH • TBL !K(V-Window) -g.dwg.
5-11-17 Function Analysis k Conic Section Graph Analysis You can determine approximations of the following analytical results using conic section graphs. • Focus/vertex/eccentricity • Latus rectum • Center/radius • x-/y-intercept • Directrix/axis of symmetry drawing and analysis • Asymptote drawing and analysis After graphing a conic section, press 4(G-SLV) to display the following graph analysis menus. u Parabolic Graph Analysis • {Focus}/{Vertex}/{Length}/{e} ...
5-11-18 Function Analysis u To calculate the focus, vertex and latus rectum [G-SLV]-[Focus]/[Vertex]/[Length] ○ ○ ○ ○ ○ Example To determine the focus, vertex and latus rectum for the parabola X = (Y – 2)2 + 3 Use the following V-Window settings. Xmin = –1, Xmax = 10, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 4(G-SLV) b(Focus) (Calculates the focus.) i 4(G-SLV) d(Vertex) (Calculates the vertex.) i 4(G-SLV) f(Length) (Calculates the latus rectum.
5-11-19 Function Analysis u To calculate the center and radius [G-SLV]-[Center]/[Radius] ○ ○ ○ ○ ○ Example To determine the center and radius for the circle (X + 2) 2 + (Y + 1)2 = 22 Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.1, Yscale = 1 4(G-SLV) b(Center) (Calculates the center.) i 4(G-SLV) c(Radius) (Calculates the radius.
5-11-20 Function Analysis i 4(G-SLV) h(Y-Icpt) (Calculates the y-intercept.) • Press e to calculate the second set of x-/y-intercepts. Pressing d returns to the first set of intercepts. u To draw and analyze the axis of symmetry and directrix [G-SLV]-[Sym]/[Dirtrx] ○ ○ ○ ○ ○ Example To draw the axis of symmetry and directrix for the parabola X = 2(Y – 1)2 + 1 Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –3.1, Ymax = 3.
5-11-21 Function Analysis u To draw and analyze the asymptotes [G-SLV]-[Asympt] ○ ○ ○ ○ ○ Example To draw the asymptotes for the hyperbola (X – 1)2 (Y – 1)2 –––––––– – –––––––– =1 2 2 22 Use the following V-Window settings. Xmin = –6.3, Xmax = 6.3, Xscale = 1 Ymin = –5, Ymax = 5, Yscale = 1 4(G-SLV) e(Asympt) (Draws the asymptotes.
