VIEWS: 48 PAGES: 8 POSTED ON: 4/7/2010 Public Domain
Introduction to Autograph 2 Modes and the grid setup Inequalities Working with points Area under a curve Linear Equations The Circle Quadratic Equations From AutoGraph to Powerpoint Statistics - Histograms 2 MODES SELECT Mode - SASWOA….. Select And See What Object Allows You need to SELECT objects before you can operate on them. To do this, left click the object ..and right click to see what you can do with the selection. Select Hand Icon POINT Mode – Enter points by left clicking on the grid. Mode (DRAG) Point Mode Zoom IN and OUT The GRID SETUP WHITEBOARD settings Go to View -> Preferences and select Whiteboard tab. There are four options presented. Click the first (No Shift for Multiple Select) and the last (Use ‘all thick lines’). Then activate Whiteboard Mode by clicking on the icon bar. STANDARD LEVEL Go to View -> Preferences and select the General tab. Click Standard Level. Either of these allows us to amend the axes Click Axes → Edit Axes OR Click the spanner Choose the Appearance Tab. Select the Graph paper theme. Make our axes stand out. Choose the Ranges Tab. You can scale the axes here. Task 1 : Working with Points Click the spanner and using the Ranges and Appearance Tab, as shown above, change the grid to graph paper with x axis numbered from 0 to 10 and y axis from 0 to 8. Place (4,6) on the grid. To label the point : Select the point, right click and choose Text Box. Plot some more points : Practise deleting them, drawing horiz/vertical lines, circles, etc;.. You can make equal spacing (equal aspect) on the x and y axes using this icon. Task 2 : Creating a Shape for Transformation Using or mark the points (5,1) (5,3) (0,5) (0,6) (5,4) (5,5) (10,5) (10,1) and (5,1) Select the points and right click or go to Object and click Group to Shape Go to Object again (or simply right click the selected shape) and find the image of the shape by (i) Reflection in x-axis (ii) Reflection in y-axis Linear Equations Entering an equation is very easy in Autograph: you just click this icon and type in the equation as given. Task 3 : The Line Enter 2x + 3y =12. Select a point on the line using . Right click to choose Text Box and display coordinates. Using the cursor move the point along the line and watch the Text Box show the co-ordinates change. Click the slow plot tortoise and re-plot the line To stop/start the line, use the space bar or the pause icon. Change the colour and appearance of the line using the Draw Option. Task 4 : Lines and Point Enter the equation x + y = 4 (L) and the point (-1,1) Select the line and point and find (i) The distance from the point to the line (ii) Draw a line parallel to x+ y = 4 through (-1,1) (K) (iii)Draw a line perpendicular to x+ y = 4 through (-1,1) (M) Does it look ? If "No" how do you adjust? (iv)Find the equations of K and M - Hint use View and click on Results or click Task 5 : Two Lines Enter the equations 2x+3y =12 (L) and x - y =4 (K). Find (i) LK (ii) Angle between L and K With equal axes, repeat for 3x + y = 6 (L ) and x-3y =3 (K) To find LK, select both lines and right click. Choose the option Solve f(x)=g(x) and mark this point of intersection (right click → text box). Quadratic Equations …. Animation and Constant controller A. Quadratic and Linear Equations On the same axis graph the functions f(x) = 5 – x – 2x2 and g(x) = 3x – 2 in the domain -4≤ x ≤ 4 and x ε R. (i) Label the functions using Text Box. Change the axes for the best display. (ii) Select each graph, right click, and choose the option for finding the points of intersection. (iii) Select the curve. Click on the tortoise and re-plot. Identify the range where f(x)≥g(x), using the space bar to stop and start at key sections. B. Controlling Quadratic Equations 1. Click on to enter an equation. Enter y = ax2 +bx +c All constants will be automatically set to 1, if you wish to change these values click on "Edit Constants". 2. Change the values of the constants by selecting each in turn then click “OK”. [ ] The View -> Constant Controller allows you to vary a , b or c 3. Click on the up and down buttons to increase or decrease the value of the constant shown or type the value into the box. Click on the left and right buttons to increase or decrease the step value or type into the box. 4. Click on the curve to select it. Right click and select Solve f(x) = 0. Try also y = (x – a)(x – b). C. To show Max, Min and Points of Inflection, you need to be in Advanced Level Go to View > Preferences > General. Select Advanced Level. 1. Click on to enter an equation and enter an equation such as y = (x - 1)(x2 – 4). (This is a good one to start with). If you do not see max and min points then move graph with hand icon. Make sure when you have finished moving the graph click on selection arrow again. 2. Select curve and click on Gradient Function Icon . The dy/dx curve will be plotted. 3. Select the dy/dx curve and then click on the Gradient Function Icon again. The d 2 y/dx 2 (line) curve will be plotted. 4. If you want to pause and analyses and re-start, click on Slow Plot Icon at the beginning. 5. N.B. During animation as the tangents are being animated they stop. Click on Space Bar to stop/start. Exercise Draw the following graphs and find the solution to each when f(x) = 0. 1 1. y = cosx 2. y = 3. y = lnx 4. y = ex 5. y = sin x x Statistics on Autograph To change to 1D statistics mode click on the toolbar. Histogram from a grouped data set To enter a Grouped Data set with equal class intervals x frequency Right click and choose Enter Grouped Data OR click . 0-10 2 Set Min, Max and Class Width. 10-20 4 In frequencies section, click Enter Manually, enter the frequencies 20-30 6 and click OK. 30-40 5 40-50 3 Click the Histogram symbol on the toolbar To enter a Grouped Data set with unequal class intervals x frequency Right click and choose Grouped Data OR click 0-10 10 In class intervals, click Enter Manually. 10-20 20 Enter the left limits and the maximum, (Class boundaries.) in this 20-40 40 case 0, 10, 20, 40, 70, 100. 40-70 50 In Frequencies section, click Enter Manually and enter the 70-100 30 frequencies. Click OK. To draw a Histogram with grouped data with unequal class intervals Enter the data in the usual way and click the Histogram icon on the toolbar. In the Edit Histogram Settings box, change Settings to Frequency Density and pick the unit width. To change the colour of the Histogram Select the Histogram, right click and select Edit Draw Options Cumulative Frequency Curve. Follow the same steps as above to enter the grouped data set then click on Cumulative frequency icon and a dialog box below will appear requesting refinements to the curve. Once the curve is drawn, select the curve, right click the mouse button and select Statistic Box or Table of Statistics to view detailed information. These results can be transferred to the Results Box. Plotting Inequalities Entering ≤ and ≥ in 3 different ways 1. Type <= for ≤, and >= for ≥. 2. Use the ≤ and ≥ keys in the dialogue box. 3. Use the On-screen Keyboard. (View → Keyboard) Plotting Convention < and > inequalities are drawn with a dotted line. ≤ and ≥ inequalities are drawn with a solid line. The Accept and Reject Regions Use View > Preferences > General to set whether the accept or reject region is shaded. The default is to shade the reject region, leaving the accept region clear to mark solution points by adding points. Task 6 – Some Inequalities Try graphing the following regions 1) y<3 3) y ≤ x² 5) 2x + 3y ≥ 6 2) x>0 4) x² + y² > 16 Finding the area under a curve Click on [ ] and enter the equation y = 4 – x² [you can type xx to get x²]. Click OK to plot the equation then use [ ] to select it. Be sure to be at Advance Level. Right mouse click and select the option "Find Area". Here you can explore rules for approximating the area, including varying the number of divisions. Ensure that the Start Point and End Point on the x-axis are set correctly for your curve. Click OK to see the result. The area under the curve appears in the status line at the bottom of the screen. To view this data more clearly, click on View then Status Box. Click in no man’s land to de-select everything. You can now click on [ ] to attach two points to the curve. Ensure both are selected: right mouse click and select the option "Find Area". This time the two selected points will denote the limits and these can be dragged around the curve. As you move one of the points the area changes dynamically in the status box. If the point crosses below the X-axis the highlighted area clearly increases but its value decreases. You can quickly investigate why some calculated areas are negative. To find the area between two curves, e.g. y = 4 – x² and y = x(x – 3) : Plot both curves then either select the two curves, "Find Area", and enter the limits yourself OR, if the limits are to be the intersection points you will need to position two points at the intersection points. To do this: position a point on one of the curves below the intersection. Select the point and the second curve then right mouse click. Select the option ‘Move to next Intersection’ The point will move to the intersection. Repeat for the second intersection. Then find the enclosed area. The Circle In Autograph you can draw a circle in three ways. Method 1. Using a point as the centre. 1. Using the icon add a point to the plane. 2. Select the point and then right click mouse button and select Circle(Radius). 3. Fill in the required radius in the dialogue box and click OK. This will create the circle. Try creating x 2 y 2 4 . (centre (0,0) and radius 2) 4. You can actively control the radius of this circle. Select the circle, use the View dropdown menu and Animate Object to see what happens as you vary the radius. ( This creates discussion and exploration on the animation features of Autograph). Method 2. Using three points. 1. Using the icon add three points to the plane. ( Pick (2,0) , (-2,0) and (0,2) to start with.). 2. Using the [ ] icon select the three points. 3. Right click mouse button and select circle (3pts). The circle will then be drawn. 4. If the drawing turns out as an ellipse you have to go to Edit Axis and click on equal aspect ratio.( See what type of discussion it generates). 5. Select the circle and the status box should show the equation as . x 2 y 2 4 6. If you select one of the points on the circle and drag it around the screen the equation in the status box will change accordingly. Method 3. By entering the equation of the circle. 1. Using the icon or by going to the Equation drop down menu enter the equation of the circle required. Enter x 2 y 2 4 . (For HL enter y (3 x 2 ) . If drawing it in class leave out the and see what type of discussion it generates. ). 2. You may notice a difference when you select the circle compared to method 1. General equation of a circle in the form of x 2 y 2 r 2 1. Using the icon or by going to the Equation drop down menu enter the equation of the circle required. x 2 y 2 r 2 (or y (r 2 x 2 ) ). This should draw a circle of centre (0,0) and radius 1. 2. Select the circle and go to the View drop down menu, select Constant Controller and then Options and you will get the following window. 3. Select Family Plot starting at 0 and finishing at 6. Then click OK. You should get a series of concentric circles. 4. If you open the Constant Controller again but this time select Animation and click OK the series of concentric circles will be animated. (You can control the speed of the animation by changing the slider button in the Constant Controller window. ( all of the above should create class discussion). General equation of a circle in the form of ( x h) 2 ( y k ) 2 r 2 Parametric form equation of a circle in the form of x r cos , y r sin Parametric form equation of a circle in the form of x h r cos , y k r sin All of these forms can be entered and the variables controlled with the Constant Controller. 6 FROM AUTOGRAPH TO POWERPOINT – Task 7 The following set of notes give step by step instructions on converting graphs drawn in Autograph and importing them, with animation into PowerPoint. 1. Drawing a graph in Autograph. Open an Autograph file. Click on the dropdown menu Equation. Click on Enter Equation. Enter equation required, remembering to start with y = (try y = x2 +3x + 2) Click OK and the graph will appear. 4 y 3 2 1 x –6 –4 –2 2 4 6 –1 –2 –3 –4 Click on the dropdown menu Page Click on Copy Graph (Metafile). The graph is now saved on to clipboard. 7 2. Pasting and manipulating the graph in PowerPoint. Open a new PowerPoint presentation and insert a blank slide. Paste the graph on to this slide, then resize if required. Select the graph image which was just pasted and right click. Click on Grouping and the Ungroup. The following dialog box appears Click Yes. Select the graph image again. Click on Grouping and the Ungroup and the image will be ungrouped click outside the image so that no object is selected. Click on the curve part of the image and it will be the only part selected. Cut this image on to clipboard. Select all the objects of the image on the slide, right click and Group these objects. Paste the curve on to the slide again. There will now be two objects on the slide a graph and a curve. Animate these two objects as you see fit. To animate the curve it is best to use Wipe from Left very slowly. 8