Geogebra: Helpful Hints (online version) geogebra.org/graphing

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Melissa Jacobs
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proper=es of the coordinate grid or the objects placed on the grid, look for the style bar in the upper right-hand corner: If an object in the algebra screen is selected, the style bar gives op=ons

1 Geogebra: Helpful Hints (online version) geogebra.org/graphing When Geogebra is opened, there are two main screens: algebra screen, on the le: where informa=on is entered graphics screen, the main sec=on where objects are displayed To change the proper=es of the coordinate grid or the objects placed on the grid, look for the style bar in the upper right-hand corner: If an object in the algebra screen is selected, the style bar gives op=ons to change the proper=es of the selected object. If no object is selected, the style bar offers op=ons to change the proper=es of the cartesian plane. In both cases, the gear box will offer further choice: Another way you can change an object s proper=es is to control+click (on a Mac) with the cursor on the object on the graphics screen. A short-list of op=ons will appear, along with the op=on to click on Object Proper,es for more detailed choices. The default senng in Geogebra adds a label for the objects you create. To turn off the labelling: Menu Op=ons Labelling No New Objects 1 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

Main Commands (top left-hand menu bar) The arrow in the first box from the le: will move selected objects on the graphics screen. The second buton from the le: creates points.

2 Main Commands (top left-hand menu bar) The arrow in the first box from the le: will move selected objects on the graphics screen. The second buton from the le: creates points. The third buton from the le: creates lines. The fourth buton from the le: creates tangent lines from a point to an object. Click on the point first, and then click on the object. The fi:h buton from the le: creates polygons. The last buton on the right gives op=ons to move the cartesian plane, or zoom in/out. To undo your last move, click on the le:-hand arrow, found in the top-right hand corner. Input equa=ons into the input bar in the algebra screen on the le:-hand side. when the circle on the le: is solid, the object is shown on the graphics screen when the circle on the right is hollow, the object is not shown to delete an object, click on the bar and then click on the x in the top right-hand corner Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

3 Implicit Curves and Tangent Lines Exploration #1 For this explora,on it is helpful to have the coordinate grid visible but not the axes. Using geogebra, create two horizontal lines on the graphics screen. It doesn t mater how far apart these are. At the midpoint of each horizontal line, draw a ver=cal line segment connec=ng the two horizontal lines. It is helpful to change the style of this line so it is appears different from the other lines. Each horizontal line represents a number line, with the midpoint of the line represen=ng zero. Determine a scale so that you can count five to the right and five to the le:. You don t need to label the number line, but should be able to pick out where the whole numbers would appear, as well as some basic frac=ons (two points are labelled in Figure 1 to illustrate the scale in this diagram). Draw a series of line segments, connec=ng the whole numbers on the top number line (1 through 5) with their reciprocals on the botom number line. For example, connect on the top to botom: 1 on the 1 Repeat on the le:-hand side of the midline as well. FIGURE 1 What is the shape that appears as the envelope of the straight lines? Determine the equa=on of this curve. What adjustments could you make to create a circle from these lines? 1 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

4 Exploration # For this explora,on it is helpful to have the coordinate grid visible and the axes. ( ) ( 7,7) ( 0,0) ( 7,7) ( ) ( 0,0) ( ) ( 7,7) Using Geogebra, draw line segments between 0,0 and, and between and. Star=ng at the point 7,7 and working down toward, place points on the line segment that are evenly spaced out. Con=nue up the line segment from 0,0 to (see Figure ). Each line segment should be divided into the same number of pieces. Connect the point at the top le: line segment, with the point at the botom of the right line segment (not the origin) by drawing a straight line (as shown in Figure ). Con=nue to connect the points in this manner, un=l all points on the le: have lines connec=ng them to points on the right. FIGURE What is the shape that appears as the envelope of the straight lines? Determine the equa=on of this curve that best matches the envelope. Explain how you found this equa=on. Show that all the line segments in your picture are actually tangent lines to the curve of your equa=on. If the line segments are not tangent lines, find an even beter matching parabola and try again. Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

5 Exploration #3 In the first two explora,ons, we started with a series of straight lines from which curves emerged. In the next two explora,ons, we will start with a curve and let Geogebra create the tangent lines. For this explora,on, it is helpful to have to have the coordinate grid visible and to have turned off the labelling in the op,ons command. 1. Enter the equa=on of a circle into the input bar of Geogebra.. Using the polygon tool, draw a square around the circle. The square will act as a frame for the string art (a frame can be any shape that the strings are atached to). You many wish to adjust the proper=es of the square before moving on (i.e. colour, opacity). 3. Create a series of points along the perimeter of the frame. These points should be about equidistant apart. Clicking on the arrow tool will allow you to reposi=on any points if you feel the need to =dy things up. 4. Using the tangent tool, create tangent lines from each point on the frame to the circle by clicking on a point and then on the circle. It is also possible to type the command tangent in the input menu bar at the botom of the screen and entering the name of the circle and the name of a point. 5. Clean up your image by doing the following: turn off the coordinate grid and the axes (it s important to do this before turning off the points, otherwise the points will reappear) turn off the circle in the input bar so you can see the envelope created by the tangent lines instead of the curve graphed by the equa=on (don t delete the equa=on of the circle or the tangents will disappear). If you don t see the circle, you may need to create more tangent lines. turn off the frame (but don t delete it) turn off each point (this is tedious, but worth it!) What would happen if you chose a different polygon to use as the frame? How does the shape of the frame affect the string art that appears? 3 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

6 String Art and Implicit Curves Assignment For this assignment you will be using Geogebra. It will be helpful to have the coordinate grid visible, and to turn off the labelling in the op,ons command. Follow the instruc=ons in Explora=on #3, but choose one of the equa=ons 1 on the following page. Decide on a possible frame to use based on the shape of the curve of your equa=on do you want a square or rectangular frame, or a polygon that mimics the shape of your curve? Decide on the spacing of the points to draw the strings from will they be evenly distributed on the frame, or scatered randomly on the frame? Once you have =died up your string art on the screen, export the image to a document in order to print a good copy of the image. In the file menu (the three horizontal bars in the top righthand corner of the screen), click on Download as and choose png. This will prompt you to choose a place on your computer to save the file. You will then be able to import the image into a document so you can decide on the sizing before prin=ng. There is an op=on to print directly from Geogebra, but some text from Geogebra will appear along with the image. Impor=ng the image into a document will give you more control over what you print. Print your string art, and present it in a visually appealing way. Consider adding a frame to your piece of art. Find the deriva=ve of your curve using implicit differen=a=on. Check your deriva=ve work using Geogebra. What have you learned about tangent lines, deriva=ves and implicit curves? How is an implicit equa=on the same/different from an explicit equa=on? How is an implicit curve the same/different as the curve of a func=on? What ques=ons do you have? 1 When typing the product of variables or factors in Geogebra, you may need to use the * symbol. Click off the original curve, the frame, the axes and the grid line. Don t delete the frame or the equa=on of the curve as your tangent lines will disappear. 4 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

7 Here is a list of some beau=ful algebraic curves in the plane: 3 Rose Curve: (! x + y ) 3 = 4x y x Hyperbola:, choose a and b a y b = 1 Nephroid: ( x + y 4a ) 3 = 108a y, choose a Lemniscate: x 4 = x y Folium of Descartes: x 3 + y 3 3axy = 0, choose a Serpen=ne Curve: Trisectrix of Maclaurin: Ampersand Curve: Bean Curve: Bicuspid Curve: Three-leaved Clover: Deltoid Curve: Devil s Curve: Limacon: Astroid: BuTerfly Curve: x 6 + y 6 = x x y + a y abx = 0, choose a and b x( x + y ) = a( 3x y ), choose a ( y x ) ( x 1) ( x 3) = 4( x + y x) x 4 + x y + y 4 = x( x + y ) ( x a ) ( x a) + ( y a ) = 0 x 4 +x y + y 4 x 3 + 3xy = 0, choose a ( x + y ) +18a ( x + y ) 7a 4 = 8a( x 3 3xy ) y ( y a ) = x ( x b ), choose a and b Hippopede: ( x + y ) = ax + by, choose a and b ( x + y ax) = b ( x + y ) ( x + y 1) 3 +7x y = 0, choose a and b, choose a 3 From Discovering the Art of Math: Calculus, Mathema,cal inquiry in the the Liberal Arts by Julian F. Fleron, Philip K. Hotchkiss and Chris=ne von Renesse with Volker Ecke 5 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

8 Checking the Equation of an Implicit Derivative Using Geogebra 1. Type the equa=on of the implicit curve into the input bar as a func=on of x and y. For example, to enter the equa=on of a circle with radius 3 ( x + y = 9 ), enter:. In a different input bar, start typing the word implicit, and click on the third command. Enter the leter f inside of the brackets and press enter (note: f is the name of the implicit equa=on entered in step 1). You should see this appear: In order to see the graphic representa=ons of the mul=variable func=ons, you need to use the 3D Graphics View. 6 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

9 3D view of f(x,y) = x + y 9 3D view of a(x,y) = x y 3D view of both f(x,y) and a(x,y) 7 Implicit Curves & Tangent Lines NWMC 017 srobinson@sd64.bc.ca

The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.

The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is. We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section