GEOGEBRA

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GEOGEBRA

• A free downloadable web based program for both
  Interactive Geometry and Coordinate Geometry.

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Finding Geogebra

• Either
• Google Geogebra
• or
• (b) www.geogebra.org
• Then click Start GeoGebra followed by
  GeoGebra Web Start

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Equations With Degree 1 or 2

• Equations of degree 1 or 2 can be entered either
  implicitly or as y
• For ease of use these equations should be given a
  name (a lower case letter)
• Eg Entering a2x3y 6 gives a straight line
  with name a

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The Command Menu

• This gives 69 different commands that can be
  applied to the diagram.
• Format
• (a) Select a command
• (b) For simple commands enter the name of the
  object in the square brackets.
• Eg To find the slope of the line, choose slope
  from the menu and type a in the brackets

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Drawing a perpendicular to a line from a point

• Choose point from the point menu (second from
  left)
• Click anywhere on the screen
• Choose perpendicular line (fourth menu from the
  left)
• Click on the point and then on the line

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Using The Object List

• Right clicking on any object in the object list
  or on the diagram gives a choice that includes
• (a) deleting the object
• (b) editing the object
• (c) changing the format of an equation
• Eg Right clicking on the equation of a straight
  line gives a choice of
• axby c y mx c or parametric form
• Using the mouse to point to any object on the
  list or on the diagram tells you what it is.

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Using Sliders

• Sliders allow you to enter a pronumeral as part
  of an equation and observe what happens as the
  pronumeral changes
• Eg Sliders can be used to observe what happens
  as values of m and c change when the equation y
  mxc is entered

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Investigating y mx c

• The slider menu is the 8th icon from the left
• Highlight this icon, then click anywhere to
  deposit the slider.
• Change setting if needed, then select Apply
• Repeat this process to create a second slider.
• Right click on the first slider, choose rename
  and change to m.
• Rename the second slider c
• Enter ay mx c Use your sliders to change
• the values of m and c

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Finding axes intercepts

• In Geogebra the name of the axes are xAxis and
  yAxis (case sensitive)
• From the command menu choose Intersect and type
  the name of your graph and the name of the axis
  in the brackets
• Eg Intersect a,xAxis
• Press enter

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Challenge Duplicate This Slide
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Investigating Circles

• Create three sliders and rename them h, k and r.
• Enter c (x-h)2(y-k)2r2
• Observe what happens as h, k and r change
• Use commands to find the centre and the radius of
  your circle

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The Circle
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More Complex Functions

• These must be entered using function notation
• A function entered as f(x) has the name f.
• For polynomials the package can
• (a) Find turning points (extremum) and
  inflexion points
• (b) Find roots
• (c) Draw first and second derivatives

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Cubics

• With the help of three sliders enter
• f(x) (x-a)(x-b)(x-c)
• Investigate the effect of changing a, b and c
• Using commands add the roots, extremum and
  inflexion point to your diagram.
• Draw the first and second derivatives
• derivativef and derivativef,2

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The Mess That You Get
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Trigonometric Functions

• Function notation must be used
• The argument of the trig function must be in
  brackets
• Right click on the x axis and choose properties
  units and select p
• Enter f(x) asin(b(x-h)) k and play
• The root in the interval 1, 2 can be found with
  rootf, 1, 2
• Derivatives can be found

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A Trigonometric Example
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Lower and Upper Sums

• Enter a slider with a range of values from 5 to
  100, label it n
• Enter your function. Eg f(x) 0.5x2 2
• To find a lower sum from 0 to 4 with n rectangles
  enter lowersumf,0,4,n
• Enter Uppersumf,0,4,n
• Play with the slider.

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Integration Example
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Demonstrating the angle in a semicircle theorem

• Right click on the axes and turn them off
• Draw a circle (Circle menu)
• Draw a line through the centre and any point on
  the circle (Line menu)
• Draw a triangle through the points of
  intersection of the line and the circle and any
  other point on the circle (polygon menu)
• Choose angle from the measure menu and click on
  any angle of the polygon
• Use the arrow (top left icon) to change the
  diagram

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