An Introduction to GeoGebra
GeoGebra is a free dynamic mathematics software that is very easy to use and
offers many features. It is much easier to use than a graphing calculator and offers
the ability to save and share work. The lead developer for this software is a high
school math teacher and contributors are programmers and teachers from all over
the world.
You can download GGB at www.geogebra.org.
When you click on Download, you’ll have several options. Webstart will put an
icon on your desktop. Applet Start will allow you to use the program on any
computer without installing anything. A student who does not have his/her own
computer can use the Applet Start anytime, anywhere, any computer. There is also
an offline installer for students who do not have internet access. This one seems to
work a little slower, but it gets the job done.
Click on Help to go to the GGB Wiki. There’s a manual and a link to tutorials.
Both are excellent sources of information. Another good source of information is
http://mathandmultimedia.com. Click on the GeoGebra link at the top of the main
page. Some of his videos are a little old, but they are pretty good.
Drop Down Menus
Move Icon
Algebra Window
Graphics Window
Input Line
Input Help
Part I – Graphing and Analyzing Functions – the short version
1. Enter a function in the Input Line.
2. Enter commands in the Input Line to tell the program what you want to do.
Fill in the inputs and press enter.
Commands:
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Zeros of a function: Root
Relative extrema of a function: Extremum
Inflection points of a polynomial function: Inflection
Points of intersection of two graphs: Intersect
Derivative of a function (including partial derivatives): Derivative
Riemann sums: RectangleSum
Upper (Lower) Sums: UpperSum (LowerSum)
Antiderivative of a function: Integral
Definite integral: Integral
Area between two curves: IntegralBetween
Limits: Limit
Asymptotes: Asymptote
Task 1: Using GGB as a calculator.
Enter an expression in the Input Line using your keyboard and press enter to
compute.


Example: 3 1  4 3  52  8
3
Task 2: Graphing a function.
Enter a function in the Input Line using your keyboard and press enter to graph the
function. Resize the Graphics window by selecting the Move Graphics View icon.
Then you can put the cursor near either axis and drag it in either direction to
change the dimensions of the window. To change the location of the origin in the
display, hover the cursor near the origin until you see the + sign, then left click to
get two perpendicular double-headed arrows. The you can drag the origin to the
desired location.
Examples:
f ( x)  2 x 3  5 x 2  4 x  3
1
g  x   x 4  3x 3  4 x 2  8 x  9
2
h  x   3x 2e 2 x
Task 3: Find zeros of a function.
Enter the function in the Input Line. Use the Root command to find zeros. Start
typing Root in the Input Line and several options will appear.
Choose Root[<Polynomial>] for a polynomial function. Highlight <Polynomial>
and replace it with the function name. Press enter on your keyboard. GGB returns
all of the zeros simultaneously.
Choose Root[<Function>, <Start x-Value>, <End x-Value>] for non-polynomial
functions. Find an interval that contains zero and use the left and right endpoints
of the interval as your start and end values.
Examples:
1
f  x   x 4  3x 3  4 x 2  8 x  9
2
x2  x  5
g  x  3
x  3x  2
h  x   x 2e  x
Task 4: Find the relative extrema of a function.
Enter the function in the Input Line. Use the Extremum command to find relative
extrema. Start typing Extremum in the Input Line and then select the appropriate
command.
For a polynomial function, choose Extremum[<Polynomial>] and press enter on
your keyboard. GGB returns all relative extrema simultaneously. Coordinates are
given in the Algebra Window. Points are labeled in the Graphics Window.
For other types of functions, choose the Extremum[<Function>, <Start x-Value>,
<End x-Value>] command and select intervals for each extremum. Find these
extrema one at a time, similar to a graphing calculator.
Examples:
1
f  x   x 4  3x 3  4 x 2  8 x  9
2
g  x
 x  3

2
x 1
Task 5: Graph the derivative of a function.
Type f '( x) in the Input Line, and GGB will graph the derivative of the function.
The software will also display the symbolic derivative in the Algebra window.
Example:
f  x
 x  3

x 1
2
Task 6: Find inflection points of polynomial functions.
Enter the function in the Input Line, then type Inflection. There is only one option.
There is no similar command for non-polynomial functions. For those, graph the
second derivative and analyze the sign of the derivative.
Examples:
1
f  x   x 4  3x 3  4 x 2  8 x  9
2
g ( x)  2 x 3  5 x 2  4 x  3
Task 7: Find any points of intersection of two equations.
Enter the two equations in the Input Line. Use the Intersect command. For two
polynomials, choose Intersect[<Object>, <Object>]. For other types of functions,
choose Intersect [<Function>, <Function>, <Start x-Value>, <End x-Value>] and
choose intervals for each point of intersection. Note that if you use the
Intersect[<Object>, <Object>] command when non-polynomial functions are
involved, GGB returns only one intersection point even if there are many.
Examples:
1
Find all points of intersection: f  x   x 4  3x3  4 x 2  8x  9 and
2
3
2
g  x   2 x  5x  4 x  3
Find all points of intersection: f  x   x e
2 x
and g  x 
 x  3

x 1
2
Task 8: Riemann sums
Enter the function in the Input Line. Use the RectangleSum command. This has
five inputs: [<Function>, <Start x-Value>, <End x-Value>, <Number of
Rectangles>, <Position for rectangle start>]. For the last input, GGB will use left
endpoints, right endpoints or midpoints – or any other location in the interval that
you specify. Use 0 for left endpoints, 0.5 for midpoints and 1 for right endpoints.
Example:
g ( x)  2 x3  5 x 2  4 x  3 on the interval [-2, 1] with 50 rectangles using (a) left
endpoints, (b) right endpoints and (c) midpoints.
Task 9: Upper sums/Lower sums
Enter the function in the Input Line. Use the UpperSum (or LowerSum) command.
These commands require four inputs: [<Function>, <Start x-Value>, <End xValue>, <Number of Rectangles>]
g ( x)  2 x3  5 x 2  4 x  3 on the interval [-2, 1] with 50 rectangles using (a) upper
sums and (b) lower sums.
Task 10: Integrals
To find an indefinite integral, enter the function in the input line. Use the Integral
[<Function>] command to find the antiderivative.
If find a definite integral, enter the function in the input line. Use the
Integral[<Function>, <Start x-Value>, <End x-Value>] command.
Examples:
  3x
3
 7 x 2  12 x  5  dx
 x 2 ln  x 2  5  
 1.68   x  32  dx


2.15
Task 11: Area between two curves
Enter the two (or more) functions into GGB. Use the IntegralBetween command.
Choose the one without Boolean Value as an input. Enter the top function first and
the bottom function second.
Examples:
Find the area between f  x   x 2  5 and g  x   x  1 between the vertical lines
x  1 and x  2 .
Find the area of the region completely bounded by f  x   x 2  6 x and
g  x   2.5 x.
Find the area of the region(s) completely bounded by
1
f  x   x 4  3x3  4 x 2  8x  9 and g ( x)  2 x3  5 x 2  4 x  3
2
Task 12: Limits
Enter the function in the Input Line. Use the Limit[<Function>, <Value>]
command.
Example:
 x 1 1
lim 

x 0
x


Task 13: Asymptotes:
Enter the function in the Input Line. Use the Asymptote[<Function>] command.
Note that this command returns slant asymptotes.
Examples:
g  x
 x  3

x 1
f  x   x 2e  x
2
Part II – Regressions
Use GGB to find linear regressions, polynomial regressions of any degree, power
regressions, exponential regressions, logistic regressions (and others).
1.
2.
3.
4.
5.
Enable the Spreadsheet View.
Enter the data that is given.
Create ordered pairs.
Create a list.
Find the regression model using a Fit command.
To create ordered pairs: Assuming data is in columns A and B at the top, enter
=(a1,b1) in cell C1 and replicate for all data, the same way you replicate data in
Excel.
To create a list: Highlight the data in ordered pair form. Click on the Create List
icon.
GGB will show the list and propose a name (usually list1). Click on create. The
list will appear in the Algebra Window.
To find a regression equation: Use a Fit command. Here are the options:
FitExp, FitGrowth, FitLine, FitLogistic, FitPoly, FitPower, FitSin. The input for
most of them is just the name of the list of points. FitPoly also asks for the degree
of the polynomial.
Example: Use the data to find linear, cubic and exponential regression models.
x 1 3 4 7 9 11
y 12 15 11 16 17 23
Example: Use the data to find linear, quadratic and exponential regression models.
x 1 2 3 4 5 6
y 27 40 46 47 48 49
To find the value for r 2 or R 2 : Type RSquare in the Input Line. Enter the list and
the function name, and GGB returns the value for r 2 or R 2 .
Optimizing a function of two variables using GGB
Find any relative extrema: f  x, y   x3  10 xy  4 y 2
Note: GGB will not solve complicated systems of equations in two variables, so
finding critical numbers may be need to be done by hand.
Useful things to know:
To get a new window: Control N or click on File and select New Window.
To change font size: Click on Options and select Font Size. I typically use 18 in
class. See what works for you.
To change the number of decimal places displayed: Click on Options and select
Rounding. Answers are usually rounded to four decimal places.
To save changes to settings: Change the settings in an open window. Then click
on Options, select Settings and choose Save Settings.
To access a list of commands: Click on Input Help, lower right corner, to the right
of the Input Line. It’s a little rectangle with an arrow head in it.
To access Spreadsheet View: Click on View and check Spreadsheet.
To de-activate something that is in the Algebra Window so that it does not display
in the Graphics Window: Click on the blue dot to the left of the item. To reactivate, click it again.
To delete something in the Algebra Window: Right click on the item and select
Delete. GGB doesn’t give an “are you sure?” prompt. Once you delete something
from the Algebra Window, it’s gone from the Graphics Window, too. But…
To review items you recently entered in the Input Line, click on the up/down
arrows at the right end of the Input Line. If you want to re-enter something, you
can select it to pull it into the Input Line.
To edit something you’ve entered: Right click on the item and select Object
Properties. On the Basic tab, there’s a line titled Value: which should list your
item. You can put your cursor in that box and edit. Once you are on that window,
you’ll see that there are many things you can change about your function.
To change colors for graphs: Right click on the function or the graph, choose
Object Properties, select Color and choose a color. The function and the graph will
change to that color.
To enter square roots: type sqrt(expression).
To enter cube roots: type (expression)^(1/3).
To return the Graphics Window to the Standard View: Put the cursor in the
Graphics Window, right click and select Standard View.
To enable the grid: Right-click inside the Graphics window and choose Grid.
To enter 3x 2 y : Use an * between x 2 and y. Multiplication of anything other than
a constant coefficient by a variable requires the use of * between the factors.
If you have a function or value for e defined in the Algebra window, and you want
to enter f  x   e x , GGB will use the definition in the window rather than
n
 1
lim 1   for e.
n
 n
If you have function f defined in the Algebra Window and you type a new function
named f in the Input Line, GGB will replace the original one; the software will not
rename your new entry.
Use the Snipping Tool in Win7 to cut and paste the GGB screen into another
document. It’s in my tray. If you can’t see it, Google “Activate Snipping Tool in
Win 7” and you’ll get plenty of advice. Or ask Dave.
You can also save or export the Graphics Window. Click on File and select Save
to see the options.