The following instructions lead through steps that build linked numeric, graphic, algebraic, and geometric models for the flight of the ball in the video above.

The instructions are also available as a PDF document.

 

 

A Numeric Representation of the Ball’s Flight

 

Click: ball flight start.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_start.html if GeoGebra is not installed on your computer.

 

The Spreadsheet View in GeoGebra provides an Excel-like spreadsheet. The numbers in columns A and B record the distances in feet, right and up from the point on the floor occupied by the ball in the first image (left end of the folded bleachers).

 

 

Graph Representation

 

Move the cursor to the graph pane and adjust the axes by:

 

Right-click > Graphics View > Axes > x and set x axis -1 to 15

                        > y and set y axis -1 to 15 and set xAxis : yAxis = 1:1

 

Highlight the A and B columns of the spreadsheet:

            Hold shift-key and click on each column

            right-click > Create List of Points

 

Problems?  Do the following to see the results of the above:

 

Click: ball flight graph.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_graph.html if GeoGebra is not installed on your computer.

 

The spreadsheet values and points are linked. Add 1.0 to any of the A column values and note the effect on the related point. Restore the A value by subtracting 1.

 

What type of curve might pass through the points?

 

 

Algebraic Representation: Equation in Standard Form: y=ax²+bx+c

 

Click: View > Algebra View (to show Algebra window)

 

Click: View > Input Bar

 

In the Input Bar type y=-3x^2 and Enter

 

This is obviously not the correct equation.  We will adjust and attempt to make the curve fit the points.

 

In the Free Objects list double-click: y=-3x²

 

Modify the equation and check fit by Enter.  Repeat to improve the fit.

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A simpler approach to modifying the equation

 

In the Input Bar type:

            a=1 Enter

            b=1 Enter

            c=1 Enter

            y=a*x^2+b*x+c  Enter

 

This puts the parameters a, b, and c in the Free Objects list and the equation y=x²+x+1 or y=ax²+bx+c  in the Dependent Objects list.

 

Click (highlight): a=1 and use the up and down cursor arrow keys to adjust the parameter, a

 

Using this approach with all three parameters adjust the curve to pass through the points.

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Alternatively:

 

            Right-click a > click Show Object

            Use the slider for a to adjust the value of the a parameter

 

Repeat for b and c

 

Use the sliders to adjust the values of all three parameters and adjust the curve to pass through the points. If the range of values is not sufficient:

            Right-click on the slider > click Object Properties…

            Adjust the minimum and/or maximum values

 

We now have linked numeric (spreadsheet) and algebraic (equation & graph) representations.

 

 

Algebraic Representation: Function in Vertex Form: f(x)=a(x-h)²+k

 

If you would like to retain an unexpanded form for the expression in x then function notation is required.

 

Create two new sliders for parameters h and k.

 

Click on the slider tool    and select Slider

 

Click on the graph window where you want the slider. Change the label to ‘h’, and Apply.

 

Repeat to get a slider for parameter ‘k’.        

 

We now have two more Free Objects h and k to use in a function

 

In the Input Bar type f(x)=a*(x-h)^2+k  and Enter         

 

Use the sliders for a, h, and k to adjust this new curve to pass through the points.

 

Problems?  Do the following to see the results of the above:

 

Click: ball flight alg curve.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_alg_curve.html if GeoGebra is not installed on your computer.

 

 

Reversing the Algebraic and Numeric Linking

 

Put a point on the curve for the function f(x)

 

            Click: New Point  

 

            Click on the curve

 

Click the Move tool 

 

Right-click on the point and select Trace to Spreadsheet.  Click and drag the point.

 

Problems?  Do the following to see the results of the above:

 

Click: ball flight numeric plus.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_numeric_plus.html if GeoGebra is not installed on your computer.

 

The algebraic (equation & graph) and numeric (spreadsheet) representations are linked.

 

 

A Statistical Modelling Approach to Constructing an Algebraic Representation

 

For some courses a statistical modelling process may be appropriate for developing a function and curve that best models the data.  GeoGebra 4.0 provides statistics tools including modelling by regression.  Return to the initial version of GeoGebra by:

 

Click: ball flight start.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_start.html if GeoGebra is not installed on your computer.

 

Highlight the numeric data in the A and B columns of the spreadsheet.  You do this by the usual method - left-click on the top left cell, hold down the left mouse button, and drag down to the bottom right cell.

Click on the Analysis icon   and in the drop down menu select Two Variable Regression Analysis

 

In the pop-up analysis window click on the Regression Model drop-down list and select Polynomial.  We do this because the scatter plot appears to take the shape of of a parabola, the curve of a second degree polynomial.  Degree 2 will automatically be selected since it is the smallest available.  If you wish to select a higher degree use the drop-down list.

 

GeoGebra will use regression analysis to generate the best fitting model using a second degree polynomial (quadratic) function and display the function and its graph.  In this case, since the points were selected to generate perfect data the model is a perfect fit.

 

In the analysis window click Options > Show Statistics.  This gives us statistical information on the x (distance) and y (height) data and the R² value of 1 tells us the model fit is perfect.

 

 

 

 

 

 

 

 

 

 

 

Additional information on using the model building feature of GeoGebra can be found on the Modelling Gas Prices page.

 

 

Building a Geometric (Conic) Model

 

Click: View > Spreadsheet  (to hide Spreadsheet window and create space)

 

We will use the Euclidean focus-directrix construction for a parabola. A parabola is the curve traced out by a point moving so that it is equidistant from a fixed point (focus) and a fixed line (directrix). (see the definition at: http://www.mathwords.com/d/directrix_parabola.htm)

 

Directrix:

 

Put a point on y-axis at approximately y=13: Click: New Point > Click on y-axis

 

Construct a line through this new point perpendicular to y-axis.  Note the steps are opposite to GSP – select the type of construction then the objects.

 

          Select: Perpendicular Line     and click the new point and the y-axis

 

Focus:

 

Place another new point below the perpendicular (horizontal) line

 

Construct the Parabola:

 

Click the Conic option  and select Parabola  

 

Click: the directrix (horizontal line) and focus (new point below the directrix)

 

Using the Move tool, adjust the locations of the focus and directrix to create a parabola that passes through the points.

 

Select the parabola and right-click.  Select Equation y=ax²+bx+c

 

Problems?  Do the following to see the results of the above:

 

Click: ball flight geometric.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_geometric.html if GeoGebra is not installed on your computer.

 

The geometric (parabola) and algebraic (equation) representations are now linked.

 

 

Using the Algebraic Representation

 

At this point all the parabolas should coincide and pass through the points.  If you are having problems:

 

Click: ball flight all.ggb and Open if you have GeoGebra on your computer

 

or

 

Click: ball_flight_all.html if GeoGebra is not installed on your computer.

 

We could use any of the models to determine the point where the ball would have hit the floor a second time if there was no net, but the function representation, f(x), provides a direct route to calculating the x-coordinate or distance to the right.

 

Click Command…

 

Scroll down the list of commands and select Root

 

Adjust the contents of the Input Bar to look read Root[f(x)] and Enter

 

In the Dependent Objects, read the x-coordinate of the new root point.

 

 

Adding Images to Connect to the Problem Context

 

You can add images (gif, jpeg, jpg, tif, png, bmp) to GeoGebra constructions using the Insert Image tool. The image may be set as background, pegged to the axes system, or may be linked to any point on the graphics plane. With this we can connect the ‘quadratic function – parabola’ activity to the original basketball context.  In the GeoGebra constructions linked below the picture of a basketball can be moved along the parabola to score a basket.   Click: ball flight bg parabola plus ball.ggb and Open if you have GeoGebra on your computer   or   Click: ball_flight_bg_parabola_plus_ball.html   if GeoGebra is not installed on your computer.