Golf ball

A golf ball is dropped and a regular strobe light illustrates its motion as follows...

 We observe that it is a parabola.

Problem:What is the equation of the parabola that the golf ball is tracing out?

ΓΙΑ ΤΟΥΣ ΜΑΘΗΤΕΣ: χρησιμοποιήστε το Excel για να σχεδιάσετε το γράφημα.
Use Excel for making the graph and finding out the equation!

We can model the motion from the data points obtained by observing the position (height) of the ball at various times.
This is the table of values time/height:

Source: http://www.intmath.com/Plane-analytic-geometry/4_Parabola.php,  http://wn.com/The_Ball_and_the_Parabola

Also the tennis ball does the same motion. Watch the video:

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Using Excel to Model Curves

You can use Microsoft Excel to module a parabola. After you plot the points, right-click on one of the points and choose "Add Trendline (Γραμμή Τάσης)".
Choose Polynomial, degree 2. In "Options (Μορφοποίηση γραμμής τάσης --> Επιλογές)" you can get Excel to display the equation of the parabola on the chart.

Here it is the excel sheet which the students created

Plot the Graph by Geogebra:

Let's get to have a look at Geogebra file! Click HERE!

Find the equation of the quadratic using Geogebra

Use the application of Geogebra (parabola_water.html) and find the equation of the .... dancing waters of the fountains of Montjuic in Barcelona!!

(Geogebra will load here. Please be patient as it takes a short time for the Java to load!)
Move the sliders to position the parabola over one of the red fountains.Let's go...

click here....

World War II Memorial

The World War II Memorial in Washington D.C.honors the 16 million who served
in the armed forces of the United States during the war and the more than 400,000
who died. It is particularly beautiful at night as the following photos show.

The fountains are also an excellent example to be used in mathematics
as you will see the Geogebra application below: 

Click on the fountain to analyze the parabola
of the dancing water

Please be patient as it takes a short time for the Java to load! 

http://jwelker.lps.org/lessons/geogebra/ww2_parabola.html

The linear equation

ψ=aχ+b

 

Any time you have a "ψ equals a number" equation (i.e. y=5, with no x in it), the graph will always be a horizontal line.

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Any time you have an equation as ψ=aχ+b with a<0, then its graph is a straight line  and the corresponding function f(x) = aχ+b is decreasing.

Changing the value of a, the angle between the x-axis and the straight line is changing.
Have a look at Excel file: y=-aχ+b.xls (Click here)
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Any time you have an equation as ψ=aχ+b with a>0 , then its graph is a straight line and the corresponding function f(x) = aχ+b is increasing.

Changing the value of b, the straight line is moving from left to right and vice versa across the axis x, changing the y-intercept.

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WORK

1. Have a look at the Excel file revised y=aχ+b.xls (Click here) and move the scrollbar in order to change the value of a and b.
What do you remark?

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2. Go to the sheet Sales of the same excel file.

What's about the line in the graph?

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3. Look at the Geogebra application (Click here)

Which is the straight line that shows that you have bought cheaper  and why?
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