Gaudi's parabolic arches

Beautiful Barcelona and the genius of Gaudi

These are catenary arches!

Under the roof of Gaudí's Casa Milá, Barcelona, Spain

The catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force
(its own weight).





Don't confuse! These are not parabolic arches!


Arches in Casa Mila, by Gaudi too!

The beloved.... parabola of Gaudi

The peaceful area inside the gardens of the Parc Güell

in Barcelona

and the Gaudi's beloved  parabola .

Parabola Monument in Barcelona

The Fossar de les Moreres (Moreres Cemetery) is a memorial plaza in Barcelona (Catalonia, Spain), adjacent to the basilica of Santa Maria del Mar. The plaza was built over a cemetery where defenders of the city were buried following the Siege of Barcelona at the end of the War of the Spanish Succession in 1714.

The plaza retains its everyday use as a public space, but also prominently features a memorial to the fallen Catalans of the war, with a torch of eternal flame and a heroic poem by Frederic Soler, "El Fossar de les Moreres". On 11th September, in this place, the Catalan people celebrate their National Day. This historical date refers back to September 11 in 1714, when Barcelona, following a 13-month siege, surrendered to the Castilian and French soldiers.

The Monument

Look at the monument!

Keep in mind that it has the shape of a parabola!

Which of the following functions corresponds to this parabola? 

• Y = x2 + 2x – 4
• Y= - x2 + 7x
• Y= -x2 + 4

 



Look at:

Analyzing Graphs of Quadratic Functions

Didn't you know that?

 

Positive - negative quadratic - Vertex

positive quadratic y = x2	negative quadratic y = - x2
If the coefficient of x2 is positive the curve looks like the mouth of a smiling child. The vertex (0,0) is down.	If the coefficient of x2 is negative the curve looks like the mouth of a joyless child. The vertex (0,0) is up.

Source: The purplemath

The meaning of a in a quadratic

The general form of a quadratic is:

y = ax² + bx + c. 

For graphing, the leading coefficient "a" indicates how "fat" or how "skinny" the parabola will be.

For | a | > 1 (such as a = 3 or a = –4), the parabola will be "skinny", because it grows more quickly (three times as fast or four times as fast, respectively (in the case of our sample values of a). 

For | a | < 1 (such as a = ¹/₃ or a = ^–1/₄ ), the parabola will be "fat", because it grows more slowly (one-third as fast or one-fourth as fast, respectively (in the case of our sample values of a). 

Also, if a is negative, then the parabola is upside-down.

You can see these trends when you look at how the curve y = ax² moves as "a" changes.

Click on the adjacent image (graph) -->

Source: The purplemath

2 straight lines

Let's plot 2 straight lines:

• y=3x+b
• y=-3x+b

Look at the Geogebra application below.

Click here!

Write your remarks?
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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.

...........................................................................................


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)
...........................................................................................

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.

...........................................................................................
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?

...........................................................................................

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

What's about the line in the graph?

...........................................................................................

3. Look at the Geogebra application (Click here)

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

Coordinate System - Glossary

Important things to remember:

• All points located in Quadrant I have positive x- and y-coordinates.
• All points located in Quadrant II have negative x-coordinates and positive y-coordinates.
• All points located in Quadrant III have negative x- and y-coordinates.
• All points located in Quadrant IV have positive x- coordinates and negative y-coordinates.
• All points located on the x-axis have the y-coordinate equal to zero.
• All points located on the y-axis have the x-coordinate equal to zero.

In this video, you will learn how to plot various points in the coordinate plane (system).

Source: Video 

Work:

Please remember to fill in the cells of the shared google spreadsheet: Coordinate system Glossary.
You need to login with your google (gmail) account!

Happy New Year 2011

May all the dreams in your eyes,
all the desires in your heart and
all the hopes in your life blend together,
to give you the most spectacular New Year ever
Happy New Year 2011!

 May all the dreams in your eyes,
All the desires in your heart and
All the hopes in your life blend together,
To give you the most spectacular New Year ever
Happy New Year 2011!