Differenze tra le versioni di "Tutorial:Input algebrico Comandi base"

Da GeoGebra Manual.
(Creata pagina con '==Suggerimenti e accorgimenti== 1. '''Assegnare un nome a un nuovo oggetto''' digitando ''nome ='' nella barra di inserimento, prima della relativa rappresentazione algebrica. {{...')
 
Riga 7: Riga 7:
 
* Segmenti, rette, circonferenze, funzioni… vengono sempre identificati con lettere minuscole. {{Example|1= circonf c: (x – 2)^2 + (y – 1)^2 = 16}}
 
* Segmenti, rette, circonferenze, funzioni… vengono sempre identificati con lettere minuscole. {{Example|1= circonf c: (x – 2)^2 + (y – 1)^2 = 16}}
 
* La variabile x in una funzione e le variabili x e y nell'equazione di una conica devono essere sempre indicate in minuscolo. {{Example|1= f(x) = 3*x + 2}}
 
* La variabile x in una funzione e le variabili x e y nell'equazione di una conica devono essere sempre indicate in minuscolo. {{Example|1= f(x) = 3*x + 2}}
−
4. If you want to use an '''object within an algebraic expression''' or command you need to create the object prior to using its name in the input bar.  
+
4. Per '''utilizzare un oggetto all'interno di un'espressione algebrica o di un comando''' è necessario creare l'oggetto prima di digitarne il nome nella barra di inserimento.  
−
* y = m x + b creates a line whose parameters are already existing values m and b (e.g. numbers / sliders).  
+
* y = m x + q genera una retta i cui parametri sono i valori già esistenti  m e q  (ad es. numeri / slider).  
−
* Line[A, B] creates a line through existing points A and B.
+
* Retta[A, B] crea la retta passante per i punti A e B già esistenti.
  
−
5. '''Confirm an expression''' you entered into the input bar by pressing the {{KeyCode|Enter}} key.
+
5. '''Confermare un'espressione''' digitata nella barra di inserimento premendo il tasto {{KeyCode|Invio}} .
  
−
6. '''Open the help window''' for using the input bar and commands by selecting ''Help'' from the [[Help Menu]] (or shortcut {{KeyCode|F1}}).
+
6. '''Aprire la finestra della Guida''' relativa alla barra di inserimento e ai comandi facendo clic su ''Guida'' nel  [[menu Guida]] (oppure premere {{KeyCode|F1}}).
  
−
7. '''Error messages''': Always read the messages – they could possibly help to fix the problem!
+
7. '''Messaggi di errore''': leggere sempre i messaggi – possono essere un valido aiuto per la risoluzione dei problemi
−
8. '''Commands''' can be typed in or selected from the list next to the [[Input Bar]]. {{hint|If you don’t know which parameters are required within the brackets of a certain command, type in the full command name and press key {{KeyCode|F1}} to open the GeoGebra Wiki.}}
+
8. '''I comandi''' possono essere digitati o selezionati direttamente dall'elenco a destra della [[barra di inserimento]]. {{hint|Se non si conoscono i parametri da indicare nelle parentesi di un comando, digitare interamente il nome del comando e premere il tasto {{KeyCode|F1}} per aprire il GeoGebra Wiki.}}
−
9. '''Automatic completion of commands''': After typing in the first two letters of a command into the [[Input Bar]], GeoGebra tries to complete the command.
+
9. '''Completamento automatico dei comandi''': Dopo la digitazione delle prime due lettere di un comando nella [[barra di inserimento]], GeoGebra esegue il completamento automatico del nome del comando.
−
* If GeoGebra suggests the desired command, hit the {{KeyCode|Enter}} key in order to place the cursor within the brackets.
+
* Se viene visualizzato il comando desiderato, premere il tasto {{KeyCode|Invio}}: il cursore verrà posizionato all'interno delle parentesi.  
−
* If the suggested command is not the one you wanted to enter, just keep typing until the suggestion matches.
+
* Se il comando visualizzato non è quello desiderato, continuare la digitazione fino ad ottenere il comando corretto.
  
−
==Constructing Tangents to a Circle (Part 1)==
+
==Costruire le tangenti ad una circonferenza (Parte 1)==
−
Open the dynamic worksheet [http://www.geogebra.org/book/intro-en/topics/files/04_Algebraic_Input_Commands_Functions/A08_Tangents_Circle.html Tangents to a Circle]. Follow the directions on the worksheet in order to find out how to construct tangents to a circle.
+
Aprire il foglio di lavoro dinamico [http://www.geogebra.org/book/intro-en/topics/files/04_Algebraic_Input_Commands_Functions/A08_Tangents_Circle.html Tangenti a una circonferenza]. Seguire le istruzioni sul foglio di lavoro, in modo da scoprire come costruire le tangenti ad una circonferenza.
  
−
===Discussion===
+
===Discussione===
−
* Which tools did you use in order to recreate the construction?
+
* Quali strumenti sono stati utilizzati per ricreare la costruzione?  
−
* Were there any new tools involved in the suggested construction steps? If yes, how did you find out how to operate the new tool?
+
* Sono stati utilizzati nuovi strumenti nel processo di costruzione suggerito? Se sì, come avete scoperto il funzionamento del nuovo strumento?  
−
* Did you notice anything about the toolbar displayed in the right applet?
+
* Avete notato la barra degli strumenti visualizzata nell'applet a destra?  
−
* Do you think your students could work with such a dynamic worksheet and find out about construction steps on their own?
+
* Pensate che i vostri studenti siano in grado di lavorare con un foglio di lavoro dinamico di questo tipo, e scoprire autonomamente i processi di costruzione?
  
 
==Constructing Tangents to a Circle (Part 2)==
 
==Constructing Tangents to a Circle (Part 2)==
Riga 37: Riga 37:
 
===Preparations===
 
===Preparations===
 
* Open a new GeoGebra window.
 
* Open a new GeoGebra window.
−
* Show the [[Algebra View]] and [[Input Bar]], as well as coordinate axes ([[View Menu]])
+
* Show the [[Vista Algebra]] and [[barra di inserimento]], as well as coordinate axes ([[View Menu]])
  
 
===Construction Steps===
 
===Construction Steps===
Riga 50: Riga 50:
  
 
{{note|GeoGebra distinguishes between [[Free, Dependent and Auxiliary Objects|free and dependent objects]]. While free objects can be directly modified either using the mouse or the keyboard, dependent objects adapt to changes of their parent objects. Thereby, it is irrelevant in which way (mouse or keyboard) an object was initially created!}}
 
{{note|GeoGebra distinguishes between [[Free, Dependent and Auxiliary Objects|free and dependent objects]]. While free objects can be directly modified either using the mouse or the keyboard, dependent objects adapt to changes of their parent objects. Thereby, it is irrelevant in which way (mouse or keyboard) an object was initially created!}}
−
{{hint|Activate Move mode and double click an object in the [[Algebra View]] in order to change its algebraic representation using the keyboard. Hit the {{KeyCode|Enter}} key once you are done.}}
+
{{hint|Activate Move mode and double click an object in the [[Vista Algebra]] in order to change its algebraic representation using the keyboard. Hit the {{KeyCode|Enter}} key once you are done.}}
−
{{hint|You can use the arrow keys in order to move free objects in a more controlled way. Activate Move mode and select the object (e.g. a free point) in either window. Press the up / down or left / right arrow keys in order to move the object into the desired direction.}}
+
{{hint|You can use the arrow keys in order to move free objects in a more controlled way. Activate Move mode and select the object (ad es. a free point) in either window. Press the up / down or left / right arrow keys in order to move the object into the desired direction.}}
  
 
{|border="1" cellpadding="15"  
 
{|border="1" cellpadding="15"  
Riga 62: Riga 62:
 
|7||d = Circle[D, C]||Circle with center D through point C
 
|7||d = Circle[D, C]||Circle with center D through point C
 
|-
 
|-
−
|8||Intersect[c, d]||Intersection points E and F of the two circles
+
|8||Intersect[c, d]||Intersection punti E and F of the two circles
 
|-
 
|-
−
|9||Line[C, E]||Tangent through points C and E
+
|9||Line[C, E]||Tangent through punti C and E
 
|-
 
|-
−
|10||Line[C, F]||Tangent through points C and F
+
|10||Line[C, F]||Tangent through punti C and F
 
|}
 
|}
  
 
===Checking and Enhancing the Construction===
 
===Checking and Enhancing the Construction===
 
* Perform the drag-test in order to check if the construction is correct.
 
* Perform the drag-test in order to check if the construction is correct.
−
* Change properties of objects in order to improve the construction’s appearance (e.g. colors, line thickness, auxiliary objects dashed,…)
+
* Change properties of objects in order to improve the construction’s appearance (ad es. colors, line thickness, auxiliary objects dashed,…)
 
* Save the construction.
 
* Save the construction.
  
Riga 78: Riga 78:
 
* Which version of the construction (mouse or keyboard) do you prefer and why?
 
* Which version of the construction (mouse or keyboard) do you prefer and why?
 
* Why should we use keyboard input if we could also do it using tools? {{hint|There are commands available that have no equivalent geometric tool.}}
 
* Why should we use keyboard input if we could also do it using tools? {{hint|There are commands available that have no equivalent geometric tool.}}
−
* Does it matter in which way an object was created? Can it be changed in the [[Algebra View]] (using the keyboard) as well as in the [[Graphics View]] (using the mouse)?
+
* Does it matter in which way an object was created? Can it be changed in the [[Vista Algebra]] (using the keyboard) as well as in the [[Vista Grafica]] (using the mouse)?
  
 
==Exploring Parameters of a Quadratic Polynomial==
 
==Exploring Parameters of a Quadratic Polynomial==
Riga 85: Riga 85:
 
# Open a '''new GeoGebra window'''
 
# Open a '''new GeoGebra window'''
 
# '''Type''' in '''f(x) = x^2''' and hit the {{KeyCode|Enter}} key. Which '''shape''' does the function graph have? Write down your answer on paper.
 
# '''Type''' in '''f(x) = x^2''' and hit the {{KeyCode|Enter}} key. Which '''shape''' does the function graph have? Write down your answer on paper.
−
# In [[Image:Tool_Move.gif‎]] Move mode, highlight the polynomial in the algebra view and use the '''↑ up and ↓ down arrow keys'''.
+
# In [[Image:Tool_Move.gif‎]] Move mode, highlight the polynomial in the Vista Algebra and use the '''↑ up and ↓ down arrow keys'''.
 
#* How does this impact the graph of the polynomial? Write down your observations.
 
#* How does this impact the graph of the polynomial? Write down your observations.
−
#* How does this impact the equation of the polynomial? Write down your observations.
+
#* How does this impact the equazione of the polynomial? Write down your observations.
−
# Again, in Move mode, highlight the function in the [[Algebra View]] and use the '''← left and → right arrow keys'''.
+
# Again, in Move mode, highlight the function in the [[Vista Algebra]] and use the '''← left and → right arrow keys'''.
 
#* How does this impact the graph of the polynomial? Write down your observations.
 
#* How does this impact the graph of the polynomial? Write down your observations.
−
#* How does this impact the equation of the polynomial? Write down your observations.
+
#* How does this impact the equazione of the polynomial? Write down your observations.
−
# In Move mode, double click the equation of the polynomial. Use the keyboard to '''change the equation''' to '''f(x) = 3 x^2.''' Use an asterisk * or space in order to enter a multiplication.
+
# In Move mode, double click the equazione of the polynomial. Use the keyboard to '''change the equazione''' to '''f(x) = 3 x^2.''' Use an asterisk * or space in order to enter a multiplication.
 
#* Describe how the function graph changes.
 
#* Describe how the function graph changes.
−
#* Repeat changing the equation by typing in different values for the parameter (e.g. 0.5, -2, -0.8, 3). Write down your observations.
+
#* Repeat changing the equazione by typing in different values for the parameter (ad es. 0.5, -2, -0.8, 3). Write down your observations.
  
 
===Discussion===
 
===Discussion===
Riga 118: Riga 118:
  
 
===Representing a Number as a Slider===
 
===Representing a Number as a Slider===
−
To display number as a [[Slider Tool|slider]] in the [[Graphics View]] you need to right click (MacOS: {{KeyCode|Ctrl}}-click) the variable in the [[Algebra View]] and select ''Show object''.
+
To display number as a [[Slider Tool|slider]] in the [[Vista Grafica]] you need to right click (MacOS: {{KeyCode|Ctrl}}-click) the variable in the [[Vista Algebra]] and select ''Show object''.
  
 
===Enhancing the Construction===
 
===Enhancing the Construction===
−
Let’s create another slider b that controls the constant in the polynomial’s equation f(x) = a x^2 + b.
+
Let’s create another slider b that controls the constant in the polynomial’s equazione f(x) = a x^2 + b.
  
 
{|border="1" cellpadding="15" col width="950"
 
{|border="1" cellpadding="15" col width="950"
−
|3||[[Image:Tool_Slider.gif‎]]||Create a slider b using the [[Slider Tool]] {{hint|Activate the tool and click on the [[Graphics View]]. Use the default settings and click ''Apply''.}}
+
|3||[[Image:Tool_Slider.gif‎]]||Create a slider b using the [[Slider Tool]] {{hint|Activate the tool and click on the [[Vista Grafica]]. Use the default settings and click ''Apply''.}}
 
|-
 
|-
 
|4||f(x) = a * x^2 + b||Enter the polynomial f {{hint|GeoGebra will overwrite the old function f with the new definition.}}
 
|4||f(x) = a * x^2 + b||Enter the polynomial f {{hint|GeoGebra will overwrite the old function f with the new definition.}}
Riga 135: Riga 135:
  
 
==Library of Functions==
 
==Library of Functions==
−
Apart from polynomials there are different types of functions available in GeoGebra (e.g. trigonometric functions, absolute value function, exponential function). Functions are treated as objects and can be used in combination with geometric constructions.
+
Apart from polynomials there are different types of functions available in GeoGebra (ad es. trigonometric functions, absolute value function, exponential function). Functions are treated as objects and can be used in combination with geometric constructions.
  
 
===Task 1: Visualizing absolute values===
 
===Task 1: Visualizing absolute values===
−
Open a new GeoGebra window. Make sure the [[Algebra View]], [[Input Bar]] and coordinate axes are shown.
+
Open a new GeoGebra window. Make sure the [[Vista Algebra]], [[barra di inserimento]] and coordinate axes are shown.
  
 
{|border="1" cellpadding="15"  
 
{|border="1" cellpadding="15"  
Riga 145: Riga 145:
 
|2||g(x) = 3||Enter the constant function g
 
|2||g(x) = 3||Enter the constant function g
 
|-
 
|-
−
|3||[[Image:Tool_Intersect_Two_Objects.gif‎]]||[[Intersect Two Objects Tool|Intersect]] both functions {{hint|You need to intersect the functions twice in order to get both intersection points.}}
+
|3||[[Image:Tool_Intersect_Two_Objects.gif‎]]||[[Intersect Two Objects Tool|Intersect]] both functions {{hint|You need to intersect the functions twice in order to get both intersection punti.}}
 
|}
 
|}
−
{{hint|You might want to close the [[Algebra View]] and show the names and values as [[Labels and Captions|labels]] of the objects.}}
+
{{hint|You might want to close the [[Vista Algebra]] and show the names and values as [[Labels and Captions|labels]] of the objects.}}
  
 
[[Image:4_absolute.PNG|center]]
 
[[Image:4_absolute.PNG|center]]
Riga 154: Riga 154:
 
(a) Move the constant function with the mouse or using the arrow keys. The y-coordinate of each intersection point represents the absolute value of the x-coordinate.
 
(a) Move the constant function with the mouse or using the arrow keys. The y-coordinate of each intersection point represents the absolute value of the x-coordinate.
  
−
(b) Move the absolute value function up and down either using the mouse or the arrow keys. In which way does the function’s equation change?
+
(b) Move the absolute value function up and down either using the mouse or the arrow keys. In which way does the function’s equazione change?
  
−
(c) How could this construction be used in order to familiarize students with the concept of absolute value? {{hint|The symmetry of the function graph indicates that there are usually two solutions for an absolute value problem.}}
+
(c) How could this construction be used in order to familiarize students with the concept of absolute value? {{hint|The symmetry of the function graph indicates that there are usually two soluzioni for an absolute value problem.}}
  
 
===Task 2: Superposition of Sine Waves===
 
===Task 2: Superposition of Sine Waves===
Riga 166: Riga 166:
 
|1||f(x) = abs(x)||Create three sliders a_1, ω_1, and φ_1 {{hint|a_1 produces an index. You can select the Greek letters from the menu next to the text field name in the Slider dialog window.}}
 
|1||f(x) = abs(x)||Create three sliders a_1, ω_1, and φ_1 {{hint|a_1 produces an index. You can select the Greek letters from the menu next to the text field name in the Slider dialog window.}}
 
|-  
 
|-  
−
|2||g(x)= a_1 sin(ω_1 x + φ_1)||Enter the sine function g {{hint|Again, you can select the Greek letters from a menu next to the [[Input Bar]].
+
|2||g(x)= a_1 sin(ω_1 x + φ_1)||Enter the sine function g {{hint|Again, you can select the Greek letters from a menu next to the [[barra di inserimento]].
 
|}
 
|}
  

Versione delle 12:14, 9 mar 2012

Suggerimenti e accorgimenti

1. Assegnare un nome a un nuovo oggetto digitando nome = nella barra di inserimento, prima della relativa rappresentazione algebrica.

Esempio: P = (3, 2) crea il punto P.

2. La moltiplicazione viene indicata digitando un asterisco o uno spazio tra i fattori.

Esempio: a*x oppure a x

3. In GeoGebra c'è distinzione tra maiuscole e minuscole Quindi lettere maiuscole e minuscole non hanno lo stesso significato.

  • I punti vengono sempre identificati con lettere maiuscole
    Esempio: A = (1, 2)
  • I vettori vengono sempre identificati con lettere minuscole
    Esempio: v = (1, 3)
  • Segmenti, rette, circonferenze, funzioni… vengono sempre identificati con lettere minuscole.
    Esempio: circonf c: (x – 2)^2 + (y – 1)^2 = 16
  • La variabile x in una funzione e le variabili x e y nell'equazione di una conica devono essere sempre indicate in minuscolo.
    Esempio: f(x) = 3*x + 2

4. Per utilizzare un oggetto all'interno di un'espressione algebrica o di un comando è necessario creare l'oggetto prima di digitarne il nome nella barra di inserimento.

  • y = m x + q genera una retta i cui parametri sono i valori già esistenti m e q (ad es. numeri / slider).
  • Retta[A, B] crea la retta passante per i punti A e B già esistenti.

5. Confermare un'espressione digitata nella barra di inserimento premendo il tasto Invio .

6. Aprire la finestra della Guida relativa alla barra di inserimento e ai comandi facendo clic su Guida nel menu Guida (oppure premere F1).

7. Messaggi di errore: leggere sempre i messaggi – possono essere un valido aiuto per la risoluzione dei problemi

8. I comandi possono essere digitati o selezionati direttamente dall'elenco a destra della barra di inserimento.

Note Suggerimento: Se non si conoscono i parametri da indicare nelle parentesi di un comando, digitare interamente il nome del comando e premere il tasto F1 per aprire il GeoGebra Wiki.

9. Completamento automatico dei comandi: Dopo la digitazione delle prime due lettere di un comando nella barra di inserimento, GeoGebra esegue il completamento automatico del nome del comando.

  • Se viene visualizzato il comando desiderato, premere il tasto Invio: il cursore verrà posizionato all'interno delle parentesi.
  • Se il comando visualizzato non è quello desiderato, continuare la digitazione fino ad ottenere il comando corretto.

Costruire le tangenti ad una circonferenza (Parte 1)

Aprire il foglio di lavoro dinamico Tangenti a una circonferenza. Seguire le istruzioni sul foglio di lavoro, in modo da scoprire come costruire le tangenti ad una circonferenza.

Discussione

  • Quali strumenti sono stati utilizzati per ricreare la costruzione?
  • Sono stati utilizzati nuovi strumenti nel processo di costruzione suggerito? Se sì, come avete scoperto il funzionamento del nuovo strumento?
  • Avete notato la barra degli strumenti visualizzata nell'applet a destra?
  • Pensate che i vostri studenti siano in grado di lavorare con un foglio di lavoro dinamico di questo tipo, e scoprire autonomamente i processi di costruzione?

Constructing Tangents to a Circle (Part 2)

What if my Mouse and Touchpad wouldn’t work?

Imagine your mouse and / or touchpad stop working while you are preparing GeoGebra files for tomorrow’s lesson. How can you finish the construction file?

GeoGebra offers algebraic input and commands in addition to the geometry tools. Every tool has a matching command and therefore could be applied without even using the mouse.

Note: GeoGebra offers more commands than geometry tools. Therefore, not every command has a corresponding geometry tool!

Preparations

Construction Steps

1 A = (0, 0) Point A
2 (3, 0) Point B
Note Suggerimento: If you don’t specify a name objects are named in alphabetical order.
3 c = Circle[A, B] Circle with center A through point B
Note Suggerimento: Circle is a dependent object
Note: GeoGebra distinguishes between free and dependent objects. While free objects can be directly modified either using the mouse or the keyboard, dependent objects adapt to changes of their parent objects. Thereby, it is irrelevant in which way (mouse or keyboard) an object was initially created!
Note Suggerimento: Activate Move mode and double click an object in the Vista Algebra in order to change its algebraic representation using the keyboard. Hit the Enter key once you are done.
Note Suggerimento: You can use the arrow keys in order to move free objects in a more controlled way. Activate Move mode and select the object (ad es. a free point) in either window. Press the up / down or left / right arrow keys in order to move the object into the desired direction.


4 C = (5, 4) Point C
5 s = Segment[A, C] Segment AC
6 D = Midpoint[s] Midpoint D of segment AC
7 d = Circle[D, C] Circle with center D through point C
8 Intersect[c, d] Intersection punti E and F of the two circles
9 Line[C, E] Tangent through punti C and E
10 Line[C, F] Tangent through punti C and F

Checking and Enhancing the Construction

  • Perform the drag-test in order to check if the construction is correct.
  • Change properties of objects in order to improve the construction’s appearance (ad es. colors, line thickness, auxiliary objects dashed,…)
  • Save the construction.

Discussion

  • Did any problems or difficulties occur during the construction steps?
  • Which version of the construction (mouse or keyboard) do you prefer and why?
  • Why should we use keyboard input if we could also do it using tools?
    Note Suggerimento: There are commands available that have no equivalent geometric tool.
  • Does it matter in which way an object was created? Can it be changed in the Vista Algebra (using the keyboard) as well as in the Vista Grafica (using the mouse)?

Exploring Parameters of a Quadratic Polynomial

In this activity you will explore the impact of parameters on a quadratic polynomial. You will experience how GeoGebra could be integrated into a "traditional" teaching environment and used for active, student-centered learning.

  1. Open a new GeoGebra window
  2. Type in f(x) = x^2 and hit the Enter key. Which shape does the function graph have? Write down your answer on paper.
  3. In Tool Move.gif Move mode, highlight the polynomial in the Vista Algebra and use the ↑ up and ↓ down arrow keys.
    • How does this impact the graph of the polynomial? Write down your observations.
    • How does this impact the equazione of the polynomial? Write down your observations.
  4. Again, in Move mode, highlight the function in the Vista Algebra and use the ← left and → right arrow keys.
    • How does this impact the graph of the polynomial? Write down your observations.
    • How does this impact the equazione of the polynomial? Write down your observations.
  5. In Move mode, double click the equazione of the polynomial. Use the keyboard to change the equazione to f(x) = 3 x^2. Use an asterisk * or space in order to enter a multiplication.
    • Describe how the function graph changes.
    • Repeat changing the equazione by typing in different values for the parameter (ad es. 0.5, -2, -0.8, 3). Write down your observations.

Discussion

  • Did any problems or difficulties concerning the use of GeoGebra occur?
  • How can a setting like this (GeoGebra in combination with instructions on paper) be integrated into a "traditional" teaching environment?
  • Do you think it is possible to give such an activity as a homework problem to your students?
  • In which way could the dynamic exploration of parameters of a polynomial possibly affect your students’ learning?
  • Do you have ideas for other mathematical topics that could be taught in similar learning environment (paper worksheets in combination with computers)?

Using Sliders to Modify Parameters

Let’s try a more dynamic way of exploring the impact of a parameter on a polynomial f(x) = a x^2 by using a slider to modify the parameter value.

Preparation

  • Open a new GeoGebra window
  • Switch to Perspectives – Algebra & Graphics

Construction Steps

1 a = 1 Create the variable a
2 f(x) = a * x^2 Enter the quadratic polynomial f
Note Suggerimento: Don’t forget to enter an asterisk * or space between a and x^2.

Representing a Number as a Slider

To display number as a slider in the Vista Grafica you need to right click (MacOS: Ctrl-click) the variable in the Vista Algebra and select Show object.

Enhancing the Construction

Let’s create another slider b that controls the constant in the polynomial’s equazione f(x) = a x^2 + b.

3 Tool Slider.gif Create a slider b using the Slider Tool
Note Suggerimento: Activate the tool and click on the Vista Grafica. Use the default settings and click Apply.
4 f(x) = a * x^2 + b Enter the polynomial f
Note Suggerimento: GeoGebra will overwrite the old function f with the new definition.

Tasks

  • Change the parameter value a by moving the point on the slider with the mouse. How does this influence the graph of the polynomial?
  • What happens to the graph when the parameter value is (a) greater than 1, (b) between 0 and 1, or (c) negative? Write down your observations.
  • Change the parameter value b. How does this influence the graph of the polynomial?

Library of Functions

Apart from polynomials there are different types of functions available in GeoGebra (ad es. trigonometric functions, absolute value function, exponential function). Functions are treated as objects and can be used in combination with geometric constructions.

Task 1: Visualizing absolute values

Open a new GeoGebra window. Make sure the Vista Algebra, barra di inserimento and coordinate axes are shown.

1 f(x) = abs(x) Enter the absolute value function f
2 g(x) = 3 Enter the constant function g
3 Tool Intersect Two Objects.gif Intersect both functions
Note Suggerimento: You need to intersect the functions twice in order to get both intersection punti.
Note Suggerimento: You might want to close the Vista Algebra and show the names and values as labels of the objects.


4 absolute.PNG


(a) Move the constant function with the mouse or using the arrow keys. The y-coordinate of each intersection point represents the absolute value of the x-coordinate.

(b) Move the absolute value function up and down either using the mouse or the arrow keys. In which way does the function’s equazione change?

(c) How could this construction be used in order to familiarize students with the concept of absolute value?

Note Suggerimento: The symmetry of the function graph indicates that there are usually two soluzioni for an absolute value problem.


Task 2: Superposition of Sine Waves

Sound waves can be mathematically represented as a combination of sine waves. Every musical tone is composed of several sine waves of the form y(t) = a sin(ω t + φ) . The amplitude a influences the volume of the tone while the angular frequency ω determines the pitch of the tone. The parameter φ is called phase and indicates if the sound wave is shifted in time. If two sine waves interfere, superposition occurs. This means that the sine waves amplify or diminish each other. We can simulate this phenomenon with GeoGebra in order to examine special cases that also occur in nature.

1 f(x) = abs(x) Create three sliders a_1, ω_1, and φ_1
Note Suggerimento: a_1 produces an index. You can select the Greek letters from the menu next to the text field name in the Slider dialog window.
2 g(x)= a_1 sin(ω_1 x + φ_1) Again, you can select the Greek letters from a menu next to the barra di inserimento.

(a) Examine the impact of the parameters on the graph of the sine functions by changing the values of the sliders.

3 Tool Slider.gif Create three sliders a_2, ω_2, and φ_2
Note Suggerimento: Sliders can only be moved when the Slider Tool is activated.
4 h(x)= a_2 sin(ω_2 x + φ_2) Enter another sine function h
5 sum(x) = g(x) + h(x) Create the sum of both functions

(b) Change the color of the three functions so they are easier to identify.

(c) Set a_1 = 1, ω_1 = 1, and φ_1 = 0. For which values of a2, ω2, and φ2 does the sum have maximal amplitude?

Note Suggerimento: In this case the resulting tone has the maximal volume.

(d) For which values of a_2, ω_2, and φ_2 do the two functions cancel each other?

Note Suggerimento: In this case no tone can be heard any more.


4 sine.PNG

en:Tutorial:Basic_Algebraic_Input,_Commands_and_Functions

Estratto da "http://wiki.geogebra.org/s/it/index.php?title=Tutorial:Input_algebrico_Comandi_base&oldid=6403"
© 2024 International GeoGebra Institute