# Tutorial:Basic Algebraic Input, Commands and Functions

## Tips and Tricks

1. **Name a new object** by typing *name =* into the input bar in front of its algebraic representation.

**Example:**P = (3, 2) creates point P.

2. **Multiplication** needs to be entered using an asterisk or space between the factors.

**Example:**a*x or a x

3. **GeoGebra is case sensitive!** Thus, upper and lower case letters must not be mixed up.

- Points are always named with upper case letters
**Example:**A = (1, 2)

- Vectors are named with lower case letters
**Example:**v = (1, 3)

- Segments, lines, circles, functions… are always named with lower case letters.
**Example:**circle c: (x – 2)^2 + (y – 1)^2 = 16

- The variable
*x*within a function and the variables*x*and*y*in the equation of a conic section always need to be lower case.**Example:**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.

- y = m x + b creates a line whose parameters are already existing values m and b (e.g. numbers / sliders).
- Line[A, B] creates a line through existing points A and B.

5. **Confirm an expression** you entered into the input bar by pressing the Enter key.

6. **Open the help window** for using the input bar and commands by selecting *Help* from the Help Menu (or shortcut F1).

7. **Error messages**: Always read the messages – they could possibly help to fix the problem!

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 F1 to open the 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.

- If GeoGebra suggests the desired command, hit the Enter key in order to place the cursor within the brackets.
- If the suggested command is not the one you wanted to enter, just keep typing until the suggestion matches.

## Constructing Tangents to a Circle (Part 1)

Open the dynamic worksheet Tangents to a Circle. Follow the directions on the worksheet in order to find out how to construct tangents to a circle.

### Discussion

- Which tools did you use in order to recreate the construction?
- Were there any new tools involved in the suggested construction steps? If yes, how did you find out how to operate the new tool?
- Did you notice anything about the toolbar displayed in the right applet?
- Do you think your students could work with such a dynamic worksheet and find out about construction steps on their own?

## 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

- Open a new GeoGebra window.
- Show the Algebra View and Input Bar, as well as coordinate axes (View Menu)

### Construction Steps

1 | A = (0, 0) | Point A |

2 | (3, 0) | Point B Hint: 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 Hint: 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!

**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 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.

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 points E and F of the two circles |

9 | Line[C, E] | Tangent through points C and E |

10 | Line[C, F] | Tangent through points 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 (e.g. 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?
**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)?

## 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.

- Open a
**new GeoGebra window** **Type**in**f(x) = x^2**and hit the Enter key. Which**shape**does the function graph have? Write down your answer on paper.- In Move mode, highlight the polynomial in the algebra view 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 equation 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**.- 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.

- 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.- 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.

### 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 Hint: 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 Graphics View you need to right click (MacOS: Ctrl-click) the variable in the Algebra View and select *Show object*.

### Enhancing the Construction

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

3 | 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. | |

4 | f(x) = a * x^2 + b | Enter the polynomial f Hint: 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 (e.g. 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 Algebra View, Input Bar 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 | Intersect both functions Hint: You need to intersect the functions twice in order to get both intersection points. |

**Hint:**You might want to close the Algebra View and show the names and values as labels of the objects.

(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?

(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.

### 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 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) | Again, you can select the Greek letters from a menu next to the Input Bar. |

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

3 | Create three sliders a_2, ω_2, and φ_2 Hint: 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 a_2, ω_2, and φ_2 does the sum have maximal amplitude?

**Hint:**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?

**Hint:**In this case no tone can be heard any more.