How to graph vector-valued functions in GeoGebra?

Graphing vector-valued functions in GeoGebra is an efficient way to visualize these dynamic mathematical expressions. GeoGebra interprets vector-valued functions as parametric curves, allowing you to plot their paths in both two and three dimensions using the Curve command.

Understanding Vector-Valued Functions and GeoGebra's Approach

A vector-valued function, often denoted as $\mathbf{r}(t)$, maps a scalar input ($t$, typically representing time) to a vector output (a position in space). In 2D, this looks like $\mathbf{r}(t) = \langle f(t), g(t) \rangle$, and in 3D, it's $\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$.

GeoGebra's Curve command is designed precisely for this purpose, treating $f(t), g(t),$ and $h(t)$ as the parametric expressions for the x, y, and z coordinates respectively.

Step-by-Step Guide to Graphing 2D Vector Functions

Graphing 2D vector functions is best done using GeoGebra Classic or the online Graphing Calculator.

Using the GeoGebra Classic App

1. Open GeoGebra: Launch GeoGebra Classic 6 or access the GeoGebra Graphing Calculator online.
2. Locate the Input Bar: This is usually at the bottom or left side of the screen.
3. Enter the Curve Command: Type Curve( followed by the components of your vector function, the parameter variable, and its range.
   • Syntax: Curve(<Expression x(t)>, <Expression y(t)>, <Parameter Variable>, <Start Value>, <End Value>)
   • Example (Circle): To graph $\mathbf{r}(t) = \langle 3\cos(t), 3\sin(t) \rangle$ for $0 \le t \le 2\pi$, type:
     Curve(3cos(t), 3sin(t), t, 0, 2pi)
4. Press Enter: GeoGebra will immediately plot the curve in the Graphics View.

Customizing Your Graph

Once your curve is plotted, you can easily adjust its appearance to enhance clarity and visual appeal:

• Access Settings:
  • In the Algebra View (left panel), locate your curve. Click the three vertical dots next to its entry.
  • Alternatively, right-click on the curve directly in the Graphics View.
  • Choose Settings from the menu.
• Change Color and Style:
  • In the Settings window, navigate to the Color tab to select a new color for your curve.
  • Go to the Style tab to modify the Line Thickness or Line Style. Making the line thicker can help it stand out.
• Zoom and Pan: Use your mouse wheel to zoom in or out on the spiral shape or any part of your graph. You can also click and drag the Graphics View to pan the canvas and explore different sections of the curve.
• Labeling: Add a Name or Caption under the "Basic" tab in Settings to label your curve.

Graphing 3D Vector Functions in GeoGebra 3D Calculator

For 3D vector functions, you'll need the GeoGebra 3D Calculator.

Accessing the 3D Environment

1. Open GeoGebra 3D Calculator: Go to GeoGebra 3D Calculator or launch the desktop app.
2. Locate the Input Bar: Similar to the 2D version.

Inputting 3D Parametric Curves

The Curve command syntax extends naturally to three dimensions:

• Syntax: Curve(<Expression x(t)>, <Expression y(t)>, <Expression z(t)>, <Parameter Variable>, <Start Value>, <End Value>)
• Example (Helix/Spiral): To graph $\mathbf{r}(t) = \langle t\cos(t), t\sin(t), t \rangle$ for $0 \le t \le 4\pi$, type:
  Curve(t*cos(t), t*sin(t), t, 0, 4pi)
• Press Enter: The helix will appear in the 3D Graphics View.

Navigating and Customizing 3D Views

The 3D environment offers additional interaction:

• Rotate View: Click and drag anywhere in the 3D Graphics View to rotate the perspective and observe the curve from different angles.
• Zoom: Use the mouse wheel to zoom in or out.
• Settings: Access Settings (via the three dots or right-click) for the curve to change its Color, Line Thickness, and Line Style, just like in 2D.
• View Options: In the 3D Graphics View toolbar, you can toggle the visibility of axes, planes, and the bounding box.

Practical Tips for Effective Visualization

• Choosing Parameter Range: The Start Value and End Value of your parameter t are crucial. Experiment with different ranges to capture the desired portion of your curve or to see its full extent.
• Adding a Point on the Curve:
  • Create a point P = Point(curveName) (e.g., P = Point(c1) if your curve is named c1).
  • You can then create a slider for a value a, and define P = Curve(a). This allows you to animate a point moving along the curve.
• Visualizing Position Vectors: To show the position vector from the origin to a point on the curve at a specific t, define P = (f(t_value), g(t_value)) and then Vector(P). You can also create a slider for t_value to see the vector change dynamically.
• Dynamic Sliders for Parameters: For more interactive exploration, create sliders for variables within your function (e.g., a in Curve(a*cos(t), a*sin(t), t, 0, 2pi)) or for the parameter range itself (e.g., Curve(cos(t), sin(t), t, 0, maxT) where maxT is a slider).

Common Vector Function Examples and Their GeoGebra Syntax

Function Type	Description	GeoGebra Command (2D)	GeoGebra Command (3D)
Circle/Ellipse	Constant radius, varying angle	Curve(r*cos(t), r*sin(t), t, 0, 2pi)	Curve(a*cos(t), b*sin(t), 0, t, 0, 2pi) (on XY plane)
Line Segment	From point (x1,y1) to (x2,y2)	Curve(x1+(x2-x1)t, y1+(y2-y1)t, t, 0, 1)	Curve(x1+(x2-x1)t, y1+(y2-y1)t, z1+(z2-z1)t, t, 0, 1)
Spiral (Archimedean)	Radius grows linearly with angle	Curve(a*t*cos(t), a*t*sin(t), t, 0, 4pi)	Not directly applicable as a basic 3D spiral
Helix	Circular motion with linear z-component	(Not applicable in 2D)	Curve(a*cos(t), a*sin(t), b*t, t, 0, 4pi)
Parabola (parametric)	y=x^2 represented parametrically	Curve(t, t^2, t, -3, 3)	Curve(t, t^2, 0, t, -3, 3) (on XY plane)

By utilizing the Curve command and customizing its appearance and interactive elements, you can effectively visualize and analyze vector-valued functions in GeoGebra.