Introduction to Creative Graphing on TI-84
Do you wish to explore mathematical art through the fun of graphing functions on your TI-84 Plus calculator? This article delves into various techniques to create visually appealing shapes, patterns, and even complex designs. From basic trigonometric functions to intricate fractals, you'll discover how to harness the power of your calculator to produce amazing results. Whether you're a math enthusiast or seeking a creative outlet, this exploration will inspire you to create a variety of stunning visual designs.
Parametric and Polar Equations for Complex Shapes
Parametric and polar equations provide a fascinating way to generate complex shapes and patterns. These equations can be explored to create visually engaging designs, enhancing your understanding of mathematical concepts in a creative manner.
1. Parametric Equations
Parametric equations can be used to create intricate and complex shapes. Let's dive into a Lissajous Curve as an example. This is a parametric curve where both x and y define the curve as a function of a third variable, t.
Example: Lissajous Curve
Set your calculator to Parametric Mode.
Enter the following equations into the and fields:
X1:10 sin(2T π/2) Y1:
10 sin(3T)
Adjust the parameters and values to create different shapes. For instance, you can try:
X1:10 sin(4T 3π/8) Y1:
10 sin(3T)
2. Polar Equations
Polar equations switch to a coordinate system where points are defined by their distance from the origin and their angle. This mode can yield beautiful patterns.
Example: Rose Curve
Set your calculator to Polar Mode.
Enter the following equation into the Rose Curve field:
R:2 sin(5θ)
Experiment with the value of n to see different patterns such as a 5-petal or 10-petal rose. You can also explore other patterns by adjusting the sine or cosine functions.
Simple Trigonometric Functions for Waves and Basic Shapes
Trigonometric functions such as sine and cosine can create beautiful wave patterns, while simple equations can define circles and ellipses. Let's explore these functions in more detail.
3. Trigonometric Functions
The following functions can create waves that are visually appealing:
Sine Wave
Enter Y1: sin(X)
Cosine Wave
Enter Y1: cos(X)
4. Basic Shapes with Equations
Use the standard window settings or solve the equations for Y to graph basic shapes like circles and ellipses.
Circle
Use the equation: (X-h)^2 (Y-k)^2 r^2
For example, to graph a circle centered at (2,3) with a radius of 4, enter:
Y1:√(4^2 - (X-2)^2) 3 Y2:
-√(4^2 - (X-2)^2) 3
Ellipse
Use the equation: (X/a)^2 (Y/b)^2 1
For example, to graph an ellipse centered at (0,0) with semi-major axis 4 and semi-minor axis 2, enter:
Y1:2 * √(1 - (X/4)^2) Y2:
-2 * √(1 - (X/4)^2)
Intricate Designs with Fractals
The world of fractals can provide infinite possibilities for intricate designs. Learn how to create and approximate the Mandelbrot Set, which is a set of complex numbers that form a boundary structure with a complex and infinitely intricate contour.
5. Fractals: Mandelbrot Set
The Mandelbrot set is a mathematical set of points whose boundary is a fractal. This complex set can be approximated through iterations.
Additional Tips for Graphing
Mastering the techniques to graph these functions effectively requires some practice. Here are a few tips to help you:
Setting Window Settings Right
Ensure that your window settings are appropriate to see the full picture of your graphs accurately.
Experimenting with Parameters
Experiment with different values and parameters to see how they affect the shape and complexity of the graphs. This process will enhance your understanding and creativity.
Exploring Different Modes
Be familiar with the Parametric Mode and Polar Mode to create more advanced and visually rich designs.
Real-World Inspiration: The LOVE Sculpture
The inspiration for creating graphing designs is not limited to abstract forms or complex equations. Even real-world sculptures can inspire your graphing projects. The LOVE sculpture in Indiana is a powerful example of how mathematical concepts can be represented in stunning physical forms.
Creating Graphs for the LOVE Sculpture
Here's a simple version of the LOVE sculpture using only two functions, right-side up:
Example Functions
First function: Y X^2 - 2X 1
Second function: Y -X^2 4X - 3
These functions can be plotted to form the iconic shape of the letter 'L' in the sculpture. Additionally, if you wanted to approximate the matching 'O' and 'V' using four functions each, you could use:
Example Functions for 'O'
Y1: (X-4)^2 (Y-2)^2 1
Y2: (X-4)^2 (Y 2)^2 1
Y1: (X 4)^2 (Y 2)^2 1
Y2: (X 4)^2 (Y-2)^2 1
Example Functions for 'V'
Y1: (X-4)^2 (Y-2)^2 0.5
Y2: (X-4)^2 (Y 2)^2 0.5
Y1: (X 4)^2 (Y-2)^2 0.5
Y2: (X 4)^2 (Y 2)^2 0.5
By combining these functions, you can create a close representation of the 'O' and 'V' in the sculpture.
If you have a TI-84 PLUS CE Graphing Calculator, try these examples and see how you can use mathematical functions to create your own unique designs or artworks inspired by real-world structures and sculptures.
Conclusion
The TI-84 Plus calculator is a powerful tool for exploring mathematical art. By leveraging parametric and polar equations, trigonometric functions, and basic shapes, you can create a wide range of visually appealing designs. Whether you're creating abstract art or inspired by real-world structures, your TI-84 calculator can help you bring your creative visions to life.