Cool Equations to Put in A Graphing Calculator

Graphing calculators are powerful tools for visualizing mathematical relationships. Whether you're a student exploring functions or an enthusiast creating artistic patterns, these equations will help you discover the beauty of mathematics through graphs.

Basic Equations to Try

Start with simple equations to get comfortable with your graphing calculator. These basic functions demonstrate key concepts like linear relationships, quadratics, and trigonometric waves.

Linear Functions

Linear functions have the form y = mx + b, where m is the slope and b is the y-intercept. Try these examples:

y = 2x + 3

y = -0.5x + 1

y = 0 (the x-axis)

Quadratic Functions

Quadratic functions have the form y = ax² + bx + c. They create parabolas that can open upwards or downwards.

y = x²

y = -x² + 4

y = 0.5x² - 2x + 1

Trigonometric Functions

Trigonometric functions like sine and cosine create wave patterns. Try these:

y = sin(x)

y = cos(x)

y = tan(x)

Parametric Equations

Parametric equations define both x and y in terms of a third variable, often t. This allows you to create complex curves and shapes.

Lissajous Curves

Lissajous curves are created by combining sine and cosine functions with different frequencies.

x = sin(2t)

y = sin(3t)

Heart Curve

This parametric equation creates a heart shape:

x = 16sin³(t)

y = 13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t)

Polar Equations

Polar equations express relationships using r (radius) and θ (angle). These can create spirals, roses, and other interesting patterns.

Spiral

A simple spiral can be created with:

r = θ

Rose Curves

Rose curves have the form r = a sin(kθ) or r = a cos(kθ). Try these:

r = 5sin(3θ)

r = 4cos(5θ)

3D Graphs

Many graphing calculators can display three-dimensional graphs. These equations create surfaces and curves in 3D space.

Sphere

The equation for a sphere with radius 2 centered at the origin:

x² + y² + z² = 4

Torus

A torus (donut shape) can be created with:

(x² + y² + z² + R² - r²)² = 4R²(x² + y²)

Where R is the distance from center to tube, and r is tube radius

Fractals

Fractals are complex patterns that display self-similarity at different scales. These equations can create stunning visualizations.

Mandelbrot Set

The Mandelbrot set is one of the most famous fractals. It's defined by the equation:

zₙ₊₁ = zₙ² + c

Where c is a complex number

Julia Set

Similar to the Mandelbrot set but with a different approach:

zₙ₊₁ = zₙ² + c

Where c is a constant complex number

FAQ

What's the easiest equation to start with?: The simplest equation to try is y = x, which creates a straight line at a 45-degree angle. This helps you understand the basic graphing interface.
Can I graph equations with multiple variables?: Most graphing calculators can handle equations with two variables (like y = mx + b) or three variables (like x² + y² + z² = 4 for a sphere). Some advanced models can handle more complex systems.
How do I adjust the viewing window?: Look for settings like "Window" or "View" in your calculator's menu. You can adjust the x and y ranges to zoom in or out on your graph.
What's the difference between parametric and polar graphs?: Parametric graphs use separate equations for x and y in terms of a third variable (usually t). Polar graphs use a single equation where r depends on θ (theta). Each creates different types of patterns and curves.
Can I save my favorite equations?: Most graphing calculators allow you to store equations in memory. Look for options like "Store" or "Save" in the equation editor.