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(x^2 + Y^2 – 1)^3 = X^2 Y^3 Desmos

4 min read Jun 17, 2024
(x^2 + Y^2 – 1)^3 = X^2 Y^3 Desmos

Exploring the Heart Curve: (x^2 + y^2 – 1)^3 = x^2 y^3 on Desmos

The equation (x^2 + y^2 – 1)^3 = x^2 y^3 might look intimidating at first glance, but it hides a beautiful and fascinating shape known as the heart curve. Using Desmos, a free online graphing calculator, we can visualize this curve and explore its characteristics.

Graphing the Heart Curve on Desmos

  1. Open Desmos: Go to and open the graphing calculator.
  2. Enter the Equation: In the input field, type the equation: (x^2 + y^2 - 1)^3 = x^2*y^3.
  3. Adjust the View: Zoom in and out using the mouse wheel or the zoom buttons. Pan the graph by dragging or using the arrow keys. You can also adjust the x and y-axis limits to focus on specific parts of the curve.

Exploring the Heart Curve

As you graph the equation, you'll notice a heart-shaped curve. This shape is due to the complex interplay between the terms in the equation. Here are some interesting observations:

  • Symmetry: The curve is symmetric about the y-axis.
  • Singular Point: The curve has a singular point at the origin (0,0). This means that the curve intersects itself at this point.
  • Asymptotes: The curve has asymptotes which means it approaches certain lines but never actually touches them. These asymptotes are located at y = ±1.

Further Exploration with Desmos

Desmos offers a wide range of tools for exploring the heart curve:

  • Sliders: You can introduce sliders for variables within the equation, allowing you to manipulate the curve in real-time. Try adding a slider for the power of 'y' and see how the curve transforms.
  • Coloring: Use different colors to highlight specific parts of the curve or different features like its asymptotes.
  • Annotations: Add text labels to explain different aspects of the curve or mark important points.

Conclusion

The heart curve is a fascinating example of how seemingly complex mathematical equations can lead to beautiful and unexpected shapes. Desmos provides a powerful and intuitive platform to visualize, explore, and gain a deeper understanding of such mathematical concepts.

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