Visualizing quadratic equations and parabolic curves.
- Author
- Sukesh Ashok Kumar
This example demonstrates the classic U-shaped curve of a parabola, defined by a quadratic equation. Parabolas appear everywhere in physics (projectile motion), engineering (suspension bridges), and mathematics.
Mathematical Concepts:
- Quadratic equation: y = ax² + bx + c (standard form)
- Coefficient 'a': determines width and direction (opens up if a>0, down if a<0)
- Coefficient 'b': controls horizontal shift/tilt
- Coefficient 'c': vertical shift (y-intercept)
- Vertex: the turning point of the parabola (minimum or maximum)
- Axis of symmetry: vertical line through vertex
- Polynomial growth: quadratic (order 2) grows faster than linear
In This Example:
- a = 0.02 (small positive value creates wide, upward-opening parabola)
- b = 0 (no tilt, perfectly symmetric)
- c = 400 (vertex near bottom of screen, opens upward)
- Centered at x = 320 (middle of window)
Parabola Properties:
- Vertex form: y = a(x - h)² + k where (h,k) is the vertex
- Our vertex: (320, 400) in screen coordinates
- Axis of symmetry: x = 320 (vertical line through center)
- Opens upward because a > 0
Programming Concepts:
- Using floating-point (double) for coefficient precision
- Coordinate transformation (centering)
- Boundary checking to prevent drawing outside window
- Mixed integer and floating-point arithmetic
- Quadratic growth: output changes faster as x increases
What you'll learn:
- How quadratic equations create curved shapes
- The effect of coefficients a, b, c on parabola shape
- Why parabolas are symmetric
- Converting continuous functions to discrete pixels
- Boundary checking for valid screen coordinates
- The difference between linear and quadratic growth
The Quadratic Formula: For y = ax² + bx + c:
- When x = 0: y = c (y-intercept)
- When x increases, x² term dominates (exponential growth)
- Symmetric: f(h + d) = f(h - d) when b = 0
Experiment Ideas:
- Change a to 0.01: wider parabola
- Change a to -0.02: opens downward
- Change c to 100: shifts parabola up
- Change b to 1: tilts the parabola
Compile:
gcc m-parabola.c gfx/simplegfx.c -o output/m-parabola -lX11
Run:
1.9.1