Parabolas

Standard Form

\(f(x) = ax^2 + bx + c\)

The coefficient \(a\) determines both width and direction:

  • \(\mathbf{a > 0}\): Opens upward
  • \(\mathbf{a < 0}\): Opens downward
  • \(\mathbf{|a| > 1}\): Narrower
  • \(\mathbf{|a| < 1}\): Wider

The constant term \(c\) controls vertical position:

  • \(\mathbf{c > 0}\): shift up
  • \(\mathbf{c < 0}\): shift down

Vertex Form

\(f(x) = a(x - h)^2 + k\)

  • \(h\) controls horizontal shift
    • \(h > 0\): shift right
    • \(h < 0\): shift left
  • \(k\) is the vertical position of the vertex
    • \(k > 0\): shift up
    • \(k < 0\): shift down

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🧭 Explore more

Parabolas appear in many real-world applications. Check out how they are used in: