Parabola Calculator

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Standard form uses coefficients a, b, c. Vertex form uses the vertex coordinates h and k. Horizontal form opens left or right.
The leading coefficient. Must be non-zero.
The linear coefficient in y = ax² + bx + c.
The constant term in y = ax² + bx + c.
Vertex
(1, 2)

The turning point of the parabola

Focus(1, 2.25)
Directrixy = 1.75
Axis of symmetryx = 1
Focal length (p)0.25
Latus rectum length1
Latus rectum endpoints(1.5, 2.25) and (0.5, 2.25)
x-intercept(s)None (real)
y-intercept(0, 3)
OpensUp
Standard formy = 1x² - 2x + 3
Vertex formy = 1(x - 1)² + 2
Domain & rangeDomain: all real x; Range: y ≥ 2
Focal length p0.25
Latus rectum length1

Parabola opens up, vertex at (1, 2)

  • The vertex (1, 2) is the turning point where the parabola changes direction.
  • The focus (1, 2.25) and directrix y = 1.75 are each 0.25 units from the vertex.
  • The latus rectum is 1 units long - this chord through the focus controls how wide the parabola is at that height.
  • The parabola does not cross the x-axis (the discriminant is negative).

Next stepCheck the domain and range: Domain: all real x; Range: y ≥ 2. For applications, the focus is the point that matters in optics, acoustics and satellite dish design.

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