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For a parametric equation of a parabola in general position see § As the affine image of the unit parabola. The implicit equation of a parabola is defined by an irreducible polynomial of degree two: + + + + + =, such that =, or, equivalently, such that + + is the square of a linear polynomial.
Regardless of the format, the graph of a univariate quadratic function () = + + is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation = + +. If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards.
Define b by the equations c 2 = a 2 − b 2 for an ellipse and c 2 = a 2 + b 2 for a hyperbola. For a circle, c = 0 so a 2 = b 2, with radius r = a = b. For the parabola, the standard form has the focus on the x-axis at the point (a, 0) and the directrix the line with equation x = −a. In standard form the parabola will always pass through the ...
In this position, the hyperbolic paraboloid opens downward along the x-axis and upward along the y-axis (that is, the parabola in the plane x = 0 opens upward and the parabola in the plane y = 0 opens downward). Any paraboloid (elliptic or hyperbolic) is a translation surface, as it can be generated by a moving parabola directed by a second ...
If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola. If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be. [17]
For example, if trying to solve the equation ... In terms of coordinate geometry, an axis-aligned parabola is a curve whose ...
This is the equation of a parabola, so the path is parabolic. The axis of the parabola is vertical. If the projectile's position (x,y) and launch angle (θ or α) are known, the initial velocity can be found solving for v 0 in the afore-mentioned parabolic equation:
While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, cosh(x), a sum of two exponential functions. One parabola is f(x) = x 2 + 3x − 1, and hyperbolic cosine is cosh(x) = e x + e −x / 2 . The curves are unrelated.