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In the theory of quadratic forms, the parabola is the graph of the quadratic form x 2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x 2 + y 2 (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form x 2 − y 2. Generalizations to more variables yield ...
The solutions of the quadratic equation + + = may be deduced from the graph of the quadratic function = + +, which is a parabola. If the parabola intersects the x -axis in two points, there are two real roots , which are the x -coordinates of these two points (also called x -intercept).
The discriminant B 2 – 4AC of the conic section's quadratic equation (or equivalently the determinant AC – B 2 /4 of the 2 × 2 matrix) and the quantity A + C (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, [14] [15] [16] as is the determinant of the 3 × 3 matrix above.
A quadratic equation is one which includes a term with an exponent of 2, for example, , [40] and no term with higher exponent. The name derives from the Latin quadrus , meaning square. [ 41 ] In general, a quadratic equation can be expressed in the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} , [ 42 ] where a is not zero (if it were ...
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form y 2 − a 2 x 3 = 0 {\displaystyle y^{2}-a^{2}x^{3}=0} (with a ≠ 0 ) in some Cartesian coordinate system .
Parabolas have only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below). A circle is an ellipse with both foci coinciding at the center.
In analytic geometry, the graph of any quadratic function is a parabola in the xy-plane. Given a quadratic polynomial of the form a ( x − h ) 2 + k {\displaystyle a(x-h)^{2}+k} the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point ) of the parabola.
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: