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f(x)=x^2
The standard equation for an upward opening parabola with its vertex at (f,g) is (x - f)2 = 4c(y - g), where c is the focal length, that is the distance of the focus from the vertex. Putting the equation shown in this form we have :- (x - 3)2 = 4 * 1(y - 2) . . . thus c = 1 The vertex is at (3,2) and as the focal length is 1 (and this is positive) then the coordinates of the focus are (3, [2 + 1]) = (3,3).
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Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
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The vertex of this parabola is at -2 -3 When the y-value is -2 the x-value is -5. The coefficient of the squared term in the parabola's equation is -3.
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For a parabola with an axis of symmetry parallel to the x-axis, the equation of a parabola is given by: (y - k)² = 4p(x - h) Where the vertex is at (h, k), and the distance between the focus and the vertex is p (which can be calculated as p = x_focus - x_vertex). For the parabola with vertex (1, -3) and focus (2, -3) this gives: h = 1 k = -3 p = 2 - 1 = 1 → parabola is: (y - -3)² = 4×1(x - 1) → (y + 3)² = 4(x - 1) This can be expanded to: 4x = y² + 6y + 13 or x = (1/4)y² + (3/2)y + (13/4)
f(x)=x^2
The coordinates will be at the point of the turn the parabola which is its vertex.
The focus of a parabola is a fixed point that lies on the axis of the parabola "p" units from the vertex. It can be found by the parabola equations in standard form: (x-h)^2=4p(y-k) or (y-k)^2=4p(x-h) depending on the shape of the parabola. The vertex is defined by (h,k). Solve for p and count that many units from the vertex in the direction away from the directrix. (your focus should be inside the curve of your parabola)
The standard equation for an upward opening parabola with its vertex at (f,g) is (x - f)2 = 4c(y - g), where c is the focal length, that is the distance of the focus from the vertex. Putting the equation shown in this form we have :- (x - 3)2 = 4 * 1(y - 2) . . . thus c = 1 The vertex is at (3,2) and as the focal length is 1 (and this is positive) then the coordinates of the focus are (3, [2 + 1]) = (3,3).
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