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The focus of a parabola is a specific point that defines its shape, while the directrix is a line used in the definition of a parabola. If the directrix is given as ( y = -2 ), the parabola opens either upwards or downwards. The focus would be located at a point above or below this directrix, depending on the orientation of the parabola. Specifically, for a parabola that opens upwards, the focus would be positioned at ( (h, k + p) ), where ( p ) is the distance from the vertex to the focus, and the vertex would be located at ( (h, -2 + p) ).
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What is the equation of the parabola with focuse (07) and the directrix y1?
To find the equation of the parabola with focus at (0, 7) and directrix ( y = 1 ), we first determine the vertex, which is the midpoint between the focus and the directrix. The vertex is at ( (0, 4) ). The distance from the vertex to the focus is 3, so the parabola opens upward. The equation of the parabola can be expressed as ( (x - h)^2 = 4p(y - k) ), where ( (h, k) ) is the vertex and ( p ) is the distance from the vertex to the focus. Thus, the equation is ( x^2 = 12(y - 4) ).
How do you find the equation of the parabola when you have the vertex focus and directrix?
To find the equation of a parabola given the vertex, focus, and directrix, start by identifying the vertex coordinates ((h, k)), the focus ((h, k + p)) for a vertical parabola (or ((h + p, k)) for a horizontal one), and the distance (p) from the vertex to the focus. The directrix will be a line located at (y = k - p) for vertical parabolas or (x = h - p) for horizontal ones. The standard form of the equation is ((x - h)^2 = 4p(y - k)) for vertical parabolas and ((y - k)^2 = 4p(x - h)) for horizontal parabolas. Substitute (p) with the distance calculated from the vertex to the focus or directrix to finalize the equation.
What is the equation of the directrix of the parabola?
The answer depends on the form in which the equation of the parabola is given. For y^2 = 4ax the directrix is x = -2a.
How do you get directrix of a parabola?
The directrix of a parabola can be found using its standard form equation. For a parabola that opens upwards or downwards, given by (y = ax^2 + bx + c), the directrix is located at (y = k - \frac{1}{4p}), where (k) is the vertex's y-coordinate and (p) is the distance from the vertex to the focus. For a parabola that opens sideways, the directrix is given by (x = h - \frac{1}{4p}), where (h) is the vertex's x-coordinate. The value of (p) can be determined based on the coefficients of the quadratic equation.
What is the equation of a parabola with vertex (0 0) and directrix x -3.?
The equation of a parabola with vertex at (0, 0) and a directrix of ( x = -3 ) opens to the right, as the directrix is a vertical line. The distance from the vertex to the directrix is 3 units. The standard form of the equation for a horizontally-opening parabola is given by ( y^2 = 4px ), where ( p ) is the distance from the vertex to the directrix. Therefore, with ( p = 3 ), the equation is ( y^2 = 12x ).
What is the equation of the quadratic graph with a focus of (3 6) and a directrix of y 4?
For a parabola with a y=... directrix, it is of the form: (x - h)^2 = 4p(y - k) with vertex (h, k), focus (h, k + p) and directrix y = k - p With a focus of (3, 6) and a directrix of y = 4, this means: (h, k + p) = (3, 6) → k + p = 6 y = k - p = 4 → k = 5, p = 1 (solving the simultaneous equations) → vertex is (3, 5) → parabola is (x - 3)^2 = 4(y - 5) which can be rearranged into y = 1/4 x^2 - 3/2 x + 29/4
Using the given equations of parabolas find the focus the directrix and the equation of the axis of symmetry of x2 -8y?
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What is the equation of the parabola with focuse (07) and the directrix y1?
To find the equation of the parabola with focus at (0, 7) and directrix ( y = 1 ), we first determine the vertex, which is the midpoint between the focus and the directrix. The vertex is at ( (0, 4) ). The distance from the vertex to the focus is 3, so the parabola opens upward. The equation of the parabola can be expressed as ( (x - h)^2 = 4p(y - k) ), where ( (h, k) ) is the vertex and ( p ) is the distance from the vertex to the focus. Thus, the equation is ( x^2 = 12(y - 4) ).
What is the relationship between the focus and directrix?
i assume this is locus you are talking about, in which case: they are both the same distance from the vertex - focal length, focus is a point: P(x,y) and directrix is a horizontal line e.g. y=-1
How do you find the equation of the parabola when you have the vertex focus and directrix?
To find the equation of a parabola given the vertex, focus, and directrix, start by identifying the vertex coordinates ((h, k)), the focus ((h, k + p)) for a vertical parabola (or ((h + p, k)) for a horizontal one), and the distance (p) from the vertex to the focus. The directrix will be a line located at (y = k - p) for vertical parabolas or (x = h - p) for horizontal ones. The standard form of the equation is ((x - h)^2 = 4p(y - k)) for vertical parabolas and ((y - k)^2 = 4p(x - h)) for horizontal parabolas. Substitute (p) with the distance calculated from the vertex to the focus or directrix to finalize the equation.
What is the equation of the directrix of the parabola?
The answer depends on the form in which the equation of the parabola is given. For y^2 = 4ax the directrix is x = -2a.
How do you get directrix of a parabola?
The directrix of a parabola can be found using its standard form equation. For a parabola that opens upwards or downwards, given by (y = ax^2 + bx + c), the directrix is located at (y = k - \frac{1}{4p}), where (k) is the vertex's y-coordinate and (p) is the distance from the vertex to the focus. For a parabola that opens sideways, the directrix is given by (x = h - \frac{1}{4p}), where (h) is the vertex's x-coordinate. The value of (p) can be determined based on the coefficients of the quadratic equation.
What is the equation of a parabola with vertex (0 0) and directrix x -3.?
The equation of a parabola with vertex at (0, 0) and a directrix of ( x = -3 ) opens to the right, as the directrix is a vertical line. The distance from the vertex to the directrix is 3 units. The standard form of the equation for a horizontally-opening parabola is given by ( y^2 = 4px ), where ( p ) is the distance from the vertex to the directrix. Therefore, with ( p = 3 ), the equation is ( y^2 = 12x ).
What is the focus and directrix of y equals 8x squared?
Need set of point parabola passes through.X = 1Y = 8----------------(1, 8)For focus,X2 = 4pYinsert values, solve for p(1)2 = 4p(8)= 1/32=======focus- 1/32=======directrix( long time since I have done this. Formula is correct, check my math )
What is the standard form of the equation of the parabola with vertex 00 and directrix y4?
Assuming the vertex is 0,0 and the directrix is y=4 x^2=0
What is the axis of symmetry for the parabola with vertex (-2 -4) and directrix y 1?
The axis of symmetry is x = -2.
How does one create a parabola?
A parabola can be created by plotting points that satisfy the quadratic equation of the form (y = ax^2 + bx + c), where (a), (b), and (c) are constants. Alternatively, it can be formed geometrically by taking a fixed point (the focus) and a fixed line (the directrix); the set of all points equidistant from both the focus and the directrix defines the parabola. To visualize it, you can also use graphing software or tools to draw it based on these principles.
