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The graph is a parabola facing (opening) upwards with the vertex at the origin.
If it passes through the origin
Since the vertex is at the origin and the parabola opens downward, the equation of the parabola is x2 = 4py, where p < 0, and the axis of symmetry is the y-axis. So the focus is at y-axis at (0, p) and the directrix equation is y = -p. Now, what do you mean with 1 and 76 units? 1.76 units? If the distance of the vertex and the focus is 1.76 units, then p = -1.76, thus 4p = -7.04, then the equation of the parabola is x2 = -7.04y.
A parabola with an equation, y2 = 4ax has its vertex at the origin and opens to the right. It's not just the '4' that is important, it's '4a' that matters. This type of parabola has a directrix at x = -a, and a focus at (a, 0). By writing the equation as it is, the position of the directrix and focus are readily identifiable. For example, y2 = 2.4x doesn't say a great deal. Re-writing the equation of the parabola as y2 = 4*(0.6)x tells us immediately that the directrix is at x = -0.6 and the focus is at (0.6, 0)
x2/a2 + y2/b2 = 1, is the equation of an ellipse with semi-major axes a and b (that's the equivalent of the radius, along the two different axes), centered in the origin.
The standard equation for a Parabola with is vertex at the origin (0,0) is, x2 = 4cy if the parabola opens vertically upwards/downwards, or y2 = 4cx when the parabola opens sideways. As the focus is at (0,6) then the focus is vertically above the vertex and we have an upward opening parabola. Note that c is the distance from the vertex to the focus and in this case has a value of 6 (a positive number). The equation is thus, x2 = 4*6y = 24y
The graph is a parabola facing (opening) upwards with the vertex at the origin.
focus , directrix
If it passes through the origin
Since the vertex is at the origin and the parabola opens downward, the equation of the parabola is x2 = 4py, where p < 0, and the axis of symmetry is the y-axis. So the focus is at y-axis at (0, p) and the directrix equation is y = -p. Now, what do you mean with 1 and 76 units? 1.76 units? If the distance of the vertex and the focus is 1.76 units, then p = -1.76, thus 4p = -7.04, then the equation of the parabola is x2 = -7.04y.
A parabola with an equation, y2 = 4ax has its vertex at the origin and opens to the right. It's not just the '4' that is important, it's '4a' that matters. This type of parabola has a directrix at x = -a, and a focus at (a, 0). By writing the equation as it is, the position of the directrix and focus are readily identifiable. For example, y2 = 2.4x doesn't say a great deal. Re-writing the equation of the parabola as y2 = 4*(0.6)x tells us immediately that the directrix is at x = -0.6 and the focus is at (0.6, 0)
x^2 + y^2 = 100
The range of a function is the set of Y values where the equation is true. Example, a line passing through the origin with a slope of 1 that continues towards infinity in both the positive and negative direction will have a range of all real numbers, whereas a parabola opening up with it's vertex on the origin will have a range of All Real Numbers such that Y is greater than or equal to zero.
When the vertex lies on the x-axis. For example x = y2, the vertex is at the origin, and the parabola is lying on its side.
The range of a function is the set of Y values where the equation is true. Example, a line passing through the origin with a slope of 1 that continues towards infinity in both the positive and negative direction will have a range of all real numbers, whereas a parabola opening up with it's vertex on the origin will have a range of All Real Numbers such that Y is greater than or equal to zero.
x2/a2 + y2/b2 = 1, is the equation of an ellipse with semi-major axes a and b (that's the equivalent of the radius, along the two different axes), centered in the origin.
It is (x - a)^2 + y^2 = a^2 or, equivalently, x^2 + y^2 = 2ax.