Find the standard and general equation of a parabola with a vertex at (0,0) and graph passing through (3,5) Axis of Symmetry at x-axis.
Since you know the vertex, your equation must be
x = a(y-0)^2 + 0 , if the axis of symmetry is horizontal
x = ay^2
but (3,5) lies on it, so
3 = a(25)
a = 3/25
x = (3/25)y^2
or
25x = 3y^2
To find the standard and general equations of a parabola, we need to use the information given about the vertex and the point on the graph.
The given vertex is (0,0), which means the parabola opens either upward or downward. And the axis of symmetry is at the x-axis, which means the parabola is symmetrical with respect to the x-axis.
Let's start by finding the standard equation of the parabola. The standard form of a parabola equation with its vertex (h, k) is:
y = a(x - h)^2 + k
Since the vertex is at (0,0), we can rewrite the equation as:
y = a(x - 0)^2 + 0
y = ax^2
Now we need to find the value of 'a'. To do so, we can substitute the coordinates of the given point (3,5) into the equation:
5 = a(3)^2
5 = 9a
a = 5/9
The standard equation of the parabola is:
y = (5/9)x^2
To find the general equation of the parabola, we can expand the equation and eliminate the fraction. Multiplying the equation by 9 will help eliminate the fraction:
9y = 5x^2
9y - 5x^2 = 0
So, the general equation of the parabola is:
9y - 5x^2 = 0