Find the Properties x=4y^2

Problem

x=4*y2

Solution

1. Identify the type of conic section. Since the equation is in the form x=a*y2 it represents a parabola that opens horizontally.

2. Rewrite the equation in the standard form (y−k)2=4*p*(x−h) Dividing both sides by 4 gives y2=1/4*x

3. Determine the vertex (h,k) By comparing y2=1/4*x to (y−0)2=4*p*(x−0) the vertex is (0,0)

4. Calculate the value of p by setting 4*p=1/4 Solving for p gives p=1/16

5. Find the focus. For a horizontal parabola opening to the right (p>0, the focus is (h+p,k) which is (1/16,0)

6. Determine the directrix. The directrix is a vertical line x=h−p which is x=−1/16

7. Identify the axis of symmetry. The axis of symmetry is the horizontal line passing through the vertex, which is y=0

Final Answer

Vertex: *(0,0), Focus: *(1/16,0), Directrix: *x=−1/16, Axis of Symmetry: *y=0

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