Hi,

     a.  y= 1/2x^2    Or   2y = x^2  Opening Up: coefficient of x^2 >0

The standard form is , where  the focus is (h,k + p)  

V(0,0), F(0,.5) directrix is y = -.5 and axis of symmetry is x= 0

     b. y^2 = 16x   below...Opening Right: coefficient of y ^2 >0,V(0,0) p = 4

The standard form is , where  the focus is (h +p,k )



See below descriptions of various conics                         

Standard Form of an Equation of a Circle is  

where Pt(h,k) is the center and r is the radius





 Standard Form of an Equation of an Ellipse is  where Pt(h,k) is the center. (a positioned to correspond with major axis)

 a and b  are the respective vertices distances from center and ±are the foci distances from center: a > b



Standard Form of an Equation of an Hyperbola opening right and  left is:

   where Pt(h,k) is a center  with vertices 'a' units right and left of center.



Standard Form of an Equation of an Hyperbola opening up and down is:

   where Pt(h,k) is a center  with vertices 'b' units up and down from center.



the vertex form of a parabola opening up or down,  where(h,k) is the vertex.

The standard form is , where  the focus is (h,k + p)



the vertex form of a parabola opening right or left,  where(h,k) is the vertex.

The standard form is , where  the focus is (h +p,k )



 

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