How do you identity if the equation #4y^2-x^2+4=0# is a parabola, circle, ellipse, or hyperbola and how do you graph it?
Please seen the explanation.
When given an equation in the form that you have been given, you can use the general Cartesian form for a conic section:
Please observe that slight rearrangement of the given equation,
Fits equation  with:
The section of the reference, entitled Discriminant, tells you how to determine what it is:
The fact that the discriminant is greater than 0 tells us that the equation describes a hyperbola.
Subtract for 4 from both sides of equation :
Divide both sides by -4:
This fits the standard form
I will fill in the equation to help you see it:
Please observe that equation  matches the variables of equation :
The form for equation  is important for graphing because of the following reasons:
- Everything is centered about the point
#(h,k) = (0,0)#
- The vertices of the hyperbola are located at
#(h-a,k) = (-2,0) and (h+a,k) = (2,0)#
- The equations of the asymptotes are
#y = -b/a(x-h)+k and y = -b/a(x-h)+k#which are the two lines # y = -1/2x and y = 1/2x#
This should help you to graph it.
Here is a graph of the hyperbola (red) with the vertices and the center (blue) and the asymptotes (green).