How Do You Determine the Graph Type and Rotation Angle for These Equations?

[LePahj]

I need help identifying the graphs of the following equations and their angles of rotation. I know I should be comparing these equations against the general equations and looking at the differences between the variables to determine the type of equation, and that I should be using the discriminant to determine the angles of rotation. However, I have never seen this done with equations with so few variables. I would really appreciate not just the answers, but explanations! Thanks.

1. y^2 + 8x = 0

I'm pretty sure this is a parabola, but I have no idea about the angle of rotation


2. 3xy - 4 = 0

hyperbola?


3. 9x^2 - 4(sqrt3)xy + 5y^2 = 15

ellipse?

 

[LePahj]

ooo and

4. xy = -5

hyperbola

 

[HallsofIvy]

I need help identifying the graphs of the following equations and their angles of rotation. I know I should be comparing these equations against the general equations and looking at the differences between the variables to determine the type of equation, and that I should be using the discriminant to determine the angles of rotation. However, I have never seen this done with equations with so few variables. I would really appreciate not just the answers, but explanations! Thanks.

1. y^2 + 8x = 0

I'm pretty sure this is a parabola, but I have no idea about the angle of rotation

If it were y= (1/8)x² would you be sure it was a parabola? y²+ 8x= 0 is the same as x= (-1/8)y² so what has happened here is that the x and y axes have been swapped. What angle is that?

2. 3xy - 4 = 0

hyperbola?

Yes, it is. What are the axes of symmetry? That should tell you how it has rotated. Have you drawn a graph?

3. 9x^2 - 4(sqrt3)xy + 5y^2 = 15

ellipse?

Now this is probably the hardest of the lot. I know a couple of methods of converting it to "principal axes" but they are very complex. There are combinations of the coefficients that will tell you whether you have ellipse, hyperbola, etc. Do you know any of those?
Of course, any time you have angles, you expect trig functions. What angles gives trig functions that include sqrt(3)?

ooo and

4. xy = -5

hyperbola

Yes. Again looking at the axes of symmetry should tell you the angle of rotation.