How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?

2. Originally Posted by AK713
How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?
x^2 +y^2 -z^2 = 0 --------(i)
Rearranging that,
x^2 +y^2 = z^2 ----------(ii)
The LHS of (ii) is a circle.
The RHS is a square of a z-coordinate. It could be the square of the radius of the LHS at the z-coordinate.

When z=0, the circle at the LHS is zero too, or no circle. It is (0,0,0), or the origin.

When z = 1, or z = -1, the circle at the LHS has a radius of 1.
When z = 2, or z = -2, the circle at the LHS has a radius of 2.
When z = 3, or z = -3, the circle at the LHS has a radius of 3.
Etc...

The figure then is that of a cone whose apex is at the (0,0,0) and opening upwards, and that of its mirror-shape that opens downward.
That is if the z-axis is the usual up-down or vertical axis in the x,y,z axes setup.
The text passage contains a mathematical question about determining the orientation of the graph for a given equation, and an answer is provided that explains how to visualize the graph as a cone.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?

2. Originally Posted by AK713
How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?
x^2 +y^2 -z^2 = 0 --------(i)
Rearranging that,
x^2 +y^2 = z^2 ----------(ii)
The LHS of (ii) is a circle.
The RHS is a square of a z-coordinate. It could be the square of the radius of the LHS at the z-coordinate.

When z=0, the circle at the LHS is zero too, or no circle. It is (0,0,0), or the origin.

When z = 1, or z = -1, the circle at the LHS has a radius of 1.
When z = 2, or z = -2, the circle at the LHS has a radius of 2.
When z = 3, or z = -3, the circle at the LHS has a radius of 3.
Etc...

The figure then is that of a cone whose apex is at the (0,0,0) and opening upwards, and that of its mirror-shape that opens downward.
That is if the z-axis is the usual up-down or vertical axis in the x,y,z axes setup.
user
The text passage contains a mathematical question about determining the orientation of the graph for a given equation, and an answer is provided that explains how to visualize the graph as a cone.

TYPE: SINGLE_QUESTION_WITH_ANSWER