Rotating 1/x to make a hyperbola?

In summary, the conversation discusses how to rotate the graph of 1/x using polar coordinates and then convert it back to cartesian coordinates. The process involves replacing the angle with a new angle to rotate the graph, using trigonometric identities to simplify the expression, and using equations to convert back to cartesian coordinates. The conversation also mentions that if the angle is replaced with its negative, the original coordinates can be obtained.
  • #1
Khan
Hey everyone, I was having trouble with this question.

The graph of 1/x is a hyperbola, but it's equation does not fit the form (x-h)/(a^2) - (y-k)/(b^2) = 1. Rotate 1/x using polar coordinates, change it back into cartesian coordinates, and write the equation in standard hyberbola notation.

I understand how to convert to polar from cartesian and vice versa, but I don't know how to rotate graphs. Any help would be great! Thanks!
 
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  • #2
When you have the expression in polar form, replace the angle θ with θ+φ to rotate by an angle φ (or θ-φ, depending on your sign conventions). You'll probably want to use the trig addition identities (for, say, sin(a+b) in terms of sines and cosines of a and b) to put your expression into a form that can be easily converted back to Cartesian coordinates.
 
  • #3
If (x,y) are the original coordinates and (x', y') are the coordinates rotated at angle θ then

x'= x cos(θ)+ y sin(θ);

y'= -x sin(θ)+ y cos(θ);

Notice that if you solve the two equations for x, y in terms of x',y', this is, inverting the change, you get

x= x' cos(θ)- y' sin(θ)

y= x' sin(θ)+ y' cos(θ)

Exactly what you would get if you replace θ by -θ

In particular, if θ= π/2, then

x= (√(2)/2) (x'- y'); y= (√(2)/2)(x'+ y')

xy= (1/2)(x'2- y'2)= x'2/2- y'2/2= 1.
 
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1. What is a hyperbola?

A hyperbola is a type of mathematical curve that is created by the intersection of a plane and a double cone. It has two branches that are symmetrical to each other and extend to infinity.

2. How is a hyperbola formed by rotating 1/x?

By rotating the function 1/x, we are essentially rotating a rectangular hyperbola around its asymptote. This creates a three-dimensional shape known as a "hyperboloid of revolution," which appears as a curved funnel shape when viewed from the side.

3. What is the significance of rotating 1/x to create a hyperbola?

Rotating 1/x allows us to visualize the hyperbola in three dimensions and see its relationship with the rectangular hyperbola. It also helps us understand how changing the orientation of a curve can affect its shape.

4. Can a hyperbola be created by rotating a different function?

Yes, a hyperbola can be created by rotating other functions such as y = 1/x^2 or y = 1/x^3. However, the resulting hyperbolas may have different shapes and dimensions depending on the function being rotated.

5. How is the equation of a hyperbola affected by rotating 1/x?

Rotating 1/x does not change the equation of the hyperbola itself. The equation of a hyperbola remains y = 1/x, but the orientation and shape of the hyperbola may change depending on the rotation angle.

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