How do I find vertices of a curve and a hyperbola?

This may help:

http://www.mathwords.com/v/vertices_of_a_hyperbola.htm

The vertex is where the tangent to the curve chages direction most rapidly. When the curve is symmetric about vertical or horizontal axis, all you have to do is find where the slope is zero or infinity (respectively), since that will be where the curve changes slope most rapidly. For functions with xy terms in the equations, the axis of symmetry will be tilted and it will be harder to do.

To find the vertices of a curve or a hyperbola, you can follow these steps:

1. Determine the equation of the curve or hyperbola. This will give you an algebraic representation of the shape.

2. Identify the axes of symmetry. For curves or hyperbolas that are symmetric about a vertical or horizontal axis, this step is easier. The axis of symmetry is the line that divides the shape into two equal halves.

3. For curves that are symmetric about a vertical axis, find the x-coordinate of the vertex. The vertex will be located where the slope of the curve is zero. To find this, set the derivative of the equation with respect to x equal to zero and solve for x.

4. For curves that are symmetric about a horizontal axis, find the y-coordinate of the vertex. The vertex will be located where the slope of the curve is zero. To find this, set the derivative of the equation with respect to y equal to zero and solve for y.

5. For hyperbolas, the equation may have xy terms, making it more difficult to find the vertices. In this case, you will need to use more advanced techniques, such as completing the square or using calculus to find the points of maximum or minimum.

Remember, it is important to understand the specific equation and characteristics of the curve or hyperbola you are working with. Practice with different examples to gain familiarity with the process.