Q: We'll start with a simple example: a hyperbola with the center of its origin. For these hyperbolas, the standard form of the equation is x2/a2 - y2/b2 = 1 for hyperbolas that extend right and left, or y2/b2 - x2/a2 = 1 for hyperbolas that extend up and down. Remember, x and y are variables, while a and b are constants (ordinary numbers). Example 1: x2/9 - y2/16 = 1 Some textbooks and teachers switch the position of a and b in these equations. Follow the equation closely so you understand what's going on. If you just memorize the equations you won't be prepared when you see a different notation. This new equation represents both asymptotes, though it will take a little more work to separate them. Example 1: x2/9 - y2/16 = 0 Factor the left hand side of the equation into two products. Refresh your memory on factoring a quadratic if you need to, or follow along while we continue Example 1: We'll end up with an equation in the form (__ __)(__ __) = 0. The first two terms need to multiply together to make x2/9, so take the square root and write it in those spaces: (x/3 __)(x/3 __) = 0 Similarly, take the square root of y2/16 and place it in the two remaining spaces: (x/3 y/4)(x/3 y/4) = 0 Since there are no other terms, write one plus sign and one minus sign so the other terms cancel when multiplied: (x/3 + y/4)(x/3 - y/4) = 0 To get the equations for the asymptotes, separate the two factors and solve in terms of y. Example 1: Since (x/3 + y/4)(x/3 - y/4) = 0, we know x/3 + y/4 = 0 and x/3 - y/4 = 0 Rewrite x/3 + y/4 = 0 y/4 = - x/3 y = - 4x/3 Rewrite x/3 - y/4 = 0 - y/4 = - x/3 y = 4x/3 We've just found the asymptotes for a hyperbola centered at the origin. A hyperbola centered at (h,k) has an equation in the form (x - h)2/a2 - (y - k)2/b2 = 1, or in the form (y - k)2/b2 - (x - h)2/a2 = 1. You can solve these with exactly the same factoring method described above. Just leave the (x - h) and (y - k) terms intact until the last step. Example 2: (x - 3)2/4 - (y + 1)2/25 = 1 Set this equal to 0 and factor to get: ((x - 3)/2 + (y + 1)/5)((x - 3)/2 - (y + 1)/5) = 0 Separate each factor and solve to find the equations of the asymptotes: (x - 3)/2 + (y + 1)/5 = 0 y = -5/2x + 13/2 ((x - 3)/2 - (y + 1)/5) = 0 y = 5/2x - 17/2
A: Write down the equation of the hyperbola in its standard form. Set the equation equal to zero instead of one. Factor the new equation. Separate the factors and solve for y. Try the same process with a harder equation.

Q: Gardeners should keep in mind that most conifer species, with the exception of yew, cannot survive a severe pruning. Although the green foliage can be trimmed, the brown areas of old growth should be left alone because these places will not regenerate if cut. Some conifers have an area in the center where no foliage grows, but this is normal and not indicative of any problems. In trees where this is the case, gardeners should avoid pruning into this area because doing so will result in a lopsided tree. The plants will not grow back to cover up the hole. Therefore, tree trimmers should check the foliage areas before making decisions about which branches to lop off Although wanting to remove the lower branches of a tree is understandable, gardeners who do so may end up with an unsightly specimen when the tree gets taller. Therefore, tree trimmers should exercise restraint and only remove the bottom branches when it becomes absolutely necessary. Evergreens should not be topped, or cut off at a certain height, because this will produce trees that are anything but attractive. Trees that are topped are also at a greater risk for disease and other ailments. Conifers should not be pruned later in the summer or in the fall. Late season pruning can result in new, lush growth that will not have a chance to mature before the cold, damaging winter weather.
A:
Be aware that most conifer species will not survive a severe pruning. Refrain from pruning into the empty, central area of the tree. Only remove the bottom branches of trees when absolutely necessary. Avoid topping evergreen trees. Do not prune conifer trees too late in the year.