Refer to the application at the beginning of the lesson. with the sun as a focus and the center at the origin, a certain comet's path follows a branch of a hyperbola. if two of the coordinates of the path are (10, 0) and (30, 100) where the units are in millions of miles, determine the equation of the path.

The equation of this hyperbola is:     ======the question says to put the center at the origin.this means the center (h,k) = (0,0).we use this equation of horizontally-opening hyperbola since it is centered at (0,0):         note that we do not use the alternate form  (which would describe an vertically opening parabola) because we are given the point (10, 0). this point is on the x-axis; a vertically opening parabola wouldn't be able to touch the x-axis.since we are given that (10, 0) is on this curved path, set x=10 and y=0 to solve for a parameter. that y=0 makes the math relatively simple:     note that we do not need to solve for "a" explicitly; we just need to find a².so our work-in-progress equation now looks like    since (30, 100) is on this curved path, set x=30 and y=100 to solve for the other parameter    finishing up to solve for b², we get    thus, with a² = 100 and b² = 1250, the equation is   

the formula of a slope:

from a table we have

substitute:

the slope of f(x) is -1.

the graph of funcfion g(x) is a horizontal line. the slope of a horizontal line is equal 0. therefore the slope of g(x) is 0.

-1 < 0

answer: the function g(x) has the greatest slope.