Question

Write an equation for rational function with given properties.

a) a hole at x = 1

b) a vertical asymptote anywhere and a horizontal asymptote along the x-axis

c) a hole at x = -2 and a vertical asymptote at x = 1

d) a vertical asymptote at x = -1 and a horizontal asymptote at y =2

e) an oblique asymptote, but no vertical asymptote

Answer 1

a) must have (x-1) in numerator and denominator

y = (x-2)(x-1)/(x-1) = (x^2 - 3x + 1)/(x-1)

b) must have (x-k) in the denominator, and not in the numerator. degree of numerator must be less than the degree of the denominator.

y = 12(x+3)/(x-5)(x+9)

c) must have (x+2) top and bottom, and have (x-1) in the bottom

y = 3(x+5)(x+2)/(x-1)(x+2) = (3x^2 + 21x + 18)/(x^2 + x - 2)

d) must have (x+1) in the bottom and have degree of top and bottom equal, with a factor of two up top.

y = (2x-5)/(x+1)

e) top degree must be bottom degree + 1

y = 3x^3/(x^2+3)
note the bottom is never 0.

Answer 2

a) should read (x^2 - 3x + 2)/(x-1)

c) should read (3x^2 + 21x + 30)/(x^2 + x - 2)

Answer 3

what is the rational equation of a function with intercepts at (8,0) and (0,0.16) and a horizontal asymptote at y=0.5 and a vertical asymptote at x=3 and a removable discontinuity at (3,-1)