Write an equation for rational function with given properties.

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.

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

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

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)

To write an equation for a rational function with the given properties, we need to consider the characteristics of the function and use that information to determine the equation.

a) For a hole at x = 1, we know that there will be a factor of (x - 1) in both the numerator and the denominator. The equation can be written as:
f(x) = (x - 1) / (x - 1) = 1, where (x - 1) in the numerator cancels out with (x - 1) in the denominator.

b) For a vertical asymptote anywhere and a horizontal asymptote along the x-axis, we know that the denominator must have a factor of x - a, where a is the x-coordinate of the vertical asymptote. The equation can be written as:
f(x) = 1 / (x - a), where a is the x-coordinate of the vertical asymptote.

c) For a hole at x = -2 and a vertical asymptote at x = 1, we need to include the factor of (x + 2) in both the numerator and the denominator, as well as the factor of (x - 1) in the denominator. The equation can be written as:
f(x) = (x + 2) / [(x + 2)(x - 1)], where (x + 2) in the numerator cancels out with (x + 2) in the denominator.

d) For a vertical asymptote at x = -1 and a horizontal asymptote at y = 2, we need the factor of (x + 1) in the denominator and a constant numerator equal to the horizontal asymptote value. The equation can be written as:
f(x) = 2 / (x + 1).

e) For an oblique asymptote without a vertical asymptote, we need to have a higher degree term in the numerator compared to the denominator. The equation can be written as:
f(x) = (4x^2 + 3x - 2) / (2x^2 + 5), where the numerator's degree is higher than the denominator's degree.

Note: These equations are just examples that satisfy the given properties. There may be multiple possible equations for each case, so the equations provided are not the only solutions.