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
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.
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.
a) should read (x^2 - 3x + 2)/(x-1)
c) should read (3x^2 + 21x + 30)/(x^2 + x - 2)