Question

Write a rational function for the given scenario. There may be more than one correct answer.

Question 1. X-intercept: none
Vertical Asymptote: X = -3
Horizontal Asymptote: y = 0

Question 2. X-intercept: none
Vertical Asymptote X=2 & x= -2
Horizontal Asymptote: y = 0

Question 3 x-intercept: (3,0) & (-3,0) Vertical Asymptote: X=-1 & x= -4
Horizontal Asymptote: y = 1

Question 4. X-intercept: (-2, 0)
Vertical Asymptote: X = 4 & x=-5 Horizontal Asymptote: y=0

Need to make a rational function for each question.

Answer 1

Q1:
`(1)/(x+3)`

Q2:
`(1)/((x-2)(x+2))`

Q3:
`((x-3)(x+3))/((x+1)(x+4))`

Q4:
`(x(x+2))/((x-4)^(2)(x+5) )`

Explanation:

If there is a vertical asymptote at x=k, then the denominator must contain a term (x-k) or (x-k) raised to a positive integer power.

If there is an x intercept at k, then the numerator must contain a term (x-k) or (x-k) raised to a positive integer power.

If there is a horizontal asymptote of 0, then the highest power of x in the denominator must exceed the highest power of x in the numerator.

If thee is a horizontal asymptote of k, then the highest power of x in the numerator and denominator must be the same and the ratio of their coefficients (numerator to denominator) must be k.