find a rational function that satisfies the given conditions vertical asymptotes x=-6,x=7 horizontal asymptotes y=10/9 x intercepts (6,0)
vertical asymptotes:
y = 1/((x+6)(x-7))
y(6) = 0, so
y = (x-6)/((x+6)(x-7))
horizontal asymptote at y = 10/9 needs a quadratic on top, so we could have
y = (10/9)(x-6)^2/((x+6)(x-7))
but that does not really cross the x-axis at x=6, but just touches it. So, how do we get that quadratic on top with no other x-intercepts? We could have a quadratic factor on top that is never zero, but then we need another linear factor underneath, such as
y = (10/9)(x-6)(x^2+1)/((x+6)^2(x-7))
See
http://www.wolframalpha.com/input/?i=(10%2F9)(x-6)(x%5E2%2B1)%2F((x%2B6)%5E2(x-7))
kind of a strange beast, but it does the job.
To find a rational function that satisfies the given conditions, we can start by determining the equation's basic form and then modify it to match the specific conditions.
A rational function is defined as the ratio of two polynomials, typically represented as f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials.
We know that the function has vertical asymptotes at x = -6 and x = 7, which means the denominator must have factors that cancel out these values. We can write the denominator polynomial as Q(x) = (x + 6)(x - 7).
The function also has a horizontal asymptote at y = 10/9, which means the degrees of the numerator and denominator polynomials must be the same. We can choose the degrees of both polynomials to be 1 to simplify the equation. Thus, the numerator polynomial becomes P(x) = k(x - 6), where k is a constant we need to determine.
Now, let's substitute these expressions into the rational function. The function becomes f(x) = (k(x - 6))/((x + 6)(x - 7)).
To find the value of k, we can use the x-intercept condition. We know that the function has an x-intercept at (6, 0), which means when x = 6, f(x) = 0. Substituting these values, we get 0 = k(6 - 6)/((6 + 6)(6 - 7)). Simplifying further, we find 0 = k/12(-1). This implies that k = 0.
Hence, our rational function is f(x) = 0/((x + 6)(x - 7)). Simplifying, we get f(x) = 0 for all x, except the values at x = -6 and x = 7, where the function has vertical asymptotes. The horizontal asymptote at y = 10/9 is also satisfied.
Therefore, the rational function that satisfies the given conditions is f(x) = 0/((x + 6)(x - 7)).