Question

Write an equation for a rational function in the form of f(x)=(ax+b)/(cx+d) , whose graph has all of the indicated features. - vertical asymptote with equation x=2 - horizontal asymptote with equation y=1 −x-intercept =−4 −y-intercept =−2

Answer 1

The equation for the rational function that satisfies the given conditions is f(x) = (x+4)/(x-2), which meets the required vertical and horizontal asymptotes, as well as the x-intercept and y-intercept values.

Step-by-step explanation:

To write an equation for a rational function with a vertical asymptote at x=2, a horizontal asymptote at y=1, an x-intercept at -4, and a y-intercept at -2, we need to incorporate these features into the function f(x) = (ax+b)/(cx+d).

Firstly, the vertical asymptote x=2 suggests that the denominator should be zero when x=2, so (cx+d) could be (x-2).

To have a horizontal asymptote at y=1, the degrees of the numerator and denominator should be the same, and the leading coefficients should form a ratio of 1. Thus, a/c should be 1, which means a = c

An x-intercept at -4 suggests that f(-4) = 0, which means that the numerator should be zero when x=-4; therefore, a(-4) + b = 0. If a=1 (as it should be from the horizontal asymptote), then b=4.

Lastly, a y-intercept at -2 means that f(0) = b/d = -2. Since we have determined b to be 4, d would need to be -2.

Putting all this together, we get the rational function f(x) = (x+4)/(x-2).