For certain constants a, b, c, and d, the graph of the functionhas a vertical asymptote of x = 7, a horizontal asymptote of y = -3, and a y-intercept of (0,6). Find the x-coordinate of the x-intercept.
To have a vertical asymptote of x = 7, the function needs a factor of (x - 7) in the denominator.
The function then can look like this: f(x) = a / (x - 7) for any non-zero value of a.
This function will have a horizontal asymptote at y = 0.
For this function to have a horizontal asymptote of y = -3, the function can be modified to be: f(x) = a / (x - 7) - 3.
Rewriting the function, using the common denominator: f(x) = [ -3x + (21 + a) ] / (x - 7)
Since the y-intercept is (0,6): 6 = (21 + a) / -7
Solving for a: a = -63
The function is now: f(x) = (-3x - 42) /(x - 7)