The graph of f(x) = (2x)/(x^2 - 5x - 14) has vertical asymptotes x = a and x = b, and horizontal asymptote y = c. Find a + b + c.
+2446 In the rational function \(f(x)=\frac{2x}{x^2-5x-14}\), we can determine the vertical asymptotes by setting the denominator equal to 0 and solving for x.
| \(x^2-5x-14=0\) | In this case, the left-hand-side quadratic is factorable, which eases the process of finding the zeros. |
| \((x-7)(x+2)=0\) | Use the zero product thereom to find the remaining zeros. |
| \(x_1=a=7\\ x_2=b=-2\) | |
The horizontal asymptote requires some observation. The horizontal asymptote lies on y=0 since the degree of the numerator is less than the degree of the denominator. Therefore, c=0. We know the unknown variables, so we can now calculate their collective sum.
\(a+b+c\\ 7-2+0\\ 5\)
