The rational function will have the next form:
[tex]f(x)=\frac{a(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}[/tex]where x1 and x2 are the x-coordinates of the x-intercepts, x3 and x4 are the vertical asymptotes and a is some coefficient.
The x-intercepts are (2,0) and (3,0), then x1 = 2, and x2 = 3
The vertical asymptotes are x = – 7 and x = -8, then x3 = -7 and x4 = -8.
Substituting this information we get:
[tex]\begin{gathered} f(x)=\frac{a(x-2)(x-3)}{(x-(-7))(x-(-8))} \\ f(x)=\frac{a(x-2)(x-3)}{(x+7)(x+8)} \end{gathered}[/tex]The y-intercept at (0,12) means when x = 0, f(x) = 12. Substituting this information into the previous formula and solving for a:
[tex]\begin{gathered} 12=\frac{a(0-2)(0-3)}{(0+7)(0+8)} \\ 12=\frac{a(-2)(-3)}{7\cdot8} \\ 12=\frac{a\cdot6}{56} \\ 12\cdot\frac{56}{6}=a \\ 112=a \end{gathered}[/tex]Finally, the equation for the rational function is:
[tex]f(x)=\frac{112(x-2)(x-3)}{(x+7)(x+8)}[/tex]
