In a rational function, m(c), we can write it in a factored form as:
[tex]m(c)=A\frac{(c-i_1)(c-i_2)\ldots}{(c-a_1)(c-a_2)\ldots}[/tex]
Where i1, i2 ... are the c-intercepts (the zeros of the functions) and a1, a2, ... are the vertical asymptotes of the function.
A we will get from m(0) = at the end.
Since we have the vertical aymptotes c = 6 and c = -6, then, the denominator is:
[tex]\begin{gathered} m(c)=A\frac{(c-i_1)(c-i_2)\ldots}{(c-6_{})(c-(-6))} \\ m(c)=A\frac{(c-i_1)(c-i_2)\ldots}{(c-6_{})(c+6)} \end{gathered}[/tex]
And, since we have the c-intercepts (1, 0) and (-5, 0), that is, c = 1 and c = -5, the numerator is:
[tex]\begin{gathered} m(c)=A\frac{(c-1_{})(c-(-5))}{(c-6_{})(c+6)} \\ m(c)=A\frac{(c-1_{})(c+5)}{(c-6_{})(c+6)} \end{gathered}[/tex]
We only need to figure "A" out now.
Since we know that m(0) = 9, we have:
[tex]\begin{gathered} m(0)=A\frac{(0-1_{})(0+5))}{(0-6_{})(0+6)}=A\frac{(-1)\cdot5}{(-6)\cdot6}=A\frac{5}{36}=\frac{5}{36}A \\ \frac{5}{36}A=9 \\ A=\frac{36\cdot9}{5}=\frac{324}{5} \end{gathered}[/tex]
This will end in the rational function:
[tex]m(c)=\frac{324}{5}\cdot\frac{(x-1)(x+5)}{(x-6)(x+6)}[/tex]
