Question

Write an equation for a rational function mm with:Use the smallest powers possible to meet the criteria.

Answer 1

In a rational function, m(c), we can write it in a factored form as:

`m(c)=A((c-i_1)(c-i_2)\ldots)/((c-a_1)(c-a_2)\ldots)`

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:

`\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}`

And, since we have the c-intercepts (1, 0) and (-5, 0), that is, c = 1 and c = -5, the numerator is:

`\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}`

We only need to figure "A" out now.

Since we know that m(0) = 9, we have:

`\begin{gathered} m(0)=A\frac{(0-1_{})(0+5))}{(0-6_{})(0+6)}=A((-1)\cdot5)/((-6)\cdot6)=A(5)/(36)=(5)/(36)A \\ (5)/(36)A=9 \\ A=(36\cdot9)/(5)=(324)/(5) \end{gathered}`

This will end in the rational function:

`m(c)=(324)/(5)\cdot((x-1)(x+5))/((x-6)(x+6))`