Respuesta :
It is required to write the equation of a rational function given vertical asymptotes, x-intercepts, and y-intercepts.
Recall that the vertical asymptotes form the factors in the denominator of the function, while the x-intercepts form the factors in the numerator.
Since the vertical asymptotes are x=-5 and x=2, it follows that the factors are (x+5) and (x-2).
Hence, the denominator will be:
[tex](x+5)(x-2)[/tex]Since the x-intercepts are x=3 and x=-6, then the factors are (x-3) and (x+6).
Hence, part of the numerator is:
[tex](x-3)(x+6)[/tex]It implies that the rational function takes the form:
[tex]y=\frac{a(x-3)(x+6)}{(x+5)(x-2)}[/tex]Where a is a constant to be determined.
It is given that the y-intercept is at 9.
Substitute x=0 and y=9 into the equation to calculate the value of a:
[tex]\begin{gathered} 9=\frac{a(0-3)(0+6)}{(0+5)(0-2)} \\ \Rightarrow9=\frac{a(-3)(6)}{(5)(-2)} \\ \Rightarrow9=\frac{-18a}{-10}=\frac{9a}{5} \\ \Rightarrow\frac{9a}{5}=9 \\ \Rightarrow9a=45 \\ \Rightarrow a=\frac{45}{9}=5 \end{gathered}[/tex]Substitute a=5 back into the function:
[tex]y=\frac{5(x-3)(x+6)}{(x+5)(x-2)}[/tex]Hence, the required rational function.
The rational function is:
[tex]y=\frac{5(x-3)(x+6)}{(x+5)(x-2)}[/tex]