Answer:
To graph rational functions:
- Find any asymptotes (draw them as dotted lines, unless they are x and y axis)
- Find any x-intercept(s) and y-intercept
- Find the values of y for different values of x
- Plot the points and connect them with a smooth curve
Part (a)
[tex]\textsf{Asymptote} \implies (x+3)^2=0 \implies x=-3[/tex]
[tex](x+3)^2\geq 0 \implies f(x) > 0 \implies \textsf{asymptote}:y=0[/tex]
[tex]\textsf{y-intercept}\implies \dfrac{1}{(0+3)^2}=\dfrac{1}{9} \implies \left(0,\dfrac{1}{9}\right)[/tex]
Part (b)
[tex]\textsf{Factor denominator} \implies x^2+2x-3=(x-1)(x+3)[/tex]
[tex]\textsf{Asymptote} \implies x-1=0 \implies x=1[/tex]
[tex]\textsf{Asymptote} \implies x+3=0 \implies x=-3[/tex]
[tex]\textsf{x-intercept} \implies x+1=0 \implies x=-1 \implies (-1,0)[/tex]
[tex]\textsf{y-intercept}\implies \dfrac{0+1}{0^2+2(0)-3}=-\dfrac{1}{3} \implies \left(0,-\dfrac{1}{3}\right)[/tex]
Part (c)
[tex]\textsf{Asymptote} \implies x-2=0 \implies x=2[/tex]
[tex]\textsf{x-intercepts} \implies x^2-9=0 \implies x=\pm 3 \implies (-3,0)\:(3,0)[/tex]
[tex]\textsf{y-intercept}\implies \dfrac{0^2-9}{0-2}=\dfrac{-9}{-2}=4.5 \implies (0,4.5)[/tex]
