Answer
Graph
Work Step by Step
$$\eqalign{
& y = {\left( {x - 1} \right)^5} \cr
& {\text{Domain: }}\left( { - \infty ,\infty } \right) \cr
& {\text{Find the }}y{\text{ intercept, let }}x = 0 \cr
& y = {\left( {0 - 1} \right)^5} \cr
& y = - 1 \cr
& y{\text{ - intercept }}\left( {0, - 1} \right) \cr
& {\text{Find the }}x{\text{ intercept, let }}y = 0 \cr
& {\left( {x - 1} \right)^5} = 0 \cr
& x = 1 \cr
& x{\text{ - intercept }}\left( {1,0} \right) \cr
& \cr
& *{\text{Find the extrema}} \cr
& {\text{Differentiate}} \cr
& y' = \frac{d}{{dx}}\left[ {{{\left( {x - 1} \right)}^5}} \right] \cr
& y' = 5{\left( {x - 1} \right)^4} \cr
& {\text{Let }}y' = 0{\text{ to find critical points}} \cr
& 5{\left( {x - 1} \right)^4} = 0 \cr
& x = 1 \cr
& *{\text{Find the second derivative}} \cr
& y'' = \frac{d}{{dx}}\left[ {5{{\left( {x - 1} \right)}^4}} \right] \cr
& y'' = 20{\left( {x - 1} \right)^3} \cr
& {\text{Evaluate }}y''\left( x \right){\text{ at }}x = 1 \cr
& y''\left( 1 \right) = 20{\left( {1 - 1} \right)^3} = 0 \cr
& {\text{Use the first derivative test}} \cr
& y'\left( 0 \right) = + ,{\text{ }}y'\left( 2 \right) = + ,{\text{ The derivative does not change}} \cr
& {\text{no relative extrema at }}x = 1 \cr
& {\text{Let }}y''\left( x \right) = 0 \cr
& 20{\left( {x - 1} \right)^3} = 0 \cr
& x = 1 \cr
& \cr
& {\text{*Calculate the asymptotes}} \cr
& {\text{No vertical asymptotes, the denominator is 1}}{\text{.}} \cr
& \mathop {\lim }\limits_{x \to \infty } {\left( {x - 1} \right)^5} = \infty \cr
& \mathop {\lim }\limits_{x \to - \infty } {\left( {x - 1} \right)^5} = - \infty \cr
& {\text{No horizontal asymptotes}} \cr
& \cr
& {\text{Graph}} \cr} $$
