  A combined analysis of the reactions $\pi^+p\to \pi^+p$, $\pi^-p\to \pi^-p$
and $\pi^-p\to \pi^0n$ is carried out with a chiral quark model. The
observations are reasonably described from the $\Delta(1232)$ resonance region
up to the $N(1440)$ resonance region. Besides the $\Delta(1232)P_{33}$, a
confirmed role of $N(1440)P_{11}$ is found in the polarizations of the
$\pi^-p\to \pi^-p$ and $\pi^-p\to\pi^0n$ reactions. It is found that the
$N(1440)N\pi$ and $\Delta(1232)N\pi$ couplings are about $1.7$ and $4.8$ times
larger than the expectations from the simple quark model, respectively, which
may suggest the unusual property of $N(1440)P_{11}$ and deficiency of the
simple quark model in the description of $N(1440)P_{11}$ and
$\Delta(1232)P_{33}$. The $t$- and $u$-channel backgrounds have notable
contributions to the $\pi^+ p\rightarrow\pi^+ p$ reaction, while in the
$\pi^-p\rightarrow \pi^- p,\pi^0 n$ reactions, the $s$-channel nucleon and $t$-
and $u$-channel backgrounds play crucial roles.
