  In this paper, we study $F(R)$ gravity by Hu-Sawicki model in
Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) background. The Friedmann
equations are calculated by modified gravity action, and then the obtained
Friedmann equations are written in terms of standard Friedmann equations. Next
behavior of bouncing cosmology is investigated in the modified gravity model,
i.e., this behavior can solve problem of non-singular in standard Big Bang
cosmology. We plot the cosmological parameters in terms of cosmic time and then
bouncing condition is investigated. In what follows, we reconstruct the
modified gravity by redshift parameter, and also graphs of cosmological
parameters are plotted in terms of redshift, in which the figures show us an
accelerated expansion of Universe. Finally, the stability of the scenario is
investigated by a function as sound speed, and the graph of sound speed versus
redshift show us that there is the stability in late time.
