Here is a quick and simple approximation to what you ask for.
Plot[(x - 3)^2 + 1, {x, 0, 5},
Axes -> None,
Frame -> True,
FrameTicks -> All,
GridLines -> Automatic,
PlotRangePadding -> {{.5, .5}, {2, 1.5}},
Epilog -> {Arrow[{{-.25, 0}, {5.25, 0}}], Arrow[{{0, -.25}, {0, 11}}]},
ImageSize -> 450]

Here is a better approximation that requires a fair bit of fooling around with the grid lines.
Plot[(x - 3)^2 + 1, {x, 0, 5},
Axes -> None,
Frame -> True,
FrameTicks -> All,
GridLines ->
{If[IntegerQ[#], {#, Black}, {#, Thin}] & /@ Range[-1, 6, 1/5],
If[EvenQ[#], {#, Black}, {#, Thin}] & /@ Range[-1, 11, 1/2]},
PlotRangePadding -> {{.5, .5}, {2, 1.5}},
Epilog -> {Arrow[{{-.25, 0}, {5.25, 0}}], Arrow[{{0, -.25}, {0, 11}}]},
ImageSize -> 450]
