sec=1/cos so, if your calculator doesn't have secant as a function but has cosine, then you enter -cos(x/4-π/3) then use the reciprocal (1/x or x-1) key to give you -3sec(x/4-π/3) because:
y=1/(-cos(x/4-π/3)/3)=-3sec(x/4-π/3).
To graph the function, make a table of some values, paying particular attention to the cyclic nature of the function (pattern repeats along the whole length of the x-axis).
Note when x/4-π/3=0, that is, when x/4=π/3, x=4π/3, because cos(0)=1, so y=-3. But cos(2πn)=0, where n is an integer, so x/4-π/3=2πn, more generally, x=4(2πn+π/3)=4(6n+1)π/3. Your table of values should contain a few examples (n=-2, -1, 0, 1, 2, for example) when y=-3. These are points to plot.
Also consider cos(x/4-π/3)=-1, that is, when x/4-π/3=π+2πn, x=4(2πn+4π/3)=4(6n+4)π/3, when y=3.
Next consider cos(x/4-π/3)=0, that is, when x/4-π/3=(2n+1)π/2,
x/4=(2n+1)π/2+π/3=(3(2n+1)π+2π)/6, x=2(6n+5)π/3.
These values of x are asymptotes for y, since y will approach ±∞ at these points. Show the asymptotes as faint vertical lines when constructing the graph.
Note also that (4(6n+1)π/3,-3) are maxima and (4(6n+4)π/3,3) are minima. Knowing this will make it clear whether y tends to + or - infinity at the asymptotes (the left of the asymptote is -∞ and the right of the asymptote is +∞).
From this information you can sketch the graph, using your table as a guide.