# How many cycles of the graph f(x)=7.8cos1.5(x-pi)+6 occur in the interval [-3pi/2,5pi]?

Aug 2, 2018

f ( x ) = 7.8 cos ( 1.5 ( x - pi ) + 6 in [ 7.8 ( - 1 ) + 6, 7.9 ( 1 ) + 6 ] = [ - 1.8, 13.8 ].

Amplitude: 7.8

Period: 2pi$/(1.5) = 4/3 pi  Phase shift = $\pi\$

Vertical shift = 6

So, the interval [ -3/2 pi, 5 pi ] = (( 5 - (- 1.5 )) pi )/(4/3 pi ) =

19.5/4 = 4.875 periods.

See graph, with all these aspects. Slide the graph uarr larr darr

rarr and see the whole amplitude and the four periods, within.
graph{(y-(7.8 cos ( 1.5 ( x - pi )) + 6))(x+1.5pi+0.001y) (x+1/6pi+0.001y)(x-7/6pi+0.001y)(x-5/2pi+0.001y)(x-23/6pi+0.001y)=0[-6 16 -2 14]}