How do you graph #f(x) = 4 cos(x - pi/2 ) + 1#?

1 Answer
Dec 4, 2015

Since #cos(x-\pi/2)=sin(x)#, we can simplify the expression into

#4sin(x)+1#

If you know the graph of #sin(x)#, then you have simply multiplied the function by #4#, and then added #1#.

Multiplying by four results in a vertical stretch. In fact, where #sin(x)# is zero, #4sin(x)# is still zero. On the other hand, the maxima and minima (which of course are #1# and #-1#), now become #4# and #-4#. All the intermediate points must follow accordingly, and so, the function result stretched.

When you add #1#, you are not associating anymore #y=f(x)#, but #y=f(x)+1#. This means that you have added one unit to the #y# coordinate, which means that you translated the graph one unit upwards. Here are the graph of the changes:

  • The fact that #cos(x-pi/2)=sin(x)#, here

  • The vertical stretch from #sin(x)# to #4sin(x)#, here

  • The upward shift from #4sin(x)# to #4sin(x)+1#, here