How do you graph and list the amplitude, period, phase shift for #y=3cos4x#?

The cosine function's general form is

#y = Acos(kx+phi)#

and the basic form we all know and love has #A=1,k=1 and phi = 0#

If we focus on functions with #phi = 0# first and adjust A, we see that at #x=0#, #y=A# because:

#y(0)=Acos(0) = A#

Hence A adjusts the amplitude of the cosine function.

Now, lets look at what k does.

If we were to put #x=pi/2# into the basic cosine, we would obtain zero.

More generally, we get #y(pi/2) = Acos((kpi)/2)#

as we increase k to some value #k>1#, we obtain values that would be further along on the basic cosine function - we are decreasing the period. The opposite is also true, decreasing k to some value #k<1# lengthens the period.

Finally, lets look at the phase constant.

#y = Acos(x+phi)#

Lets say #phi# is some arbitrary number #0<=phi<=2pi#. A positive phase means that a larger x value is going into the cosine function so it will produce a value that we would expect further to the right - positive phase shifts the graph to the left!

So, lets look at our function:

#y(x) = 3cos(4x)#

We can see that #A = 3, k = 4 and phi = 0#

We know that it will start where we expect since there is no phase shift. The amplitude is now 3, so the graph will be stretched in the y direction - ranging from #-3<=y<=3#.

#k=4 implies #period is 4 times shorter than the basic function. So there will be 4 wavelengths in the period #[0,2pi)#.

graph{3cos(4x) [-2.7, 13.32, -3.57, 4.44]}

As you can see from the graph, this is true. The amplitude is 3 and there are 4 wavelengths between 0 and #2pi#