Graph `y=8cos(-x/4) ` and list the maximum, the minimum, and the intercepts:
The graph of `y=Acos(Bx) ` will have amplitude |A| and period `(2pi)/|B| ` . (For a sinusoid, if B<0 the graph is reflected across the vertical axis. Since the cosine is symmetric about the y-axis, `y=8cos(-x/4) ` and `y=8cos(x/4) ` have the same graph.)
Thus the amplitude is 8. This is the distance from the midline. Since the graph has not been translated vertically, the midline is y=0. Then the maximum will be 8 and the minimum -8.
The period is `p=(2pi)/(1/4)=8pi ` .
Since the graph has not been translated horizontally, the maximum will occur at` ` `x=8npi,n in ZZ ` (n an integer -- i.e. the maximum occurs at `...,-16pi,-8pi,0,8pi,16pi,... `
The minimum will occur at `x=4pi + 8npi,n in ZZ ` (i.e. the minimums occur at `...,-4pi,4pi,12pi,... `
The y-intercept is at (0,8).
The x-intercepts are at `x=2pi+4npi,n in ZZ ` .
Here is the graph -- the vertical ticks are in 2's while the horizontal ticks are by ` 2pi ` :
Take note of the form y = Acos(Bx) where Amplitude = |A| and period = |2pi/B|.
Therefore, Amplitude = 8, and period `= |2pi/(-1/4)| = |2pi*-4| = |-8pi| = 8pi`
The B value of -1/4 tells us that a complete cycle of the graph will be from 0 to 8pi.
I attached the graph of a given function the max will be at the points (0, 8) and (8pi, 8),
The x-intercepts are (2pi, 0) and (6pi, 0).
The y-intercept is (0, 8).
That's it! :)