Graph y=8cos(-x/4) Include max, min, intercepts, and make sure you LABEL!

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embizze eNotes educator| Certified Educator

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 ` :

violy eNotes educator| Certified Educator

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! :)

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