For the given cosine function
we were asked to determine the amplitude,
period, and phase shift.
After we do this, we will also verify
the results graphically.
For a quick review, when we have the sine
and cosine functions in this form,
the absolute value of A is the amplitude.
If A is less than zero, we have a reflection
across the midline.
Two pi divided by B is the period.
C indicates a vertical translation.
Where C is greater than zero or positive,
the function is translated up C units.
If C is less than zero or negative,
the function is translated down
the absolute value of C units,
and finally D in this form
indicates a horizontal translation
where if D is greater than zero,
the function is translated right D units,
and if D is less than zero or negative,
the function is translated left
the absolute value of D units.
Notice how when D is less than zero or negative,
this would be in the form of X plus D
because X minus negative D
is equivalent to two X plus D.
So going back to our example,
notice A is equal to negative seven,
B is equal to positive three,
and D in this form is equal to positive pi over five.
Also notice that C is equal to zero.
Because C is zero, there is no vertical shift,
and therefore the midline remains the x-axis
or Y equals zero.
So because A is equal to negative seven,
we know we have a reflection across the midline,
and since the absolute value of A
is equal to the absolute value of negative seven,
which is positive seven,
the amplitude is positive seven.
The period is equal to two pi divided by B,
and B is equal to positive three,
and therefore the period is two pi divided by three radians
or 2/3 pi radians,
and then finally D is equal to positive pi over five,
again, because it has to be in the form of X minus D,
and because D is positive,
the phase shift, or horizontal shift,
is pi over five units to the right.
And let's verify the results graphically.
Since C is equal to zero,
there is no vertical translation,
and therefore the midline remains the x-axis
or Y equals zero.
When graphing a basic cosine function,
we normally graph one period of the function
from zero to two pi radians,
but in this case, because we have a phase shift
or horizontal shift of pi over five units
to the right, we will graph one period
of the given function starting at X equals pi over five
indicated by this red vertical line,
and then because the period is 2/3 pi radians,
if we add 2/3 pi radians
to pi over five radians, we get 13/15 pi radians
indicated by this red vertical line.
So between these two vertical lines
we have one complete graph
of the given cosine function,
and because the amplitude is seven
and we don't have a vertical translation,
we know we have a maximum function value
of positive seven and a minimum function value
of negative seven, and then when we graph
the basic cosine function,
we divide the period into four equal subintervals,
and then the pattern for the cosine function
is maximum, midline, minimum, midline, maximum,
but because we have a reflection across the midline,
the pattern for our function over one period
is minimum, midline, maximum, midline, and a minimum,
and once we have this piece of the graph,
we can simply copy and paste this graph
over a larger interval.
So this graph does verify our results are correct.
I hope you found this helpful.