College Algebra & Trigonometry, 2018a

9.5. COMBINING THE TRANSFORMATIONS 425
9.5 Combining the Transformations
In the previous sections, we have seen how the various transformations act on
the trigonometric functions and we have worked with the first three (amplitude,
period and vertical shift) in combination with each other. Combining the phase
shift with the other transformations is tricky because of the way that the period
and the phase shift interact with each other.
Now we have two standard equations for the sinusoid:
y = A sin(Bx + C)+D
and
y = A cos(Bx + C)+D
A and D, the amplitude and the vertical shift affect the y-axis, while B and C
affect the x-axis.
y-axis
Amplitude= |A|
Vertical Shift= D
x-axis
Period= 2π B
Phase Shift=− C B
Let’s look at an example in which we need to combine a change in the period of
the graph with a phase shift.
Example 1
Graph at least one period of the given function. Indicate the important values
along the x and y axes.
y =cos(4x + π)
The transformations in this example only affect the x-axis. The period of the
function is 2π B =2π 4 =π 2
. So, the function will complete one full cycle over a distance
of π 2
along the x-axis.