A quick review of 4.2:
Graphing y = A sin Bx and y = A cos Bx from the equation. Graph one full period and mark the scales on your axes.
y = 3 cos 4x
y = ­2 sin πx
Section 4.3
How do the graphs of y= k + cos x and y = k + sin x compare to y = cos x and y = sin x?
Try some.
1.
y = 2 + sin x
2.
y = ­3 + cos x
What about if we put more changes in?
3.
y = ­3 + 2 sin x
4.
y = 2 ­ 0.5 sin x
5.
y = ­1 + 2 sin 2x
A phase shift happens if the graph is moved left or right.
What happens to the sine function when we change it to y = sin (x + π/4)?
y = sin (x ­ π/3)?
What happens to the cosine function when we change its argument by adding or subtracting a number?
y = cos (t ­ π)
Summarize the result of adding or subtracting a number to the argument of the sine or cosine function.
It's slightly more complicated when we have both a period change and a phase shift change.
Check out the graph for this function.
Look at the graph to help you find the phase shift and period.
y = period =
phase shift = How about y = cos (3x + π )?
Is there a better way to find phase shift?
There are two ways to find the phase shift for sin (Bx + C) or cos (Bx + C).
1.
Set the argument equal to zero and solve:
2.
Factor out the B inside the argument.
Let's summarize.
y = k + A sin (Bx + C) or y = k + A cos (Bx + C)
Amplitude = |A|
Period = Vertical shift = k
Phase shift (horizontal) Bx + C = 0 and solve
x = is the phase shift
Let's try it! Find the amplitude, period, vertical shift, and phase shift for each of the following and use them to graph one complete cycle.
Challenge Problem: Find an equation for this graph.
π
y = 1 ­ sin 2x is one possibility.
y = ­1 + 2 sin (3x ­ π/2)