f(x) = a*cos(bx + c) + d
in blue and
f(x) = a*cos(bx) + d
in red as shown in the figure below. This is to explain the effects of c on the phase shift.
You may also want to consider another tutorial on the trigonometric unit circle .
Once you finish the present tutorial, you may want to go through a self test on trigonometric graphs .
Interactive Tutorials
Two possibilities to explore graph of functions of the form f(x) = a*cos(bx + c) + d
1) Using a cosine Function HTML5 applet
2) Using a Java applet
Click on the button above "click here to start" and maximize the window obtained.
Explore how the 4 coefficients a,b,c and d affect the graph of f(x)?

Use the scrollbar to set a=1,b=1,c=0 and d=0. Write down f(x) and take note of the amplitude, period and phase shift of f(x)? Now change a , how does it affect the graph?

Set a=1,c=0,d=0 and change b. Find the period from the graph and compare it to 2pi/b. How does b affect the graph of f(x)?

set a=1,b=1,d=0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to
c/b.

set a=1,b=1,d=0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to
c/b.

repeat 3 and 4 above for b=2,3 and 4.

set a,b and c to non zero values and change d. What is the direction of the shift of the graph when d is positive and when d is negative?
Links to topics related to cosine function

