Cosine Function

General Cosine Function

Explore interactively the general cosine function
f(x) = a cos(b x + c) + d
and its properties such as amplitude, period and phase shift by changing the parameters a, b, c and d.
Particular exploration of the phase shift is undertaken by plotting 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.
cosine function with and without 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 Tutorial

Explore graph of functions of the form f(x) = a cos(b x + c) + d
cos(x)
a =
b =
c =
d =
>
Explore how the 4 coefficients a,b,c and d affect the graph of f(x)?
  1. 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?
  2. 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)?
  3. 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.
  4. 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.
  5. repeat 3 and 4 above for b = 2,3 and 4.
  6. 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?

More References to Cosine Functions