Cosine Function

This applet helps you explore the general cosine function

f(x) = a*cos(bx + c) + d

and its properties such as amplitude, period and phase shift by changing the parameters a, b, c and d. Particular exloration 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 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

Your browser is completely ignoring the <APPLET> tag!
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)?

  1. 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?

  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?

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Updated: February 2015

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