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?

Links to topics related to cosine function


Step by Step Calculator to Find Domain of a Function New ! Free Trigonometry Questions with Answers Interactive HTML5 Math Web Apps for Mobile LearningNew !
Free Online Graph Plotter for All Devices
Home Page -- HTML5 Math Applets for Mobile Learning -- Math Formulas for Mobile Learning -- Algebra Questions -- Math Worksheets -- Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests -- GRE practice -- GMAT practice Precalculus Tutorials -- Precalculus Questions and Problems -- Precalculus Applets -- Equations, Systems and Inequalities -- Online Calculators -- Graphing -- Trigonometry -- Trigonometry Worsheets -- Geometry Tutorials -- Geometry Calculators -- Geometry Worksheets -- Calculus Tutorials -- Calculus Questions -- Calculus Worksheets -- Applied Math -- Antennas -- Math Software -- Elementary Statistics High School Math -- Middle School Math -- Primary Math
Math Videos From Analyzemath
Author - e-mail


Updated: 2 April 2013

Copyright © 2003 - 2014 - All rights reserved