Key Concepts
amplitude: the
amplitude of y = a sin x and y = a
cos x is half the distance between the maximum and minimum
values of the function and is given by:
period: let b
be a positive real number, then the period of y = a
sin bx and y = a cos bx is 
periodic function: a function f is periodic if there exists a positive real number c such that f(t + c) = f(t) for all t in the domain of f
even function: a function that is symmetric about the y-axis OR a function f such that f(t) = f(-t)
odd function: a function that is symmetric about the origin OR a function f such that f(-t) = -f(t)
note: Cosine is an even function and sine is an odd function.
Click here for an applet that generates the sine graph using the unit circle.
Click here for an applet
that generates the cosine graph using the unit circle.
Click here for a GSP sketch that allows you to manipulate and see the effects of changing a, b, c, and d, in the function f(x) = a sin(b(x-c))+d.
Begin with the first tab at the bottom left of the screen to see the effects of changing a, the second tab to see the effects of changing b, the third tab to see the effects of changing c, the fourth tab to see the effects of changing d, and the last tab to see the big picture.
Click here for a GSP sketch that allows you to manipulate and see the effects of changing a, b, c, and d, in the function f(x) = a cos(b(x-c))+d.
Follow the same process as before.
In this lesson, we have relied heavily on technology to illustrate the sine and cosine graphs. GSP is a powerful tool that gives students hands-on experience with the changes in the graphs.