SINE AND COSINE FUNCTIONS GRAPHS OF SINE AND COSINE FUNCTIONS

We are interested in the graph of y = f(x) = sin x Start with a "t" chart and let's choose values from our unit circle and find the sine values. plot these points x y 1 - 1 x y = sin x We are dealing with x's and y's on the unit circle to find values. These are completely different from the x's and y's used here for our function.

y = f(x) = sin x choose more values x y = sin x plot these points join the points x y 1 - 1 If we continue picking values for x we will start to repeat since this is periodic.

Here is the graph y = f(x) = sin x showing from -2 to 6 Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function)

What are the x intercepts? Where does sin x = 0? …-3, -2, -, 0, , 2, 3, 4, . . . Where is the function maximum? Where does sin x = 1?

Where is the function minimum? Where does sin x = -1?

Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin x + 2 looks like? This is often written with terms traded places so as not to confuse the 2 with part of sine function y = 2 + sin x The function value (or y value) is just moved up 2. y = sin x

Thinking about transformations that you've learned and knowing what y = sin x looks like, what do you suppose y = sin x - 1 looks like? y = sin x The function value (or y value) is just moved down 1. y = - 1 + sin x

Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = sin (x + /2) looks like? y = sin x This is a horizontal shift by - /2 y = sin (x + /2)

Thinking about transformations that you learned and knowing what y = sin x looks like, what do you suppose y = - sin (x )+1 looks like? y = 1 - sin (x ) This is a reflection about the x axis (shown in green) and then a vertical shift up one. y = - sin x y = sin x

What would the graph of y = f(x) = cos x look like? We could do a "t" chart and let's choose values from our unit circle and find the cosine values. plot these points x y 1 - 1 x y = cos x We could have used the same values as we did for sine but picked ones that gave us easy values to plot.

y = f(x) = cos x Choose more values. x y = cos x plot these points y 1 - 1 cosine will then repeat as you go another loop around the unit circle

Here is the graph y = f(x) = cos x showing from -2 to 6 Here is the graph y = f(x) = cos x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function)

Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here

What are the x intercepts? Where does cos x = 0? Where is the function maximum? Where does cos x = 1? …-4, -2, , 0, 2, 4, . . .

Where is the function minimum? Where does cos x = -1? …-3, -, , 3, . . .

Let's try y = 3 - cos (x - /4) y = - cos x y = cos x You could graph transformations of the cosine function the same way you've learned for other functions. moves right /4 moves up 3 Let's try y = 3 - cos (x - /4) y = - cos x y = cos x reflects over x axis y = 3 - cos x y = 3 - cos (x - /4)

amplitude of this graph is 2 What would happen if we multiply the function by a constant? All function values would be twice as high y = 2 sin x amplitude is here amplitude of this graph is 2 y = 2 sin x y = sin x The highest the graph goes (without a vertical shift) is called the amplitude.

For y = A cos x and y = A sin x, A  is the amplitude. What is the amplitude for the following? y = 4 cos x y = -3 sin x amplitude is 3 amplitude is 4

The last thing we want to see is what happens if we put a coefficient on the x. y = sin 2x y = sin 2x y = sin x It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes .

What do you think will happen to the graph if we put a fraction in front? y = sin 1/2 x y = sin x The period for one complete cycle is twice as long or 4

The period T = So if we look at y = sin x the  affects the period. This will be true for cosine as well. What is the period of y = cos 4x? y = cos x This means the graph will "cycle" every /2 or 4 times as often y = cos 4x

absolute value of this is the amplitude Period is 2 divided by this

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au