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Transforming the Graphs of Sine and Cosine - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The coefficients A and B in y=Asin(Bx) or y=Acos(Bx) each have a different effect on the graph. If A and B are 1, both graphs have an amplitude of 1 and a period of 2pi. For sine and cosine transformations, when A is larger than 1, the amplitude increases and is equal to the value of A; if A is negative, the graph reflects over the x-axis. When B is greater than 1, the period decreases; use the formula 2pi/B to find the period.

I want to talk about transformations of the sine graph. To begin with we're going to discuss transformations of the form y equals a sine bx, and I want to talk about what those transformations do, what the coefficients a and b actually do to the graph of sine. Let's begin by reviewing the graph of sine, I've got it drawn here over the interval from 2 pi to 2 pi. Remember that sine has an amplitude of 1 and a period of 2 pi and when weÂ’re graphing sine and cosine functions in the future we would really want to be, we want to know the key points of the sine graph and the cosine graph. For the sine graph the key points are these points 0, 0 pi of a 2, 1, pi 0 3pi over 2 negative 1 and 2 pi, 0. These 5 points are really important and we'll use them a lot when we're graphing sine and cosine functions. well let's take a look at a demonstration that'll show us what these transformations will do.
Okay so we're in geometer sketch pad here this is the kind of function that I want to discuss. g of x equals a times the sine of bx, I've got a graph of the regular sine function in blue here and a graph of the transformed sine function in red. And up here I've got sliders that allow me to control the values of a and b. So right now a is 2 and b is 1, b is 1 this is basically just 2 times the sine of x. If I increase the value of a you can see what happens it increases the amplitude, increases the maximum and minimum values. I can also make a less than 1 in which case you get a vertical compressions. So this is a vertical compression, this is a vertical stretch.
Now if I make a negative, I get in addition to vertical stretch and compression I get reflection across the x axis. So that's something we'll have to watch out for. Okay now what happens if I adjust the b value? Do I also get stretches and compressions the answer is yes but it's not the way you'd expect. When you increase b you get a horizontal compression so for example when b is 2 my period is exactly half of what it used to be. Now my period is pi, when b is a half the period is twice what it used to be this is going to be 4 pi. Now the way to figure out period is to use this formula 2 pi over b where b is the coefficient in front of the x 2 pi over b is my new period.
Alright let's review what we just learned for y equals a sine bx, the amplitude is not just a because we could have a reflection across the x axis we have to use the absolute value of a you saw that I could make the a value negative but the amplitude is always positive. It's always the maximum minus the minimum divided by 2. So it's the absolute value of this number that gives me amplitude and period as we saw in the demonstration is 2 pi over b. So just to recap when you're graphing functions of this form remember the amplitude is the absolute value of a, period is 2 pi over b and remember your key points 0, 0 pi over 2, 1, pi 0, 3 pi over 2 negative 1 and 2 pi, 0. These key points will get you through a lot of graphing.

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