Horizontal ``Stretching''

We now come to the final type of graphical adjustment that can be made to the basic sine and cosine functions. It is known as horizontal scaling or ``stretching''. At this point, we have seen addition to the function on the outside, as well as addition within the argument of the function. Both of these operations cause a shift or translation, either vertically or horizontally. We have also seen multiplication on the outside of the function and noted that this brings about vertical scaling. We will now see the related multiplication and note its affect.

If we multiply the argument of the function by a constant, we find that horizontal stretching takes place. Note from our previous discussion that this will affect the period of the function in question. Let's turn to an example to see this affect.

Consider the graphs of the functions and . (Remember that naturally appeared in the double angle formulas above.) How do the graphs of these two differ? Let's look at the graphs and determine the answer to this question.

We see that one cycle of is achieved after a distance of on the x-axis, whereas one cycle of requires a distance of . Hence, we see that the period of is rather than . You may not see the pattern yet, but we will note it here officially.