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Vertical and Horizontal Stretches

Another common way that the graphs of trigonometric functions are altered is by stretching the graphs. Stretching a graph involves introducing a coefficient into the function, whether that coefficient fronts the equation as in y = 3 sin(x) or is acted upon by the trigonometric function, as in y = sin(3x) . Though both of the given examples result in stretches of the graph of y = sin(x) , they are stretches of a certain sort. The first example creates a vertical stretch, the second a horizontal stretch.

Vertical Stretches

To stretch a graph vertically, place a coefficient in front of the function. This coefficient is the amplitude of the function. For example, the amplitude of y = f (x) = sin(x) is one. The amplitude of y = f (x) = 3 sin(x) is three. Compare the two graphs below.

Figure %: The sine curve is stretched vertically when multiplied by a coefficient
The amplitude of the graph of any periodic function is one-half the absolute value of the sum of the maximum and minimum values of the function.

Horizontal Stretches

To horizontally stretch the sine function by a factor of c , the function must be altered this way: y = f (x) = sin(cx) . Such an alteration changes the period of the function. For example, continuing to use sine as our representative trigonometric function, the period of a sine function is , where c is the coefficient of the angle. Usually c = 1 , so the period of the sine function is 2Π . Below are pictured the sine curve, along with the following functions, each a horizontal stretch of the sine curve: y = f (x) = sin(2x) and y = f (x) = sin() .

Figure %: The sine function is stretched horizontally when the angle is multiplied by a scalar