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Vertical and Horizontal Stretches

Vertical and Horizontal Stretches

Vertical and Horizontal Stretches

Vertical and Horizontal Stretches

Vertical and Horizontal Stretches

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