91Ó°ÊÓ

In trigonometric functions like sine and cosine, amplitude represents the height from the center line to the peak of the curve. It essentially shows how tall the wave is. For the function given in the exercise, $$y=-2 \text{sin}(\pi x-\text{pi})$$, the amplitude can be deduced from the coefficient in front of the sine function. Here, the coefficient (a) is -2. Don't worry about the sign; amplitude is always positive. Therefore, the amplitude is $$|a| = |-2| = 2.$$ This means the sine wave oscillates 2 units above and 2 units below its central axis (y = 0). Understanding amplitude helps you visualize the wave's height and how it scales vertically.

The period of a trigonometric function is the length of one complete cycle of the wave. For sine and cosine functions, the period is determined by the coefficient of x inside the function. In our exercise, $$y=-2 \text{sin}(\text{pi} x- \text{pi})$$, the coefficient (b) in front of x is π. The formula to calculate the period is $$ \text{Period} = \frac{2\text{pi}}{b}.$$ Plugging in our value of $$ b = \text{pi}, $$ we get $$ \text{Period} = \frac{2π}{π} = 2.$$So, each complete wave cycle spans 2 units along the x-axis. Knowing the period helps you understand how frequently the wave repeats itself within a given length along the x-axis.

The phase shift of a trigonometric function tells you how much the graph of the function is shifted horizontally from its usual position. To find the phase shift, we use the expression inside the sine function. For $$ y = -2 \text{sin}(\text{pi} x- \text{pi} ), $$ the expression inside the sine is $$ (\text{pi} x-\text{pi}).$$ The phase shift formula is given by $$-\frac{c}{b}.$$ In this case, $$c=Ï€ $$ and $$b=Ï€.$$ Plugging these values in, we have $$ \text{Phase shift} = -\frac{Ï€}{Ï€} = -1.$$ Since the result is negative, the phase shift is 1 unit to the left. The minus sign in front of the result indicates the direction of the shift. Understanding phase shift is crucial for knowing how the graph of the trigonometric function is translated horizontally.