Phase Shifts: Videos & Practice Problems

Understanding the transformations of sine and cosine functions is essential in trigonometry, particularly when it comes to phase shifts. A phase shift refers to a horizontal shift of the graph, which can occur to the left or right, similar to how vertical shifts move the graph up or down.

To illustrate a phase shift, consider the cosine function, y = \cos(x). When a number is added or subtracted inside the cosine function, such as in y = \cos(x - \frac{\pi}{2}), it results in a shift of the graph. For example, evaluating points for this function shows that the graph starts at 0 instead of 1, indicating a rightward shift of \frac{\pi}{2} units. This transformation can be generalized: if the function is in the form y = \cos(bx - h), the graph shifts to the right by \frac{h}{b} units when h is positive, and to the left when h is negative.

For instance, in the function y = \cos(1x - \frac{\pi}{2}), the value of h is \frac{\pi}{2} and b is 1, resulting in a rightward shift of \frac{\pi}{2} units. This shift can make the cosine graph resemble a sine graph, as both functions share similar wave patterns.

To further explore phase shifts, consider the function y = \sin(2x + \pi). The period of a sine function is determined by the formula 2\pi/b, where b is the coefficient of x. Here, b is 2, giving a period of \pi. The phase shift can be calculated by recognizing that the function is in the form y = \sin(bx + h), where h is negative, indicating a leftward shift. Thus, h/b = \frac{\pi}{2}, leading to a shift of \frac{\pi}{2} units to the left.

In summary, phase shifts are a crucial aspect of graphing sine and cosine functions, allowing for horizontal transformations that can significantly alter the appearance of the graph. By understanding how to identify and calculate these shifts, students can effectively analyze and graph trigonometric functions.

To graph the function \( y = 3 \sin(x + \pi) \), we start by identifying the general form of the sine function, which is expressed as \( y = a \sin(b(x - h)) + k \). In this case, we can see that the amplitude \( a \) is 3, indicating that the graph will oscillate between 3 and -3 on the y-axis. The absence of a vertical shift means \( k = 0 \).

Next, we analyze the term \( x + \pi \). To fit the general form, we rewrite this as \( x - (-\pi) \). This reveals that \( h = -\pi \). The phase shift can be calculated using the formula \( \text{Phase Shift} = \frac{h}{b} \). Here, \( b = 1 \) (since there is no coefficient in front of \( x \)), leading to a phase shift of \( \frac{-\pi}{1} = -\pi \). This indicates a shift of \( \pi \) units to the left.

Now, we can graph the basic sine function \( y = 3 \sin(x) \). The standard sine graph starts at the origin, reaches a peak of 3 at \( \frac{\pi}{2} \), crosses the x-axis at \( \pi \), and reaches a valley of -3 at \( \frac{3\pi}{2} \). However, due to the phase shift of \( \pi \) units to the left, we adjust the key points accordingly:

• The peak at \( \frac{\pi}{2} \) shifts to \( -\frac{\pi}{2} \).
• The x-intercept at \( \pi \) shifts to \( 0 \).
• The valley at \( \frac{3\pi}{2} \) shifts to \( \frac{\pi}{2} \).

Thus, the graph of \( y = 3 \sin(x + \pi) \) will start at the center (0), peak at \( -\frac{\pi}{2} \) with a value of 3, cross through 0 at \( -\pi \), and reach a valley at \( -\frac{3\pi}{2} \) with a value of -3. The graph will continue to oscillate in this manner, maintaining the amplitude of 3 and the periodic nature of the sine function.

In summary, the graph of \( y = 3 \sin(x + \pi) \) exhibits a phase shift of \( \pi \) units to the left, with an amplitude of 3, resulting in a wave pattern that oscillates between 3 and -3.