Problem 5 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\sin \left(x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{2}\) to the right.

Step 1: Identify Amplitude

The amplitude of a sine function, written in the form \(y = a \sin(bx + c)\), is given by \(|a|\). For \(y = \sin(x - \frac{\pi}{2})\), the coefficient \(a\) is 1, since there is no number multiplying the sine function. Therefore, the amplitude is 1.

Step 2: Determine Period

The period of a sine function is calculated as \(\frac{2\pi}{|b|}\), where \(b\) is the coefficient of \(x\) in the function \(y = \sin(bx + c)\). For this function, \(b = 1\), hence the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Calculate Phase Shift

The phase shift of the sine function \(y = \sin(bx + c)\) is found using \(-\frac{c}{b}\). Here, \(c = -\frac{\pi}{2}\) and \(b = 1\). Therefore, the phase shift is \(-\frac{-\frac{\pi}{2}}{1} = \frac{\pi}{2}\). This means the graph of the function is shifted to the right by \(\frac{\pi}{2}\) units.

Step 4: Graph the Function

To graph \(y = \sin(x - \frac{\pi}{2})\), start with the basic sine curve \(y = \sin x\), which has key points at \((0,0), (\frac{\pi}{2},1), (\pi,0), (\frac{3\pi}{2},-1), (2\pi,0)\). Shift all of these points \(\frac{\pi}{2}\) units to the right due to the phase shift. Now, the points correspond to \((\frac{\pi}{2},0), (\pi,1), (\frac{3\pi}{2},0), (2\pi, -1), (\frac{5\pi}{2},0)\). Plot these points and draw a smooth curve to complete the graph.

Key Concepts

Amplitude

When we talk about the amplitude of a sine function, we are discussing the measure of how "tall" or "short" the wave peaks are. Think of it as the height from the centerline to the top of the wave. In the general sine function format, \( y = a \sin(bx + c) \), the amplitude is given by the absolute value, \( |a| \). This means you take whatever number is multiplying the sine function and use its positive form. For the function \( y = \sin(x - \frac{\pi}{2}) \), the coefficient \( a \) is 1. That means our amplitude is simply 1. This shows us that the highest and lowest points the graph reaches are 1 and -1, respectively.

• Amplitude indicates the maximum displacement from the rest position.
• Amplitude of 1 means peaks at 1 and troughs at -1 on the y-axis.

Period

The period of a sine function tells you how long it takes for the function to complete one full cycle. It's about how "wide" the waves are when they complete their path from crest to crest. In general, for the function \( y = a \sin(bx + c) \), the period is calculated by \( \frac{2\pi}{|b|} \). In our example, \( b = 1 \), which simplifies things. Calculating it, we get \( \frac{2\pi}{1} = 2\pi \). Thus, every \( 2\pi \), the sine wave repeats itself. This consistent cycle length is what defines sine and cosine functions.

• The period reflects how frequently the sine waves cycle through.
• A period of \( 2\pi \) means it repeats every full circle of the unit circle.

Phase Shift

Phase shift is a horizontal movement of the entire graph to the left or right. This occurs due to the \( c \) term in our general formula \( y = a \sin(bx + c) \). We calculate it by \( -\frac{c}{b} \). For our specific function \( y = \sin(x - \frac{\pi}{2}) \), \( c = -\frac{\pi}{2} \) and \( b = 1 \). Thus, the phase shift is \( -\frac{-\frac{\pi}{2}}{1} = \frac{\pi}{2} \). This shows that our graph shifts to the right by \( \frac{\pi}{2} \) units. This shift means that point zero on the sine curve aligns with \( \frac{\pi}{2} \).

• Phase shift represents the left or right movement of the graph.
• A shift by \( \frac{\pi}{2} \) units moves it to the right on the x-axis.

Graphing Sine Functions

Graphing sine functions can be simple once you know the key features, such as the amplitude, period, and phase shift. First, recognize the basic shape of \( y = \sin x \), which smoothly oscillates between 1 and -1. You can begin plotting basic points like \((0,0), (\frac{\pi}{2},1), (\pi,0), (\frac{3\pi}{2},-1), (2\pi,0)\). For the function \( y = \sin(x - \frac{\pi}{2}) \), the graph will be adjusted by the phase shift. This means moving all these basic points \( \frac{\pi}{2} \) units to the right. Your key points will now include \( (\frac{\pi}{2},0), (\pi,1), (\frac{3\pi}{2},0), (2\pi, -1), (\frac{5\pi}{2},0) \). Connect these points with a smooth wave that replicates the sine shape.Remember:

• Consider amplitude for the vertical reach of your graph.
• Apply the period to capture how frequently waves repeat.
• Incorporate the phase shift to horizontally align your wave properly.

By combining these three elements, you can effectively graph any sine function, ensuring accuracy in depicting its periodic nature.