Problem 37 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=\sin \left(x+\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Sketch the sine wave starting at \((-\frac{\pi}{2}, 0)\), passing through \( (0,1)\), and ending at \((\frac{3\pi}{2},0)\).

Step 1: Identify the Standard Sine Function

The standard sine function is given by \( y = \sin(x) \). It has a period of \( 2\pi \), an amplitude of 1, and oscillates between -1 and 1. The graph starts at \( (0,0) \), rises to \( (\frac{\pi}{2},1) \), returns to \( (\pi,0) \), goes to \( (\frac{3\pi}{2},-1) \), and completes the cycle at \( (2\pi,0) \).

Step 2: Determine the Phase Shift

The given function is \( y = \sin\left(x + \frac{\pi}{2}\right) \). The term \( \frac{\pi}{2} \) inside the sine function indicates a phase shift. Phase shifts occur in the opposite direction of the sign inside the parentheses, so \( +\frac{\pi}{2} \) represents a shift to the left by \( \frac{\pi}{2} \) units.

Step 3: Apply the Phase Shift to Standard Points

Apply the phase shift of \( -\frac{\pi}{2} \) to each key point of the standard sine cycle. Original points: \( (0,0), (\frac{\pi}{2},1), (\pi,0), (\frac{3\pi}{2},-1), (2\pi,0) \). New points after shift: \( (-\frac{\pi}{2},0), (0,1), (\frac{\pi}{2},0), (\frac{\pi}{1},-1), (\frac{3\pi}{2},0) \).

Step 4: Sketch One Cycle of the Shifted Function

Start plotting the shifted points: begin at \( (-\frac{\pi}{2},0) \), rise to \( (0,1) \), return to \( (\frac{\pi}{2},0) \), decrease to \( (\pi,-1) \), and complete the cycle at \( (\frac{3\pi}{2},0) \). Draw a smooth curve through these points to finalize the graph, ensuring symmetry around the origin and continuity of waves.

Key Concepts

Period

The period of a trigonometric function refers to the length of one complete cycle of the wave. For the basic sine function, this is the distance over which the wave begins to repeat itself.
The standard sine function, denoted as \( y = \sin(x) \), has a period of \( 2\pi \). This means that every \( 2\pi \) units along the x-axis, the sine wave starts a new cycle.
For functions of the form \( y = \sin(bx) \), the period is calculated using the formula:

• Period = \( \frac{2\pi}{b} \)

In our exercise, the function \( y = \sin(x + \frac{\pi}{2}) \) has no coefficient \( b \) other than 1 in front of \( x \), so the period remains \( 2\pi \). Remember that the key to understanding periods is realizing how they help in determining how frequently the wave patterns occur along the graph.

Amplitude

Amplitude in trigonometric functions, such as the sine function, refers to the wave's height from the middle line (axis) to its peak. It indicates the range of the function's oscillations.
For the basic sine function, \( y = \sin(x) \), the amplitude is always 1. This is because the sine function peaks at 1 and troughs at -1 in its standard form, meaning it moves 1 unit above and 1 unit below the horizontal axis.
If a sine function is modified to \( y = A \cdot \sin(x) \), the amplitude becomes \( |A| \). In our case, since the function provided in the exercise is \( y = \sin(x + \frac{\pi}{2}) \), the amplitude remains 1.
Grasping the concept of amplitude is crucial for understanding the vertical stretch or compression of trigonometric graphs.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric graph along the x-axis. If you add or subtract a constant from the variable \( x \) in a sine function, it causes a phase shift.
The function in the exercise is \( y = \sin(x + \frac{\pi}{2}) \). Here, the \( \frac{\pi}{2} \) added to \( x \) causes the graph to shift horizontally. Specifically, a positive addition inside the sine function, like \( x + \frac{\pi}{2} \), implies a shift to the left. Conversely, a subtraction (e.g., \( x - \frac{\pi}{2} \)) would shift the graph to the right.

• Positive phase shift: leftward movement
• Negative phase shift: rightward movement

For the given graph, the phase shift is to the left by \( \frac{\pi}{2} \) units. Understanding phase shift is essential for accurately predicting where the graph of the function starts its cycle along the x-axis.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting these functions based on their key characteristics such as period, amplitude, and phase shift. It's about translating mathematical descriptions into visual graphs.
To plot a trigonometric function like the one in our exercise, \( y = \sin(x + \frac{\pi}{2}) \), we consider each of the transformations:

• Start with the standard sine curve that begins at the origin for \( y = \sin(x) \).
• Apply a phase shift of \( \frac{\pi}{2} \) units to the left, adjusting each of the critical points such as the intercepts, peaks, and troughs.
• Check amplitude, which remains 1, indicating no vertical stretch or compression.
• Complete the cycle at \( 2\pi \), as the period isn’t altered.

This function's graph will begin at \( -\frac{\pi}{2} \) due to the phase shift, rise to a peak at 0, and continue through \( \frac{3\pi}{2} \). Ensuring a smooth curve, symmetrical and continuous, aids in clear visualization. Such visual graphing helps students understand the dynamics of trigonometric functions in a concrete manner.