Problem 31 - Contemporary Precalculus

Question

Sketch a complete graph of the function. $$h(t)=3 \sin (2 t+\pi / 2)$$

Step-by-Step Solution

Answer

Answer: The amplitude is 3, the period is \(\pi\), the phase shift is \(\frac{\pi}{4}\) to the left, and the vertical shift is 0.

Step 1: Identify the Amplitude

The amplitude of a sine function is the absolute value of the coefficient of the sine function. In this case, the coefficient is 3, so the amplitude is \(|3|=3\). This means that the graph will oscillate between 3 and -3.

Step 2: Identify the Period

The period of a sine function is the length of one complete cycle of the graph. The coefficient of t in the argument of the sine function affects the period of the graph. In this case, the coefficient is 2, so the period is given by: $$P = \frac{2\pi}{|2|} = \pi$$ This means that the graph will complete one full cycle in the interval \([0, \pi]\).

Step 3: Identify the Phase Shift

The phase shift is the horizontal shift of the graph. If the argument of the sine function is \((2t + \phi)\), then the phase shift is given by: $$\frac{\phi}{2}$$ In this case, the phase shift is \(\frac{\pi / 2}{2} = \frac{\pi}{4}\). The graph will shift to the left by \(\frac{\pi}{4}\).

Step 4: Identify the Vertical Shift

Since there is no constant term added or subtracted from the sine function, the vertical shift is 0. This means that the graph is not shifted up or down.

Step 5: Sketch the Graph

Using the properties found in the previous steps, we can sketch the graph of the function \(h(t) = 3\sin(2t+\pi/2)\): 1. The amplitude is 3, so the graph will oscillate between 3 and -3. 2. The period is \(\pi\), so the graph will complete one full cycle in the interval \([0, \pi]\). 3. The phase shift is \(\frac{\pi}{4}\), so the graph will shift to the left by \(\frac{\pi}{4}\). 4. The vertical shift is 0, so the graph is not shifted up or down. With these properties, we can now sketch the graph by plotting the key points (maximum, minimum, intercepts) and connecting them smoothly. Since the sine function starts from 0, it will be shifted to the left by \(\frac{\pi}{4}\) and then oscillate between 3 and -3 with a period of \(\pi\).

Key Concepts

Understanding Amplitude

Amplitude represents the height of the peaks and the depths of the troughs in a trigonometric function like sine. For the function \( h(t) = 3 \sin(2t + \pi/2) \), the amplitude is the coefficient of the sine function, which is 3.
This means:

• The graph will reach as high as 3 and as low as -3.
• Amplitude is always expressed as a positive value, representing distance from the central axis.

This helps in predicting how much the waveform will move above and below its central position.

Period of a Function Explained

The period of a sine function refers to the distance required for the graph to complete one full cycle. Understanding the period lets you know how often the waveform repeats.

In the function \( h(t) = 3 \sin(2t + \pi/2) \), the coefficient 2 affects the period.

• Use the formula \( P = \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( t \).
• Here, \( P = \frac{2\pi}{2} = \pi \).

This means the sine wave will repeat every \( \pi \) units along the horizontal axis, compressing the typical sine wave which generally completes a cycle in \( 2\pi \) units.

Grasping Phase Shift

Phase shift describes the horizontal movement of the graph along the x-axis. It tells us how the starting point of the sine wave shifts from its usual position.

In \( h(t) = 3 \sin(2t + \pi/2) \), the phase shift is determined by the equation inside the sine function:

• The general form \( (bt + \phi) \) determines the phase shift using \( \frac{\phi}{b} \).
• In this case, \( \frac{\pi/2}{2} = \frac{\pi}{4} \).

Thus, the graph will move left by \( \frac{\pi}{4} \), altering where the sine wave begins its cycle.

Exploring the Sine Function

The sine function is a fundamental trigonometric function forming smooth, wave-like graphs. This function is periodic and symmetric.

For the equation \( h(t) = 3 \sin(2t + \pi/2) \), the sine function:

• Starts at 0, but shifts in this case due to the phase shift.
• Has a regular shape with peaks and troughs defined by the amplitude.
• Completes its cycles as determined by the period.

The smooth oscillation of the sine wave between its maximum and minimum values makes it a useful tool in modeling many natural phenomena.