Problem 10 - Precalculus Mathematics for Calculus

Question

Graph the function. $$g(x)=2 \sin x$$

Step-by-Step Solution

Answer

The graph is a sine wave with amplitude 2, period \(2\pi\), oscillating between -2 and 2.

Step 1: Identify the Function Type

The given function is a trigonometric function, specifically a sine function, represented as \( g(x) = 2 \sin x \). Recognizing it as a sine function is crucial as the graph has a specific wavelike shape.

Step 2: Determine Amplitude

The amplitude of a sine function \( a \sin x \) is given by the absolute value of \( a \). In this case, \( a = 2 \), so the amplitude is 2.

Step 3: Determine the Period

The period of a sine function \( \sin(bx) \) is calculated as \( \frac{2\pi}{b} \). Here \( b = 1 \), so the period is \( 2\pi \). This means one complete wave occurs over an interval of length \( 2\pi \) on the x-axis.

Step 4: Identify Key Points in One Period

For one period of \( 2\sin x \), key points are at \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). The values of \( g(x) \) at these points are \( 0, 2, 0, -2, 0 \) respectively.

Step 5: Sketch the Graph

Start plotting points using key points from Step 4 on a coordinate plane. Connect these points smoothly to form the wave of the sine function. Repeat this pattern for further cycles both to the left and right of the initial period.

Key Concepts

Amplitude

In the context of trigonometric functions, particularly the sine function, the amplitude plays a key role in understanding how the graph behaves. The "amplitude" refers to the maximum distance that the graph of the function stretches above and below the horizontal axis, which is also known as the center line or equilibrium position of the sine wave.

For a function such as \( g(x) = a \sin x \), the amplitude is determined by the coefficient \( a \) in front of the sine function. Specifically, the amplitude is the absolute value of \( a \), denoted by \(|a|\). In our example \( g(x) = 2 \sin x \), \( a = 2 \), thus the amplitude is 2. This means the wave will reach a high point of 2 and a low point of -2.

It is important to note that the amplitude only tells us about the vertical stretch of the graph; it does not affect the period or horizontal stretch of the wave.

Period of Sine Function

The period of a sine function is crucial for knowing how frequently the wave pattern repeats along the x-axis. It essentially gives us the length of one complete cycle of the wave.

In a general sine function given by \( \sin(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \). Here, \( b \) represents the coefficient of \( x \), which affects the horizontal stretching or compression.

In the specific case of our function \( g(x) = 2 \sin x \), the coefficient \( b \) is 1. Thus, the period of this function is simply \( 2\pi \). This tells us that the wave completes one full cycle from start to finish every \( 2\pi \) units along the x-axis. As a result, the same pattern of peaks and troughs will repeat, allowing us to predict the behavior and structure of the graph over longer stretches of the x-axis.

Key Points of Sine Function

Identifying the key points in one complete period of a sine wave is essential for sketching the graph accurately. These key points correspond to the positions along the x-axis where the function reaches significant values such as maximum, minimum, and intercepts.

For our function \( g(x) = 2 \sin x \), within the period \( 0 \) to \( 2\pi \), the primary key points are:

• At \( x = 0 \), the function is \( g(x) = 0 \).
• At \( x = \frac{\pi}{2} \), the function reaches its maximum value, \( g(x) = 2 \).
• At \( x = \pi \), it crosses back through zero, \( g(x) = 0 \).
• At \( x = \frac{3\pi}{2} \), the function reaches its minimum value, \( g(x) = -2 \).
• At \( x = 2\pi \), it returns to zero, \( g(x) = 0 \).

Connecting these points with a smooth, wavy line will produce the graph of the function for one complete cycle. Understanding these key points is essential for accurately continuing the pattern over more than one cycle, both forward and backward along the x-axis.