Problem 12 - Algebra and Trigonometry

Question

Graph the function. $$ g(x)=4-2 \sin x $$

Step-by-Step Solution

Answer

Graph \(g(x) = 4 - 2 \sin x\) with a vertical shift, dilation, and a changed midline at \(y = 4\).

Step 1: Identify the Base Sine Function

The given function is based on the sine function. Let's start by identifying the base function. In this case, it is \( \sin x \). The sine function oscillates between -1 and 1 and completes a full cycle over the interval \( [0, 2\pi] \).

Step 2: Apply the Vertical Dilation

The function \( -2\sin x \) involves a vertical dilation of the sine wave by a factor of 2, meaning each point of \( \sin x \) is stretched away from the x-axis by two units and then flipped over the x-axis. This modifies the range of \( \sin x \) from \([-1, 1]\) to \([-2, 2]\).

Step 3: Apply the Vertical Shift

Now, consider the vertical shift. The function is \(4 - 2\sin x\). This shifts the entire graph of \(-2\sin x\) up by 4 units. As a result, the midline of the sine wave is now \(y = 4\) and the range becomes \([2, 6]\).

Step 4: Plot Key Points and Sketch the Graph

Identify key points within one period (e.g., 0, \(\pi/2\), \(\pi\), \(3\pi/2\), \(2\pi\)) and calculate \(g(x)\) values to get: \(g(0)=4\), \(g(\pi/2) = 4\), \(g(\pi) = 6\), \(g(3\pi/2) = 4\), \(g(2\pi) = 2\). Plot these points and sketch the curve which will be a sine wave that has been flipped, vertically stretched and shifted.

Key Concepts

Sine Function Transformation

Understanding sine function transformations is essential when graphing trigonometric functions. The basic sine function is represented as \( \sin x \) and it has distinct characteristics:

• It oscillates between -1 and 1.
• A full cycle is completed over the interval \([0, 2\pi])\).
  

A transformation involves changing the sine function in some manner, such as stretching, compressing, or shifting its graph. For example, when the function includes a coefficient, such as \(-2\sin x\), this indicates additional transformations taking place. Each transformation affects the shape and position of the sine curve on the coordinate plane.

Vertical Shift

The concept of vertical shift involves moving a function up or down along the y-axis. In the context of the sine function, a vertical shift can change the midline of the wave pattern.
When applying a vertical shift, we focus on the constant added or subtracted from the sine function. For example, in the function \( g(x) = 4 - 2\sin x \), the subtraction of the sine term indicates an upward shift. This shift changes the midline from the original line at \( y = 0 \) to \( y = 4 \). As a result, the entire graph is translated four units upwards. The minimum and maximum values are also adjusted, with the range now extending over \([2, 6]\).

• The midline serves as the center of oscillation for the graph after shifting.
• Vertical shifts do not affect the amplitude or frequency, which are associated with other types of transformations.

Vertical Dilation

Vertical dilation influences the amplitude, or height, of a sine wave. This transformation is dictated by a coefficient in front of the sine function. In the function \(-2\sin x\), the coefficient of -2 indicates two operations:

• First, the magnitude of 2 suggests a vertical stretch. Originally, \( \sin x \) varies from -1 to 1, but multiplying by 2 stretches this range to \([-2, 2]\).
• The negative sign implies a reflection over the x-axis, flipping the graph upside down.

These adjustments alter how the graph peaks and troughs within each period, rather than affecting its horizontal span. Understanding vertical dilation helps in predicting the wave pattern's appearance after these adjustments.