Problem 14 - Algebra and Trigonometry

Question

\(3-16\) Graph the function. $$ g(x)=4-2 \sin x $$

Step-by-Step Solution

Answer

Graph the function with amplitude 2, vertical shift 4, and reflection.

Step 1: Understand the Function

The function given is a transformation of the basic sine function. The general form of the function is given by \( g(x) = 4 - 2 \sin x \). Here, the sine function \( \sin x \) has been modified by multiplying by -2 and adding 4.

Step 2: Identify the Transformations

- The amplitude of the function is \( |A| = | -2 | = 2 \).- The vertical shift is upwards by 4 units (from \(0\) to \(4\)).- The negative sign in front of 2 indicates a vertical reflection of the sine curve.These transformations impact the graph's appearance.

Step 3: Determine Key Points

To correctly form the graph, determine key points:- At \( x = 0 \): \( g(0) = 4 - 2\sin(0) = 4 \)- At \( x = \frac{\pi}{2} \): \( g(\frac{\pi}{2}) = 4 - 2\sin(\frac{\pi}{2}) = 2 \)- At \( x = \pi \): \( g(\pi) = 4 - 2\sin(\pi) = 4 \)- At \( x = \frac{3\pi}{2} \): \( g(\frac{3\pi}{2}) = 4 - 2\sin(\frac{3\pi}{2}) = 6 \)- At \( x = 2\pi \): \( g(2\pi) = 4 - 2\sin(2\pi) = 4 \)

Step 4: Plot the Key Points

Using the calculated key points, plot them on a coordinate plane:- \((0, 4)\)- \(( \frac{\pi}{2}, 2)\)- \(( \pi, 4)\)- \(( \frac{3\pi}{2}, 6)\)- \(( 2\pi, 4)\)These points demonstrate one full cycle of the sine function transformation.

Step 5: Draw the Curve

With the points plotted, draw a smooth curve connecting the points, making sure to reflect the repeated wave pattern of the sine function.- The graph will oscillate between \(2\) and \(6\), centered around \(y = 4\).- Remember the curve should start at \((0, 4)\), go through the minimum at \((\frac{\pi}{2}, 2)\), back up to \(( \pi, 4)\), then maximum at \(( \frac{3\pi}{2}, 6)\), and finish one cycle at \((2\pi, 4)\).

Key Concepts

Sine Function Transformation

When graphing trigonometric functions, it's crucial to understand how transformations affect the sine function. The basic sine function, given by \( y = \sin x \), has a wave-like pattern, oscillating between 1 and -1. Transformations allow us to adjust its appearance in various ways.

In this exercise, the function \( g(x) = 4 - 2 \sin x \) transforms the basic sine curve. There are two main changes: a vertical shift and a reflection. Such modifications can stretch, compress, or move the wave without altering its periodic nature. Identifying each transformation will help in plotting the graph accurately.

Amplitude

Amplitude refers to the height of the wave from the center line to its peak. For a standard sine function \( \sin x \), the amplitude is 1.

In the function \( g(x) = 4 - 2 \sin x \), the coefficient in front of the sine function is \(-2\). The amplitude is the absolute value of this coefficient, which is \( | -2 | = 2 \). This indicates that the wave will rise 2 units above and fall 2 units below its central axis.

• The peak reaches 6 (4 + 2)
• The trough goes down to 2 (4 - 2)

Amplitude controls the wave's "tallness," and larger values stretch the wave vertically.

Vertical Shift

The vertical shift of a trigonometric function moves the graph up or down on the Cartesian plane. For the function \( g(x) = 4 - 2 \sin x \), the constant term "4" signifies a vertical shift.

• The graph is shifted 4 units upwards. This means every point on the sine wave from the horizontal axis (y=0) is moved to y=4.
• The central axis, originally y=0 for \( \sin x \), becomes y=4.

This shift is crucial for determining the midline of the graph, around which the wave oscillates. Moving the center line changes the overall position of the graph.

Reflection of Sine Curve

Reflection involves flipping the graph over a specific axis, altering its orientation. In \( g(x) = 4 - 2 \sin x \), the negative sign before the 2 reflects the sine curve about the center line.

This results in:

• Turning what would have been peaks into valleys and vice versa.
• Preserving the amplitude and vertical shift, while reversing the direction of the wave's oscillation.

The reflection means that as \( \sin x \) would typically rise, the function \(-2 \sin x\) instead dips. Consequently, understanding this transformation aids in correctly sketching the wave's lifecycle from peak to trough.