Problem 10 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of vertical translations to graph at least two cycles of the given functions. $$f(x)=\sin x-2$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\sin x-2\) will be a sine curve starting from the point (0,-2), going to point (1, \(\pi/2 - 2\)), then to (0, \(\pi-2\)), next to (-1, \(3\pi/2 - 2\)), and finally returning to (0, \(2\pi-2\)). This pattern is repeated for all subsequent cycles.

Step 1: Understanding the basic sine function

The basic sine function, \(\sin x\), has a period of \(2\pi\) and oscillates between -1 and 1 on the y-axis. It starts from the origin (0,0), goes up to (1, \(\pi/2\)), then back to (0, \(\pi\)), down again to (-1, \(3\pi/2\)), and finally back to (0, \(2\pi\)) to complete one cycle.

Step 2: Applying a vertical shift to the sine function

Given function \(f(x)=\sin x-2\) is the sine function shifted down by 2 units. This means every y-coordinate of the original sine function will be subtracted by 2. Hence, the graph starts from (0,-2), goes up to (1, \(\pi/2 - 2\)), then back to (0, \(\pi-2\)), down again to (-1, \(3\pi/2 - 2\)), and finally back to (0, \(2\pi-2\)) to complete one cycle.

Step 3: Sketching the graph

Now, sketch the graph of the function \(f(x)=\sin x-2\). Start by marking the points identified in step 2 on the x-y plane, then connect these points smoothly to draw at least two cycles of this translated sine function.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in geometry and calculus. These functions describe the relationships between the angles and sides of triangles. The most common trigonometric functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)).

These functions are periodic and repeat values in regular intervals. For instance, the sine and cosine functions have a period of \( 2\pi \), meaning their values repeat every \( 2\pi \) radians. This periodic nature makes them extremely useful for modeling phenomena such as waves, sound, and light.

Additionally, trigonometric functions have specific characteristics such as amplitude, period, and vertical shifts. Amplitude refers to the height of the wave from the centerline to the peak. The period is the distance between repeating patterns. Vertical shifts occur when the entire graph is moved up or down on the y-axis, which we'll explore further in graph transformations.

Exploring the Sine Function

The sine function is one of the primary trigonometric functions. It is defined for a real variable \( x \) as \( f(x) = \sin x \). The graph of the sine function is a smooth and continuous wave-like pattern that oscillates above and below the x-axis.

Key characteristics of the sine function include:

• Amplitude: The maximum vertical distance from the x-axis, typically 1 for the basic sine function.
• Period: A complete cycle of the sine wave occurs over \( 2\pi \).
• Range: The sine function values fluctuate between -1 and 1.

To visualize the sine function, consider its critical points within one cycle:

• Starts at origin: \( (0, 0) \).
• Rises to a peak: \( (\pi/2, 1) \).
• Returns to center: \( (\pi, 0) \).
• Falls to a trough: \( (3\pi/2, -1) \).
• Completes cycle: \( (2\pi, 0) \).

These points illustrate the repetitive nature of the sine function.The sine function is versatile and can be adjusted to fit different real-world contexts through various transformations.

Graph Transformations: Vertical Translations

Graph transformations involve altering the position or shape of a graph on the coordinate plane. Vertical translations are a type of transformation where the graph of a function is shifted up or down without changing its shape.

For example, in the given function \( f(x) = \sin x - 2 \), the sine function \( \sin x \) is shifted vertically downward by 2 units. This means every point on the graph of \( \sin x \) is moved 2 units lower along the y-axis.

This vertical shift affects the sine wave as follows:

• Origin point moves from \( (0, 0) \) to \( (0, -2) \).
• Peak moves from \( (\pi/2, 1) \) to \( (\pi/2, -1) \).
• Center returns to \( (\pi, -2) \).
• Trough moves from \( (3\pi/2, -1) \) to \( (3\pi/2, -3) \).

This vertical translation results in the entire graph moving down, altering its minimum and maximum values but retaining the same periodic pattern. Understanding these transformations is key for graphing modified functions accurately.