Problem 14 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal translations to graph at least two cycles of the given functions. $$f(x)=\sin (x+\pi)$$

Step-by-Step Solution

Answer

The function \(f(x) = \sin (x + \pi)\) is a sine function that has been horizontally translated \(\pi\) units to the left. Two cycles can be graphed starting at \(- \pi\) and ending at \(3\pi\).

Step 1: Understand the basic sine function

The basic sine function has the shape of a wave that oscillates between -1 and 1. It completes one full cycle in a 2\(\pi\) interval starting from 0 to 2\(\pi\). It starts at (0,0), reaches a maximum at \(\pi/2, 1), then comes down to \(\pi, 0\), reaches its minimum at \(3\pi/2, -1), and ends the cycle at \((2\pi, 0).\)

Step 2: Apply the horizontal translation

A positive value inside the parentheses with the \(x\), such as \(x + \pi\), translates the function to the LEFT by the given amount. Therefore, each point of one cycle of the function will move \(\pi\) units to the left.

Step 3: Graph the translated function

Start by graphing one cycle of the sine function, beginning from \(-\pi\) instead of 0, due to the left shift. The points will be, \((- \pi, 0)\), \((- \pi/2, 1)\), \((0, 0)\), \((\pi/2, -1)\), \((\pi, 0)\). Repeat this pattern to graph a second cycle right after the first one.

Key Concepts

Understanding the Sine Function

The sine function, often written as \( f(x) = \sin(x) \), is one of the fundamental trigonometric functions. It creates a smooth, wave-like pattern that we call a sinusoidal wave. The sine wave oscillates between values of -1 and 1. This means whenever you use the sine function, the highest and lowest points on the graph are always these two values.

Key characteristics of the sine wave:

• Amplitude: The height from the centerline to the peak (or trough) is 1 in the basic form \( \sin(x) \).
• Period: The length of one complete cycle is \( 2\pi \). It spans from \( 0 \) to \( 2\pi \).
• Starting Point: Initially, the sine starts at point \( (0, 0) \).

Understanding these features helps us graph functions more effectively, predicting how shifts or changes will influence the wave.

Trigonometric Functions Overview

Trigonometric functions are crucial in mathematics, connecting angles to ratios of a right triangle's sides. The most common trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions are cyclical, repeating their values in predictable patterns called cycles or periods.

Important features of trigonometric functions:

• They are periodic, meaning that they repeat their values at regular intervals. For \( \sin(x) \) and \( \cos(x) \), this interval is every \( 2\pi \).
• A complete cycle of sine and cosine functions creates wave-like patterns that are mirrored horizontally and vertically.
• These functions are defined for all real numbers, and their values range between -1 and 1.

Trigonometric functions are widely used in physics, engineering, and many others, thanks to their ability to model repetitive phenomena.

Graphing Translations of Trigonometric Functions

When graphing trigonometric functions, particularly sine waves, understanding translations is key. Horizontal translations shift the graph left or right. This shift is determined by the number added or subtracted inside the function's parentheses with the variable \( x \).

For example, with the function \( f(x) = \sin(x + \pi) \):

• The '+\( \pi \)' inside the function means the graph of \( \sin(x) \) will move to the LEFT by \( \pi \) units.
• Applying this to the graph, every point of the sine curve shifts left by \( \pi \). This results in the starting point moving from \( (0, 0) \) to \( (-\pi, 0) \).

The translated function will continue to oscillate between -1 and 1, maintaining the original shape but simply shifted horizontally. Understanding how translations work enables one to accurately predict and graph altered trigonometric functions.