Problem 46 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Graph the function. \(g(x)=-\cos (x+\pi)-2\)

Step-by-Step Solution

Answer

The graph of the function \(g(x) = - \cos(x+ \pi) -2\) is an upside-down cosine wave that has been shifted \(\pi\) units to the left and 2 units downward.

Step 1: Understand the original function

The base function is the cosine function (\(\cos(x)\)), which has a range of -1 to 1 and creates a periodic wave-like pattern. The period is \(2\pi\). The cosine function starts at 1 at \(x=0\), then decreases to -1 at \(x=\pi\), and then goes back to 1 at \(x=2\pi\).

Step 2: Understand the transformations

The given function \(g(x)\) has two transformations applied to the cosine function. The \(x+ \pi\) term inside the cosine function means that the function is shifted \(\pi\) units to the left. The '-' sign before cosine flips the wave along the x-axis, this is also known as reflection. Finally, the '-2' at the end moves the entire function down by 2 units, this is a vertical shift.

Step 3: Draw the graph

Start by drawing the graph of \(g(x) = - \cos(x)\). Reflect the entire graph of \(\cos(x)\) over the x-axis for this. The next step is to move the result \(\pi\) units to the left, and then slide it down by 2 units. The resulting graph will still look like a wave, but it will be upside-down compared to the original cosine function, its peaks and troughs are moved to the left, and all its points are 2 units lower.

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions can feel like unraveling puzzles, as they involve complex transformations. Understanding how these transformations affect the graph is crucial. When graphing, you often begin with the basic function, like \(\cos(x)\), which is likely familiar. This cosine function has certain characteristics:

• Amplitude: The "height" of the wave from the center line is 1.
• Period: The length for one complete cycle to repeat is \(2\pi\).
• Starting Point: Begins at its maximum value, 1, when \(x = 0\).

By consistently applying changes to these basic features, you can vividly see how transformations like shifts and reflections modify the trigonometric graph pattern. Ensuring each step clear enables drawing complex graphs accurately.

Cosine Function

At its core, the cosine function, \(\cos(x)\), traces a smooth wavy path that swings between 1 and -1. This is an even, periodic function characterized by its symmetry and elegance. Consider the following features:

• Even Function: Symmetrical about the y-axis, i.e., \(\cos(-x) = \cos(x)\).
• Key Values: Peaks at \(x = 0, 2\pi, 4\pi, \ldots\), troughs at \(\pi, 3\pi, 5\pi, \ldots\).
• Repeating Pattern: Every \(2\pi\) units, the function completes a full cycle and repeats.

By knowing its structure, any further transformations can be more precisely interpreted and executed. Recognize how each change affects these innate qualities of the cosine wave.

Phase Shift

The term "phase shift" refers to horizontal shifts in a trigonometric graph. It adjusts the start of the wave along the x-axis. For the equation \(g(x) = -\cos(x+\pi) - 2\), the \(x + \pi\) inside the cosine function signifies a leftward shift of \(\pi\) units. Imagine pulling the whole wave left, which recompenses the usual angle positions:

• A phase shift left indicates moving the wave opposite to the positive x-direction.
• Each point on \(\cos(x)\) shifts left by \(\pi\), altering when peaks and valleys occur.

This transformation ensures the original cosine breath smoothly transitions to a new angular start. Grasping phase shifts aids in predicting the new locations of peaks and troughs promptly.

Reflection

Reflection is a crucial transformation that flips the graph over an axis. In the function \(g(x) = -\cos(x+\pi) - 2\), the negative sign before cosine indicates reflection across the x-axis. What does this mean in practical terms?

• Up and down flip: Peaks turn into troughs, and troughs flip to peaks.
• Visual effect: It inversely mirrors the original cosine wave.
• Transition points (like crossing the x-axis) remain fixed.

This reflection fundamentally alters the graph's orientation, making it appear "upside-down." Understanding this helps you avoid common mistakes when predicting graph behaviors.

Vertical Shift

A vertical shift involves moving the entire trigonometric function up or down along the y-axis. In \(g(x) = -\cos(x+\pi) - 2\), the \(-2\) at the end lowers the whole graph by two units. This type of transformation effectively changes the baseline of the wave.

• Every point on the graph moves down two units.
• Midline shift: The graph's centerline, usually at y=0, drops to y=-2.
• Amplitude remains unaffected, only the baseline shifts.

These changes result in the graph maintaining its wave shape but adjusted downwards. Recognizing vertical shifts is essential in analyzing modified trigonometric functions thoroughly.