Problem 26 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Graph the function. \(g(x)=\cos x-4\)

Step-by-Step Solution

Answer

The graph of the function \(g(x) = \cos x - 4\) is derived from the standard cosine function and is moved 4 units downward on account of the 'minus 4' in the equation. The resulting graph has the same shape as the original cosine wave, but it is entirely below the x-axis due to the downward translation.

Step 1: Basic Cosine Graph

First, draw the basic cosine graph \(y = cos x\). This function has a period of \(2\pi\). Therefore, the function produces a complete wave within this range. The peak value is 1 and the lowest value is -1. The wave is symmetrical about the x-axis.

Step 2: Downward Shift

Next, shift the graph of \(y = cos x\) down by 4 units to yield \(g(x) = \cos x - 4\). This means that the highest value becomes -3 and the lowest value becomes -5.

Step 3: Draw Final Graph

Finally, draw the graph of \(g(x) = \cos x - 4\) based on the original cosine function and the downward shift. The graph will have the same shape as the cosine function, but it will lie entirely below the x-axis. The peaks and valleys of the cosine wave will correspond to the values of -3 and -5, respectively.

Key Concepts

Graphing Functions

Graphing functions is a way of visually representing a mathematical function. It involves plotting points on a coordinate plane to show the shape and behavior of a function. For trigonometric functions like the cosine function, these graphs can help us see the periodic nature of the function.

• Each point on the graph corresponds to an input-output pair, or ( x, y ).
• The x-axis represents the input values, usually measured in radians for trigonometric functions.
• The y-axis represents the output of the function.

For a function like \( g(x) = \cos x - 4 \), first sketch the basic curve of the cosine. This helps identify transformations applied to it. By steadily and systematically graphing, you'll observe and understand how different operations, such as shifts or stretches, affect the graph.

Cosine Function

The cosine function, denoted as \( y = \cos x \), is a fundamental trigonometric function that maps angles to values between -1 and 1. Its graph is a smooth, periodic wave-like curve.

• Period: The cosine function has a period of \(2\pi\), meaning it repeats every \(2\pi\) units.
• Amplitude: This is the height from the center line to the peak or trough of the wave. For \( \cos x \), the amplitude is 1.
• Symmetry: Its graph is symmetric with respect to the y-axis.

When exploring functions like \( g(x) = \cos x - 4 \), it is essential to start with the basic cosine graph. This understanding helps predict changes due to modifications in the function, such as shifts or stretches. Recognizing these standard properties makes it easier to handle transformations.

Vertical Translations

Vertical translations refer to moving a graph up or down in the coordinate plane. This transformation translates every point of the function vertically by a specified amount.

• When a constant is subtracted from the function, \( y = f(x) - c \), the graph shifts downward by \( c \) units.
• When a constant is added, \( y = f(x) + c \), the graph shifts upward by \( c \) units.

For the function \( g(x) = \cos x - 4 \), we shift the basic cosine graph down by 4 units. This affects the range of the function, shifting the maximum and minimum values. Instead of oscillating between 1 and -1, the graph of \( g(x) \) oscillates between -3 and -5.This translation doesn't change the shape or period of the cosine wave, but it alters its vertical position. Understanding vertical translations allows you to accurately graph functions that start from a known shape, such as the cosine function, and adjust its vertical position appropriately.