Problem 37 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Describe the transformation of the graph of \(f\) represented by the function \(g\). \(f(x)=\cos x, g(x)=2 \cos \left(x-\frac{\pi}{2}\right)+1\)

Step-by-Step Solution

Answer

The function \(g(x)\) represents a cosine wave that is vertically stretched by a factor of 2, shifted to the right by \(\frac{\pi}{2}\) units, and translated upwards by 1 unit.

Step 1: Identify the Original Function

The original function given is \(f(x)=\cos x\), which represents a cosine wave. It completes a full cycle over the interval \([0, 2\pi]\) and has an amplitude of 1.

Step 2: Recognize the Transformation

The function \(g(x)=2 \cos \left(x -\frac{\pi}{2}\right)+1') is derived from the function \(f(x)=\cos x\). The number 2 stretches the graph vertically. The term \(-\frac{\pi}{2}\) inside the cosine function shifts the graph to the right by \(\frac{\pi}{2}\) units, while the +1 outside the cosine function vertically translates the graph upwards by 1 unit.

Step 3: Describe the Transformed Function

The function \(g(x)\) is therefore a cosine wave, just like the original function \(f(x)\), that is vertically stretched by a factor of 2, shifted to the right by \(\frac{\pi}{2}\) units, and translated upwards by 1 unit. The function also completes a full cycle over the interval \([0, 2\pi]\), but now the amplitude is 2, the wave is displaced to the right and is translated 1 unit higher than the original cosine wave.

Key Concepts

Cosine Function

The cosine function, denoted as \(f(x) = \cos x\), represents a periodic wave that oscillates between -1 and 1. It is one of the core trigonometric functions often used to describe oscillations, waves, and circular motion.
The basic cosine wave completes a full cycle over the interval \([0, 2\pi]\). During this interval, it starts from 1, decreases to -1, and returns to 1.
Key features of the cosine function include:

• Amplitude: This is the height from the center line to the peak (or trough). The standard cosine function has an amplitude of 1.
• Period: The length of one full cycle of the wave, which is \(2\pi\) for the basic cosine function.
• Symmetry: The cosine function is an even function, meaning that \(\cos(-x) = \cos(x)\).

Understanding the basic shape and properties of the cosine function is essential before applying transformations like stretch, shift, or translation.

Vertical Stretch

A vertical stretch in a trigonometric function alters the amplitude, effectively making the peaks and troughs further from the midline of the function.
In the function \(g(x) = 2 \cos(x - \frac{\pi}{2}) + 1\), the coefficient "2" in front of the cosine function indicates a vertical stretch. This changes the amplitude from 1 to 2.
Here's what that means:

• The graph is pulled vertically away from the x-axis.
• Every y-value from the original \(f(x) = \cos x\) is multiplied by 2, resulting in taller peaks and deeper troughs.

This transformation does not affect the period of the wave; it still completes a full cycle over \([0, 2\pi]\) but with greater vertical reach.

Phase Shift

A phase shift refers to a horizontal movement of a trigonometric function along the x-axis. It determines the starting point of the function's cycle.
In \(g(x) = 2 \cos(x - \frac{\pi}{2}) + 1\), the term \(-\frac{\pi}{2}\) inside the cosine function signifies a phase shift. This shifts the entire graph to the right by \(\frac{\pi}{2}\) units. Let's break this down:

• Shifting rightwards implies the wave starts its cycle further along the x-axis.
• It means the cosine wave's peaks and valleys are delayed compared to the original function.

Despite the phase shift, the shape and size of the cycles remain unchanged except for the horizontal displacement.

Vertical Translation

Vertical translation moves the graph up or down on the y-axis without changing its shape. It's like sliding the entire graph vertically.
In our transformed function \(g(x) = 2 \cos(x - \frac{\pi}{2}) + 1\), the "+1" represents a vertical translation. This shifts the graph up by 1 unit. Here's what this modification means:

• All y-values of the graph increase by 1.
• The midline of the cosine wave moves from y = 0 to y = 1.
• As a result, every feature of the wave, including peaks and troughs, is also shifted upward by 1 unit.

Vertical translation does not affect the amplitude or the period; it simply repositions the wave higher on the plane.