Problem 32 - Calculus for Biology and Medicine

Question

Explain how the following functions can be obtained from \(y=\cos x\) by basic transformations: (a) \(y=1+2 \cos x\) (b) \(y=-\cos \left(x+\frac{\pi}{4}\right)\) (c) \(y=-\cos \left(\frac{\pi}{2}-x\right)\)

Step-by-Step Solution

Answer

(a) Vertical shift up 1, and stretch by 2; (b) Reflect over x-axis, shift left by \(\frac{\pi}{4}\); (c) Reflect \(\sin x\) over x-axis.

Step 1: Identify Vertical Transformation in (a)

The function is given by \(y = 1 + 2\cos x\). Start by identifying the transformation applied to \(y = \cos x\). Here, the function is vertically shifted and vertically stretched. The term \(+1\) indicates the graph is shifted upwards by 1 unit. The term \(2\cdot \cos x\) means the amplitude of the cosine function is doubled, stretching it vertically by a factor of 2.

Step 2: Identify Transformation in (b)

The function in part (b) is \(y = -\cos\left(x+\frac{\pi}{4}\right)\). First, the negative sign in front of the cosine indicates the graph is reflected over the x-axis. Next, the term \(x+\frac{\pi}{4}\) indicates the graph is horizontally shifted to the left by \(\frac{\pi}{4}\) units.

Step 3: Convert into Known Form for (c)

The function given is \(y = -\cos \left(\frac{\pi}{2}-x\right)\). This can be rewritten by using the co-function identity \( \cos(\frac{\pi}{2} - x) = \sin x \). Thus, the function transforms to \(y = -\sin x\), which indicates a reflection over the x-axis compared to \(y = \sin x\).

Key Concepts

Vertical Transformations

Vertical transformations involve moving the graph of a function up or down on the coordinate plane. When dealing with trigonometric functions like cosine or sine, a vertical transformation can be identified by constants added or multiplied outside the function.
For example, in the function \(y = 1 + 2\cos x\), two vertical transformations occur:

• The constant \(+1\) shifts the entire graph of the function up by 1 unit. This is because every value of the function \(y = \cos x\) increases by 1, lifting the graph vertically.
• Furthermore, the coefficient \(2\) in front of \(\cos x\) scales the amplitude. This stretches the graph vertically by a factor of 2, making peaks taller and troughs deeper compared to the standard cosine graph.

Take these simple steps to recognize vertical transformations: Look at the constants directly modifying your trigonometric function and note their effects on the graph in terms of shifts and stretches.

Horizontal Transformations

Horizontal transformations involve shifting the graph of a function left or right along the x-axis. In trigonometric functions, these shifts are often introduced inside the function's argument.
Consider the function \(y = -\cos(x + \frac{\pi}{4})\):

• The term \((x + \frac{\pi}{4})\) indicates a shift to the left by \(\frac{\pi}{4}\) units. This happens because the regular position of the cosine curve is moved backward along the x-axis.

Subtle changes to the input of trigonometric functions translate into horizontal movements. Recognize a horizontal shift by changes within the parentheses and remember that adding means moving left, while subtracting means moving right. Understanding this type of transformation is crucial, especially as it can interact with other transformations such as reflections, which we see in the negative sign transforming \( -\cos(x + \frac{\pi}{4})\).

Amplitude Changes

Amplitude changes describe how the height of a wave is altered in trigonometric functions. The amplitude is the distance from the midline of the graph to its peak. In equations, this is represented by the coefficient multiplying the trigonometric function — for example, the coefficient \(a\) in \(a\cos x\) or \(a\sin x\).
Let's revisit \(y = 1 + 2\cos x\):

• The amplitude of \(\cos x\) is normally 1. In \(2\cos x\), the amplitude doubles to 2. This results in peaks and troughs that are twice as far from the midline compared to \(\cos x\).

Amplitude directly affects how vertically large or small the waves appear. It's important to recognize not only for visualizing the function but also for understanding physical phenomena described by the cosine and sine functions in fields like physics and engineering.