Problem 13 - Contemporary Precalculus

Question

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=-\cos t$$

Step-by-Step Solution

Answer

Answer: The required transformation is a reflection across the x-axis.

Step 1: Identify the given functions and their behavior

The given functions are \(f(t) = \cos t\) and \(g(t) = -\cos t\). We know that the cosine function has a periodic behavior with a period of \(2\pi\). The function \(f(t)\) is an even function, which means it's symmetric with respect to the y-axis. The function \(g(t)\) is an odd function, since negating the input value negates the output value.

Step 2: Compare the functions and identify the primary transformation

Comparing \(f(t)\) and \(g(t)\), we can see that the functions are similar, except for a negative sign in \(g(t)\). This means that the primary transformation to change the graph of \(f(t)\) into the graph of \(g(t)\) is a reflection across the x-axis. In other words, we need to multiply the output value of \(f(t)\) by -1 to change its graph to \(g(t)\).

Step 3: List the required transformations

The transformation required to change the graph of \(f(t)\) into the graph of \(g(t)\) is a reflection across the x-axis.

Key Concepts

Reflection Across the x-axis

Understanding how the graph of a function changes with various transformations is a fundamental concept in algebra and calculus. One such transformation is a reflection across the x-axis. This occurs when each point \( (x,y) \) on the original graph is transformed to \( (x,-y) \) on the new graph. In essence, the y-coordinates of all points are multiplied by -1, essentially 'flipping' the graph over the x-axis.

The reflection has a distinct effect on the symmetry of a function. For instance, if the original function had a peak at a certain point, after a reflection across the x-axis, that peak turns into a trough. This concept is not just abstract; it can have practical implications in fields like physics, where such transformations can represent changes in direction of movement.

In our exercise, the transformation needed to change the graph of \(f(t) = \text{cos}\, t\) into the graph of \(g(t) = -\text{cos}\, t\) is precisely this reflection across the x-axis. The negative sign in front of the cosine function indicates that each y-coordinate is multiplied by -1.

Even and Odd Functions

A deeper dive into the symmetry of functions introduces us to the concepts of even and odd functions. An even function is symmetrical about the y-axis, meaning that for every point \( (x,y) \) on the graph, there is a corresponding point \( (-x,y) \) that also lies on the graph. Mathematically, a function \(f(x)\) is even if \(f(x) = f(-x)\) for all x in the domain of f.

On the other hand, an odd function has rotational symmetry around the origin. This means that for every point \( (x,y) \) on an odd function’s graph, the point \( (-x,-y) \) is also on the graph. A function \(f(x)\) is odd if \(f(-x) = -f(x)\) for all x.

Why is it important to distinguish between even and odd functions?

It's not merely an academic distinction. In calculus, the properties of even and odd functions can simplify integration over symmetric intervals, and in Fourier analysis, they determine which types of Fourier series represent a given function. In our exercise, \(f(t) = \text{cos}\, t\) is an even function, and the transformed function \(g(t)\), resulting from a reflection over the x-axis, is considered an odd function due to the negative sign.

Cosine Function

Finally, let's dive into the essence of the cosine function, denoted by \( \text{cos} \) and often represented mathematically as \(f(x) = \text{cos}\, x\). Cosine is a periodic function, meaning it repeats its values in regular intervals along the x-axis. Its period is \(2\pi\), which means that the function completes one cycle every \(2\pi\) units.

The cosine function is also an even function, signifying that its graph is symmetrical across the y-axis. When graphed, it produces a wave-like pattern that starts at a maximum value of 1 when \(x = 0\) and oscillates between 1 and -1. The peaks and troughs of the cosine function are points where the rate of change is zero – a concept that is critical when considering the function's derivatives in calculus.

Applications in Different Fields

The applications of the cosine function extend beyond mathematics. In physics, it describes wave motion and oscillations. In engineering, it is used in signal processing to represent waveforms. Understanding the properties of the cosine function is essential not only in academics but also in practical, real-world problems that involve periodic behavior.