Problem 23 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\cos 2 x\\\ &g(x)=-\cos 2 x \end{aligned}$$

Step-by-Step Solution

Answer

The graphs of \(f(x) = \cos(2x)\) and \(g(x) = -\cos(2x)\) have the same amplitude and period and neither graph has a shift. However, \(g(x) = -\cos(2x)\) is a reflection of \(f(x) = \cos(2x)\) across the x-axis.

Step 1: Identify Amplitude

Firstly identify the amplitudes of \(f(x)\) and \(g(x)\). For a function in the form \(y = A\cos (Bx)\), the amplitude is |A|. In this case, the absolute value of A is equal to 1 for both \(f(x)\) and \(g(x)\). Therefore, the amplitudes of both functions are the same (i.e., 1).

Step 2: Identify Period

The period of \(y = A\cos (Bx)\) is \(2\pi / B\). The B value in both functions is 2 so the period for both \(f(x)\) and \(g(x)\) is \(\pi\). Therefore, the periods of both functions are the same.

Step 3: Identify Shift

In this form \(y = A\cos (Bx + C)\), if C ≠ 0, the graph is shifted along the x-axis. However, in both functions \(f(x)\) and \(g(x)\), there is no value added or subtracted from \(2x\). Therefore, there are no horizontal or vertical shifts in either function.

Step 4: Identify the relationship

So far, the amplitudes and periods of both functions are the same and there is no shift in either function. However, there is a difference between \(f(x)\) and \(g(x)\). In the standard form \(y = A\cos(Bx)\), the graph of \(y = \cos(Bx)\) is reflected over the x-axis if A < 0. Therefore, \(g(x) = -\cos(2x)\) is a reflection of \(f(x)\) across the x-axis.

Key Concepts

Amplitude

The amplitude of a trigonometric function gives us an idea of how tall or short the waves of the graph are. When we look at functions like \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), the formula \(y = A\cos(Bx)\) helps us determine the amplitude. Here, \(A\) is the coefficient of the cosine, which, in both cases, is 1.

The amplitude is calculated as the absolute value of \(A\), so for both functions, the amplitude is \(|1| = 1\). This means the peaks of the waves reach a height of 1 and the troughs go down to -1. Since the amplitudes of both \(f(x)\) and \(g(x)\) are identical, their vertical stretches and compressions match perfectly. You can think of amplitude like the volume dial on a radio, controlling how 'loud' or 'soft' the peaks and valleys of the wave appear.

Period

The period of a trigonometric function like cosine tells us how long it takes for the wave to complete one full cycle. The general form to follow is \(y = A\cos(Bx)\). In this form, the period is calculated as \(\frac{2\pi}{B}\).

For both \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), the value of \(B\) is 2. Therefore, their periods are both \(\frac{2\pi}{2} = \pi\). This means that it takes a distance of \(\pi\) along the x-axis for both these functions to achieve one full cycle. In other words, they have evenly spaced waves that repeat every \(\pi\) units, so their oscillations appear at the same intervals.

Reflection

Reflection in trigonometric graphs can alter the graph's orientation without affecting its amplitude or period. When comparing \(f(x) = \cos 2x\) to \(g(x) = -\cos 2x\), the negative sign in front of the cosine function in \(g(x)\) plays a crucial role.

In the formula \(y = A\cos(Bx)\), if \(A\) is negative, the graph reflects over the x-axis. This means that all the high points (peaks) of the wave of \(f(x)\) become low points (troughs) in \(g(x)\), and vice versa. Essentially, \(g(x)\) looks like a mirror image of \(f(x)\) across the x-axis. This reflection causes the waves to move in opposite directions while maintaining the same amplitude and period, creating a neat pattern flip.