Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{aligned} &f(x)=\sin x\\\ &g(x)=-\frac{1}{2} \sin x \end{aligned}$$

Step-by-Step Solution

Answer

The functions \( f(x) = \sin x \) and \( g(x) = -\frac{1}{2} \sin x \) both have the same period (\(2\pi\)) and start at the same point on the x-axis (no shift). However, \( g(x) \) has half the amplitude of \( f(x) \), and it is a reflection of \( f(x) \) over the x-axis.

Step 1: Identify the Amplitude

The amplitude of \( f(x) = \sin x \) is 1, and the amplitude of \( g(x) = -\frac{1}{2} \sin x \) is \(\frac{1}{2}\). Thus, the amplitude (highest point) of \( g(x) \) is half of that of \( f(x) \). The negative sign means that the graph of \( g(x) \) is reflected through the x-axis compared to \( f(x) \).

Step 2: Compare the Periods

The period of \( f(x) = \sin x \) is \(2\pi\) and the period of \( g(x) = -\frac{1}{2} \sin x \) is also \(2\pi\). Thus, both functions have the same period, which means they repeat their patterns at the same rate.

Step 3: Analyze the Shifts

Both \( f(x) = \sin x \) and \( g(x) = -\frac{1}{2} \sin x \) do not have any shifts since there isn't any additional or subtracted constant applied to x. So, both of the functions graphs start at the same point on the x-axis.

Key Concepts

Amplitude

In the realm of trigonometric functions, amplitude represents the height of the wave from the middle line (equilibrium) to its peak. For the function \( f(x) = \sin x \), the amplitude is 1. This means that the wave of \( \sin x \) rises to a height of 1 above and below the equilibrium. However, for the function \( g(x) = -\frac{1}{2} \sin x \), the amplitude is \( \frac{1}{2} \). This indicates that the wave's height is halved compared to \( f(x) \).

### Key Points About Amplitude:

• The amplitude is always a positive number, representing the magnitude of the wave's peak.
• The negative sign in \( g(x) = -\frac{1}{2} \sin x \) affects the graph's direction, not its amplitude. Thus, the wave is upside down, but its amplitude remains \( \frac{1}{2} \).
• Understanding amplitude is crucial for graphing and interpreting waves, especially in physics and engineering contexts.

Graph Reflection

Graph reflection can often be misconstrued as just flipping the graph over. However, it specifically refers to a mirror image over an axis. In trigonometric functions, when a negative sign multiplies the function, it reflects across the x-axis. For instance, \( g(x) = -\frac{1}{2} \sin x \) is a reflection of \( \sin x \). Instead of cresting up first, it dips down.

### Essential Insights on Graph Reflection:

• Reflection through the x-axis means every point \((x, y)\) on \( f(x) \) has a corresponding point \((x, -y)\) on \( g(x) \).
• This mirror effect impacts only the vertical orientation of the graph, not the wave's shape or period.
• Graph reflections are common in optics and can help in understanding wave inversions in real-world scenarios.

Periodicity

Periodicity refers to the repeating nature of wave functions over intervals. The sine function, such as \( f(x) = \sin x \), has a period of \(2\pi\). This means the pattern of the sine wave starts repeating after every \(2\pi\) units along the x-axis.

In our comparison, \( g(x) = -\frac{1}{2} \sin x \) retains the same period of \(2\pi\) despite its amplitude and reflection changes. This identical periodicity allows the waves to have synchronized cycles.

### Core Details on Periodicity:

• Knowing the period helps in predicting the wave's future behavior over time.
• Unlike changes in amplitude or reflections, altering the periodicity requires a modification in frequency or wavelength.
• Periodicity is a fundamental concept in signal processing, helping identify cyclical phenomena like sound waves or tidal patterns.