Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitudes, periods, and shifts. $$\begin{array}{l} f(x)=\sin x \\ g(x)=5 \sin (-x) \end{array}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = 5 \sin(-x)\) is a vertical stretch and a reflection across the y-axis of \(f(x) = \sin(x)\). The amplitude of \(g(x)\) is 5 times that of \(f(x)\), and both functions have the same period of \(2\pi\).

Step 1: Analyze amplitudes

The amplitude of \(f(x) = \sin(x)\) is 1, as there is no coefficient before the \(\sin\) term, i.e., it varies from -1 to 1. The amplitude of \(g(x) = 5 \sin(-x)\) is 5, which is suggested by the coefficient 5 before the \(\sin\) term, i.e., it varies from -5 to 5.

Step 2: Identify periods

The period of both functions \(f(x)\) and \(g(x)\) is \(2\pi\) as there is no coefficient with \(x\), hence they repeat their values every \(2\pi\) units.

Step 3: Notice the Reflection

The function \(g(x) = 5 \sin(-x)\) is a reflection of the function \(f(x) = \sin(x)\) along the y-axis. The negative sign in the argument \(-x\) of \(g(x)\) is responsible for the reflection.

Step 4: Summarize the observations

The graph of \(g(x)\) is a stretched and reflected version of the graph of \(f(x)\) along the y-axis. It is stretched due to the larger amplitude and reflected due to the negative sign of the argument.

Key Concepts

Amplitude

The term "amplitude" in trigonometric graphs refers to the height of the wave-like pattern. For a sine function like \( f(x) = \sin(x) \), the amplitude is 1. This means that the maximum and minimum values the function can reach are 1 and -1 respectively.

• The amplitude shows how "tall" the wave is.
• It's determined by the coefficient in front of the sine or cosine term.

In our exercise, the second function \( g(x) = 5 \sin(-x) \) changes the amplitude. The coefficient here is 5, making the amplitude 5.
This means the function ranges from -5 to 5, essentially making the wave taller compared to \( f(x) \). By visualizing or sketching these functions, you will see that the taller wave of \( g(x) \) can't go unnoticed. Think of amplitude like the volume knob; increasing it makes everything louder, or in this case, taller.

Period

The "period" of a trigonometric function refers to how often the wave pattern repeats itself. For functions like \( \sin(x) \) and \( \cos(x) \), the standard period is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the function repeats its values.

• A smaller period means the wave cycles occur more frequently.
• Conversely, a larger period spreads the cycles out.

In the case of \( f(x) = \sin(x) \) and \( g(x) = 5 \sin(-x) \), since there is no altering coefficient inside the sine term, both maintain the original period of \(2\pi\).
Thus, for both \( f(x) \) and \( g(x) \), every \(2\pi\) interval will replicate the wave pattern making them identical in terms of periodic behavior despite differences in amplitude and reflections.

Reflection

Reflection in trigonometric graphs involves flipping the wave over an axis. For \( g(x) = 5 \sin(-x) \), this involves the y-axis.

• The negative sign within the function \(-x\) leads to the reflection.
• This effectively changes the direction of the wave pattern.

For \( g(x) = 5 \sin(-x) \), the sine wave is mirrored across the y-axis as opposed to \( f(x) = \sin(x) \), which has no reflection.
Understanding reflection deeply involves visualizing how points on the graph swap positions symmetrically about the y-axis.
For instance, points that were on the right of the y-axis for \( f(x) \) flip to the left for \( g(x) \) and vice-versa.
This reflection creates a mirror image effect, essentially flipping the graph horizontally. Yet, our periods and amplitudes still hold true, making these reflections consistent wave transformations.