Problem 11 - Algebra and Trigonometry

Question

\(3-16\) Graph the function. $$ g(x)=-\frac{1}{2} \sin x $$

Step-by-Step Solution

Answer

Graph of \(g(x)=-\frac{1}{2} \sin x\) is a sine wave, shortened and flipped vertically, with amplitude \(\frac{1}{2}\) and period \(2\pi\).

Step 1: Identify the Parent Function

The parent function is \( \sin x \), which is a standard sine wave that oscillates between -1 and 1 with a period of \(2\pi\).

Step 2: Apply Vertical Stretching and Reflection

The function \( g(x) = -\frac{1}{2} \sin x \) indicates a vertical stretch by a factor of \( \frac{1}{2} \), followed by a reflection across the x-axis. This means the usual sine wave peaks at \(-\frac{1}{2}\) and troughs at \(\frac{1}{2}\).

Step 3: Determine Key Points and Amplitude

For the sine function, key points within one period (\(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)) have specific outputs. With \( g(x) \), the transformations result in outputs \(0, -\frac{1}{2}, 0, \frac{1}{2}, 0\). The amplitude is \(\frac{1}{2}\) due to the vertical stretch.

Step 4: Sketch the Graph

Draw the x and y axes. Begin plotting the transformed key points: at \(x=0\), \(g(0)=0\); at \(x=\frac{\pi}{2}\), \(g(\frac{\pi}{2})=-\frac{1}{2}\); at \(x=\pi\), \(g(\pi)=0\); at \(x=\frac{3\pi}{2}\), \(g(\frac{3\pi}{2})=\frac{1}{2}\); ending at \(x=2\pi\), \(g(2\pi)=0\). Connect these points smoothly with a sinusoidal curve.

Step 5: Analyze the Graph Characteristics

The graph of \( g(x) = -\frac{1}{2} \sin x \) has the same period \((2\pi)\) as \(\sin x\), with a maximum of \(\frac{1}{2}\) and a minimum of \(-\frac{1}{2}\). Each period features one complete wave cycle, reflecting the amplitude and the effect of the negative sign flipping the graph vertically.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that is key to understanding many wave-like phenomena in mathematics and physics.
It is denoted as \( \sin(x) \). In its simplest form, the graph of \( \sin(x) \) is a smooth, continuous wave.

• The basic sine wave goes through regular cycles, peaking at 1 and troughing at -1.
• The wave starts at zero, rises to its maximum at \( x = \frac{\pi}{2} \), falls back to zero at \( x = \pi \), dips to its minimum at \( x = \frac{3\pi}{2} \), and returns to zero at \( x = 2\pi \).

Graphs made from the sine function are periodic, which means they repeat over regular intervals.
Understanding these properties helps in graphing more complex sine functions after applying transformations.

Amplitude

Amplitude is a critical concept when studying trigonometric functions.
It defines how far a wave stretches vertically from its central axis or middle value.

• The amplitude of a basic sine function, \( \sin(x) \), is 1.
• In transformed functions, the amplitude can change, which stretches or compresses the wave.

For the given function \( g(x) = -\frac{1}{2}\sin(x) \), the amplitude is \( \frac{1}{2} \).
This means the wave will reach a maximum height of \( \frac{1}{2} \) and drop to a minimum height of \(-\frac{1}{2} \).
Understanding amplitude is crucial for correctly plotting points on the graph.

Vertical Stretching and Reflection

Vertical stretching and reflection are transformations that alter the graph of a sine function.
These transformations involve multiplying the function by a constant and possibly reflecting it across an axis.
In the case of \( g(x) = -\frac{1}{2}\sin(x) \):

• The factor \(-\frac{1}{2}\) indicates both a vertical stretch and a reflection.
• Vertical stretching by \(\frac{1}{2}\) compresses the graph so the peaks and troughs move closer to the center line.
• The negative sign reflects the wave upside down across the x-axis.

Due to these transformations, the typical wave structure is altered, impacting how the function graph is sketched.

Wave Period

The wave period of a sine function refers to the length over which it repeats.
A complete period means going through one full wave cycle, consisting of reaching the maximum, returning to zero, reaching the minimum, and back to zero.

• The basic sine function \( \sin(x) \) has a period of \( 2\pi \).
• For the transformed function \( g(x) = -\frac{1}{2}\sin(x) \), the period remains \( 2\pi \) as it is not affected by vertical transformations.

Therefore, despite the transformation changing the amplitude and orientation, the period of the graph remains the same.
This periodicity means that the function keeps repeating its wave pattern uniformly along the x-axis.