Problem 19 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=10 \sin \frac{1}{2} x $$

Step-by-Step Solution

Answer

Amplitude is 10, period is \(4\pi\).

Step 1: Identify the Sine Function Format

The given function is in the form of \( y = A \sin(Bx) \). Here, \( A = 10 \) and \( B = \frac{1}{2} \). Both of these parameters will help us determine the amplitude and period.

Step 2: Determine the Amplitude

The amplitude of a sine function \( y = A \sin(Bx) \) is given by the absolute value of \( A \). Thus, the amplitude is \( |A| = |10| = 10 \). This means the graph of the function will oscillate 10 units above and below the horizontal axis.

Step 3: Calculate the Period

The period of the sine function is calculated using the formula \( \frac{2\pi}{|B|} \). Substitute the value of \( B \):\[\text{Period} = \frac{2\pi}{\left| \frac{1}{2} \right|} = \frac{2\pi}{\frac{1}{2}} = 4\pi\].Therefore, the function completes one full cycle every \( 4\pi \) units.

Step 4: Sketch the Graph

To sketch the graph of the function, note that it is a sine wave with an amplitude of 10 and a period of \( 4\pi \). The graph will start at the origin \((0,0)\), reach its maximum at \((\pi, 10)\), return to the midline at \((2\pi, 0)\), reach its minimum at \((3\pi, -10)\), and complete one cycle at \((4\pi, 0)\). Repeat this pattern to illustrate additional cycles.

Key Concepts

Understanding Amplitude

Amplitude is a crucial concept in trigonometry, particularly when working with trigonometric functions like sine and cosine. It represents the maximum extent or height of the wave from its central axis:

• In the sine function format, given by \( y = A \sin(Bx) \), the amplitude is represented by \(|A|\), the absolute value of \(A\).
• The amplitude tells us how far up and down the wave goes from its central axis - in this case, the x-axis.
• For the exercise problem \( y=10 \sin \frac{1}{2}x \), the amplitude is 10. This means the wave rises 10 units above and falls 10 units below its central axis, indicating the maximum and minimum points on the graph.

Understanding amplitude helps students visualize the 'loudness' or 'intensity' of the sine wave. It's like understanding how high and low the peaks and troughs of a wave reach.

Exploring the Sine Function

The sine function is one of the fundamental trigonometric functions and plays a pivotal role in representing periodic oscillations:

• It is represented in the form \( y = A \sin(Bx) \), where \(A\) is the amplitude, and \(B\) helps determine the function's period.
• The sine function starts at 0, climbs to a peak, returns to 0, descends to a trough, and then returns to 0 to complete a full cycle.
• For our function \( y=10 \sin \frac{1}{2}x \), it signifies that the oscillation starts from the origin \((0,0)\) and has peaks and troughs determined by the amplitude and the specific positions where these occur within the period.

The sine wave is smooth and continuous, making it handy for modeling cyclic phenomena like sound waves, light patterns, and even day-night cycles.

Determining the Period of a Function

The period of a trigonometric function indicates the length of one complete cycle of the wave. It shows how frequently the function repeats itself:

• For a sine function in the form \( y = A \sin(Bx) \), the period is calculated by the formula \( \frac{2\pi}{|B|} \).
• The period determines the horizontal length of one full wave cycle on the graph.
• In the exercise, with \( B = \frac{1}{2} \), the period calculates to \( 4\pi \), which means the sine wave completes one full cycle every \( 4\pi \) units along the x-axis.

By understanding the period, learners can infer how rapidly or slowly the wave oscillates across the horizontal plane, which is essential in applications like signal processing and other scientific analyses.