Problem 21 - Precalculus Mathematics for Calculus

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: Understand the Sine Function

The general form of the sine function is \(y = A \sin(Bx + C) + D\). In this form, \(A\) determines the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical shift. Comparing this with \(y = 10 \sin \frac{1}{2} x\), we have \(A = 10\), \(B = \frac{1}{2}\), \(C = 0\), and \(D = 0\).

Step 2: Determine the Amplitude

The amplitude of a sine function is the absolute value of \(A\). In our function, \(A = 10\), so the amplitude is \(|10| = 10\).

Step 3: Calculate the Period

The period of the sine function is determined by the formula \(\frac{2\pi}{B}\). For our function, \(B = \frac{1}{2}\), so the period is \[\frac{2\pi}{\frac{1}{2}} = 4\pi\].

Step 4: Sketch the Graph

To sketch the graph, start by plotting the sine function \(y = 10 \sin \frac{1}{2}x\). The function starts at \(0\), reaches its maximum of \(10\) at \(x = 2\pi\), returns to \(0\) at \(x = 4\pi\), reaches \(-10\) at \(x = 6\pi\), and returns to \(0\) at \(x = 8\pi\). This completes one cycle over the period of \(4\pi\). Consider marking the important points such as maximum, minimum, and zero crossings to aid in sketching.

Key Concepts

Amplitude

The amplitude of a sine function tells us how far up and down the wave moves from its central position. It represents the maximuma deviation from the zero or equilibrium position on the graph. In simpler terms, it's how 'tall' or 'short' the waves of the sine function will appear. Here's how you can identify the amplitude in a mathematical expression:

• The amplitude is represented by the absolute value of the coefficient in front of the sine function, usually labeled as 'A' in the general equation form.
• For instance, in our given function: \(y = 10 \sin \frac{1}{2}x\), the value of 'A' is 10.
• Thus, the amplitude is \(|10| = 10\), meaning the waves will peak at +10 and trough at -10 on the graph.

Understanding amplitude helps you visualize how "tall" each cycle of the sine wave appears, so you can better predict its behavior on a graph.

Period of a Function

The period of a function describes the length of one complete cycle of the wave before it starts repeating. For a basic sine function, this period is determined by the coefficient of 'x' inside the sine. Let's delve into how you calculate the period:

• In the general sine function form \(y = A \sin(Bx + C) + D\), the period is given by the formula \(\frac{2\pi}{B}\).
• In our specific function \(y = 10 \sin \frac{1}{2}x\), the 'B' value is \(\frac{1}{2}\).
• Substitute this value into the period formula: \(\frac{2\pi}{\frac{1}{2}} = 4\pi\).

So, the period here is \(4\pi\), which means the sine wave will complete one full cycle or pattern every \(4\pi\) units along the x-axis. This knowledge allows you to predict exactly where and when the waves will peak and return to zero.

Graph Sketching

Graph sketching involves plotting key points of the function and connecting them smoothly to form the sine wave. Let's break it down step by step:

• Start at the origin (0,0) since there's no phase shift or vertical movement in our function.
• Because the amplitude is 10, the wave will reach its maximum at 10 and minimum at -10.
• Using the calculated period of \(4\pi\), plot critical points: maximum at \(x = 2\pi\), crossing zero at \(x = 4\pi\), minimum at \(x = 6\pi\), and back to zero at \(x = 8\pi\).
• Remember, the pattern repeats every \(4\pi\), so you can extend or shorten the graph while repeating the same cycle.

Considering these strategies, you can easily plot a smooth and consistent sine wave that visually represents the function \(y = 10 \sin \frac{1}{2}x\), making the abstract mathematics tangible on paper.