Problem 99 - Precalculus Essentials

Question

Determine the amplitude and period of \(y=3 \sin \frac{1}{2} x\) Then graph the function for \(0 \leq x \leq 4 \pi\) (Section 4.5, Example 3)

Step-by-Step Solution

Answer

The amplitude of the function \(y = 3 \sin \frac{1}{2}x\) is 3 and the period is \(4\pi\). The graph is a sine wave that starts at the origin, reaches a maximum at \(x=\pi\), goes back to zero at \(x=2\pi\), reaches a minimum at \(x=3\pi\), and completes a single oscillation at \(x=4\pi\).

Step 1: Identify the Amplitude

The coefficient of the sine function gives us the amplitude. In the given equation \(y = 3 \sin \frac{1}{2}x\), we see that the Amplitude is 3 because it is the absolute value of the coefficient of the sine function.

Step 2: Calculate the Period

The period of the function is given by \( \frac{2\pi}{B}\). Here, \(B = \frac{1}{2}\), so substituting B = 1/2 into the formula for the period gives us Period = \( \frac{2\pi}{\frac{1}{2}} = 4\pi\).

Step 3: Plot the Graph

To plot the graph, we first put the origin, then a point at \(x=4\pi\) (because that’s the period), and then points in the middle where the sine function reaches its maximum and minimum. The maximum is reached at \(x=\pi\) and minimum at \(x=3\pi\). Then joining these points, we get a wave that completes a single oscillation for 0 to 4π.

Key Concepts

Amplitude

When we talk about amplitude in the context of a trigonometric function, we're referring to how far the function moves up and down from its central horizontal axis. This middle line is usually the x-axis when dealing with the sine function.

For the function given by the equation \(y = 3 \sin \frac{1}{2}x\), the amplitude is determined by the coefficient of the sine function. In this case, the coefficient is 3.

What this means is that the sine wave will reach a maximum of 3 above the x-axis and dip a minimum of 3 below. This makes the maximum height of the wave from the centerline a total of 3.

• Understanding amplitude helps in visualizing how tall or short the oscillations of the wave will be.
• A larger amplitude means a taller wave, while a smaller amplitude means a shorter wave.

Remember, when graphing the sine function, the amplitude is always a positive value as it represents distance.

Period

The period of a sine function tells us how long it takes to complete one full cycle of the wave. This is like measuring the length of a full oscillation.

In mathematical terms, for a sine function of the form \(y = A \sin Bx\), the period is calculated using the formula \(\frac{2\pi}{B}\).

For our function \(y = 3 \sin \frac{1}{2}x\), the value of \(B\) is \(\frac{1}{2}\). Plugging this into the formula gives us:
\[ \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi \]
This tells us that it will take an interval of \(4\pi\) along the x-axis for the function to complete one full cycle.

• The period informs us about how stretched or compressed the wave is along the x-axis.
• A larger period means the wave takes longer to repeat, thereby being more spread out.
• A smaller period means the function repeats more frequently, appearing compressed.

Sine Function

The sine function is a fundamental wave-like pattern in trigonometry. It's represented mathematically with the expression \(y = A \sin Bx\), where \(A\) determines the amplitude, and \(B\) affects the period.

The sine function is known for its smooth, repetitive oscillations. It starts at zero, rises to a maximum, returns to zero, dips to a minimum, and returns to zero once more, completing one full cycle.

In our example \(y = 3 \sin \frac{1}{2}x\), the sine wave:

• Begins at the origin, as sine of 0 is zero.
• Rises to its maximum of 3 at \(x=\pi\).
• Returns to zero at \(x=2\pi\).
• Reaches its minimum at \(x=3\pi\).
• Finishes the cycle at \(x=4\pi\).

This cyclic behavior—rising and falling uniformly—makes the sine function essential in modeling periodic phenomena, like sound waves and tides, where a smooth, repeating pattern is observed.