Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=5 \sin x$$

Step-by-Step Solution

Answer

The graph has a regular period of 2\(\pi\) with an amplitude of 5. It oscillates between -5 and 5 with zero at \( \pi \) intervals.

Step 1: Identify amplitude and period

The amplitude of this function is 5, this is because of the coefficient of the sine function. The period is 2\(\pi\)

Step 2: Draw the sinusoidal axis

Draw a horizontal line halfway between the maximum and minimum values of the function (here it will be at y=0) to represent the sinusoidal axis [The line y=0].

Step 3: Mark the period on the x-axis

Mark the period of 2\(\pi\) on the x-axis and divide it into four equal parts which corresponds to \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\).

Step 4: Draw the wave

Starting from the sinusoidal axis at the first \(0\), go up to the maximum of 5 at \(\frac{\pi}{2}\), back down to the sinusoidal axis at \(\pi\), then go down to the minimum of -5 at \(\frac{3\pi}{2}\), and finally, return back to the sinusoidal axis at \(2\pi\).

Step 5: Repeat the wave

Repeat the wave as established in Step 4 for a second full period.

Step 6: Verify with a graphing utility

Finally it's advised to confirm the sketch using a graphing utility. This can assist in verifying the accuracy of the period and amplitude.

Key Concepts

Amplitude of Sine Function

The amplitude of a sine function represents the maximum distance from the sinusoidal axis to the peak or trough of the wave. In mathematical terms, if our sine function is written as \( y = A \sin(Bx + C) + D \), where \( A \), \( B \), \( C \), and \( D \) are constants, the amplitude is the absolute value of \( A \).

In the given exercise, \( y = 5 \sin x \), the amplitude is the coefficient of the sine function, which is 5. This means the graph will peak at 5 units above the sinusoidal axis and dip to 5 units below it. Higher amplitude results in a taller wave, and conversely, a smaller amplitude results in a flatter wave. It's crucial for students to recognize that amplitude is always a positive value as it denotes the extent of oscillation regardless of the function's direction.

Period of Sine Function

The period of a sine function is the length of one complete cycle of the wave. Mathematically, for the general sine function \( y = A \sin(Bx + C) + D \), the period is calculated as \( \frac{2\pi}{|B|} \).

In our example, \( y = 5 \sin x \), the value of \( B \) is implicitly 1, making the period \( 2\pi \). This means that the sine wave repeats itself every \( 2\pi \) units along the x-axis. Understanding the period is crucial for graphing because it indicates where the wave pattern starts and stops before repeating. During graphing, students should mark the x-axis with increments that divide the period into segments of \( \frac{\pi}{2} \) for ease of plotting the major points of the sine wave.

Sinusoidal Axis

The sinusoidal axis of a sine function is the horizontal line that represents the 'middle' of the wave. It's the axis around which the graph oscillates, acting as an equilibrium line between the maximum and minimum values. For any function formatted as \( y = A \sin(Bx + C) + D \), the sinusoidal axis is the line \( y = D \).

In our example, since there is no vertical shift (\( D = 0 \)), the sinusoidal axis is the x-axis, or \( y = 0 \). A clear understanding of the sinusoidal axis helps in sketching an accurate sine graph, as students need to know the central line from which the amplitude is measured and around which the graph oscillates.

Graphing Utility Verification

Using a graphing utility can greatly enhance a student's comprehension and confidence in sketching sine functions. After manually plotting the graph, verification with technology ensures that the amplitude and period—and indeed, the full wave shape—are accurately represented.

For our exercise, a student is advised to input the function \( y = 5 \sin x \) into a graphing calculator or software and compare the output to their sketch. The graphing utility serves as a reference point to double-check the plotted points, axis scaling, wave shape, and especially the accuracy of the amplitude and period. This step solidifies the connection between the theoretical understanding of sine functions and their practical graphical representations.