91Ó°ÊÓ

The amplitude of a trigonometric function like the sine function is a critical concept in understanding its behavior. Simply put, the amplitude represents how far the graph extends above and below its central axis (usually the x-axis for the sine function). Amplitude is given by the absolute value of the coefficient in front of the sine function.

In our example, the function is defined as \( f(x) = 5 \sin \frac{2x}{5} \). Here, the coefficient is 5. So, the amplitude is \( |5| = 5 \). This means the graph of the sine function will oscillate between 5 and -5, peaking at these values.

Knowing the amplitude helps us understand how "tall" the wave will appear in its graph. Essentially, it helps in plotting the maximum and minimum points of a sine wave on the coordinate plane.

The period of a function, particularly in trigonometric contexts, defines how long it takes for the function to repeat itself. For sine functions, this period can be computed from the formula \( \frac{2\pi}{|b|} \) where \( b \) is the coefficient of \( x \) inside the sine function.

In our case, the function given is \( f(x) = 5 \sin \frac{2x}{5} \), where \( b = \frac{2}{5} \). Thus, the period will be calculated as follows:

• Period = \( \frac{2\pi}{\frac{2}{5}} = 5\pi \)

This calculation tells us that the sine wave takes \( 5\pi \) units along the x-axis to complete one full cycle.

Understanding the period is crucial when sketching the graph, as it determines the extent along the x-axis you need to consider for capturing a full wave. In this exercise, we need to plot two full periods, thus stretching the graph over \( 10\pi \) units.

Trigonometric graphs offer a visual representation of functions like sine, cosine, and tangent that exhibit periodic behavior. These graphs are invaluable in modeling a variety of physical phenomena, like sound waves or tides.

The sine function, in particular, is known for its smooth, wave-like shape, beginning at the origin, oscillating upwards to its maximum, down to its minimum, and then back again.

• The basic wave pattern is determined by the amplitude, which tells you how "high" or "low" the wave goes.
• The period dictates the "length" of one complete wave cycle on the x-axis.

In the above exercise, sketching the sine graph involves understanding these characteristics to properly scale your graph according to \( y = 5 \sin \frac{2x}{5} \). First, identify key points such as the maximums, minimums, and where the graph crosses the x-axis.

Graphically, starting from the origin at \( (0,0) \), you should sketch one complete oscillation every \( 5\pi \) units, rising to a maximum of 5 and dipping to a minimum of -5. This periodic pattern is repeated, ensuring your understanding of trigonometric graphs goes beyond mere calculations, aiding you in correctly visualizing their application.