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In trigonometric functions such as the cosine function, a vertical stretch occurs when the output values (y-values) are multiplied by a constant number. For our function, \(y = \frac{5}{2} \cos \frac{x}{4}\), the constant \(\frac{5}{2}\) represents this vertical stretch.

This modification affects the amplitude of the function, which is the height from the central axis to the peak or trough.

• In the basic cosine function, \(\cos(x)\), the amplitude is 1, so the values range from -1 to 1.
• When we multiply the cosine function by \(\frac{5}{2}\), the amplitude becomes \(\frac{5}{2}\), increasing the range from -1 to 1 to -\(\frac{5}{2}\) to \(\frac{5}{2}\).

Thus, a vertical stretch makes the peaks higher and the troughs deeper, visually pulling the graph vertically away from the x-axis.

The period of a trigonometric function is the interval over which the function completes one full cycle before starting to repeat.

For the cosine function, \(y = \cos(x)\), the period is \(2\pi\) radians (or 360 degrees). This means that every \(2\pi\) units along the x-axis, the function repeats its pattern.

In our function \(y = \frac{5}{2} \cos \frac{x}{4}\), the input is \(\frac{x}{4}\). This division by 4 extends the period because it slows down the function's oscillation.

• To find the new period, multiply the original period by 4: \(4\times 2\pi = 8\pi\).

Therefore, the period of this transformed cosine function is \(8\pi\) radians, which means the function repeats every \(8\pi\) units along the x-axis.

Trigonometric functions are mathematical functions that relate angles of a triangle to the lengths of the triangle's sides. They are essential in many areas of mathematics and applied sciences. The most commonly used trigonometric functions are sine, cosine, and tangent.

The cosine function, \(y = \cos(x)\), is known for its wave-like pattern, which oscillates between -1 and 1 across the x-axis. It is widely used in physics, engineering, and signal processing to model periodic phenomena like sound waves and light waves.

In the function \(y = \frac{5}{2} \cos \frac{x}{4}\), we're specifically using the cosine function, which showcases important concepts such as vertical stretch and period adjustments.

• These transformations help explore and represent real-world situations where the amplitude and frequency of waves change.

Understanding how to manipulate these parameters allows us to model more complex periodic behaviors effectively.