Problem 47 - 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.) $$y=\cos \frac{x}{2}$$

Step-by-Step Solution

Answer

The function \(y = \cos(x/2)\) is a horizontally stretched version of the basic function \(y = \cos x\) by a factor of 2, causing it to complete one full cycle in a range of \(4\pi\) instead of \(2\pi\). The graph of the function would include two full periods, and must show peaks at \(0, 4\pi, 8\pi\) and valleys at \(2\pi, 6\pi\). This can be verified by using a graphing utility or software.

Step 1: Understand the Function

The given function is \( y = \cos (x/2) \). Compared to the basic function \( y = \cos x \), the 'x' inside the cosine function is halved. When 'x' is halved, it results in a horizontal stretching of the function by a factor of 2. This means that the period of the regular cosine function, which is \(2\pi\), will be doubled for our function to \(4\pi\). So, the cosine wave will complete one full cycle over a range of \(4\pi\) instead of \(2\pi\).

Step 2: Sketch the Graph

Start at the origin point (0,1) because the cosine function starts at 1 when x = 0. The function reaches its minimum (-1) at \(2\pi\), returns to maximum (1) at \(4\pi\). Now, continue this pattern to complete two full periods of the function. Beyond \(4\pi\), the function will repeat its values, that is, it will reach minimum at \(6\pi\) and then maximum at \(8\pi\). So, after sketching, the graph would have two complete waves from 0 to \(8\pi\) with peaks at \(0, 4\pi, 8\pi\) and valleys at \(2\pi, 6\pi\).

Step 3: Verify with a Graphing Utility

Utilise a graphing utility or software to plot the function \(y = \cos(x/2)\). Examine if the plotted graph aligns with the manually sketched graph covering two full periods from 0 to \(8\pi\). Also, the plotted graph should show peaks at \(0, 4\pi, 8\pi\) and valleys at \(2\pi, 6\pi\).

Key Concepts

Cosine Function

The cosine function, symbolized as \( \cos x \), is a fundamental trigonometric function that describes the cosine of an angle \( x \). It's one of the building blocks of trigonometry helping to model periodic phenomena such as sound waves, light waves, and mechanical vibrations.

• The basic cosine function, \( y = \cos x \), starts at a maximum value of 1 when \( x = 0 \).
• As \( x \) increases, the cosine value decreases to reach a minimum of -1, and then increases back to a maximum of 1, leading to a wave-like pattern.

This smooth, wave-like nature is what makes the cosine function valuable in modeling cycles. For instance, in our function \( y = \cos(\frac{x}{2}) \), halving \( x \) causes the cosine wave to stretch out. This feature allows the function to complete one full cycle over a larger interval of \( 4\pi \), changing the cycle length effectively."

Graphing Utility

Graphing utilities, such as software programs or graphing calculators, are crucial tools for students and educators alike. They graph functions quickly and accurately, helping verify manually sketched plots. When graphing the function \( y = \cos(\frac{x}{2}) \), these steps are useful:- Enter the function into the graphing utility.- Set the viewing window to display at least two full periods, from 0 to \( 8\pi \).- Compare the utility's graph pattern to your manual sketch. Ensure peaks occur at \( 0, 4\pi, \text{and } 8\pi \) and troughs at \( 2\pi \text{and } 6\pi \).Making use of graphing utilities can reduce errors and confirm understanding, giving students confidence in their solution."

Period of a Function

The period of a trigonometric function is the interval over which it repeats. For cosine, a cycle completes in \( 2\pi \) units for the basic function \( y = \cos x \). However, when the input is adjusted, as in \( y = \cos(\frac{x}{2}) \), the period changes.

• In \( y = \cos(\frac{x}{2}) \), inside the cosine, \( x \) is divided by 2.
• Dividing \( x \) by 2 scales the period by a factor of 2, turning the standard period from \( 2\pi \) to \( 4\pi \).

This means the function \( y = \cos(\frac{x}{2}) \) completes one full cycle over \( 4\pi \) rather than the usual \( 2\pi \). By understanding this property, students can manipulate and recognize transformed trigonometric functions in various applications."