Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of \(y=\cos b x\) for \(b=\frac{1}{2}, 2\) and \(3 .\) How does the value of \(b\) affect the graph? How many complete cycles of the graph of \(y\) occur between 0 and \(2 \pi\) for each value of \(b ?\)

Step-by-Step Solution

Answer

On the graph of \(y=\cos(bx)\), the value of \(b\) affects the frequency of the cosine function. Specifically, a higher value of \(b\) results in more cycles completed in a given range. Between 0 and \(2\pi\), the function \(y=\cos\left(\frac{1}{2}x\right)\) completes 0.5 cycles, \(y=\cos(2x)\) completes 2 cycles, and \(y=\cos(3x)\) completes 3 cycles.

Step 1: Sketch the graph for \(b=\frac{1}{2}\)

Plot the function \(y=\cos\left(\frac{1}{2}x\right)\). The period of the graph of \(\cos x\) is \(2\pi\), so when \(b=\frac{1}{2}\), the period of the graph of \(y=\cos\left(\frac{1}{2}x\right)\) is going to be twice as big, thus completing 0.5 cycles between 0 and \(2\pi\).

Step 2: Sketch the graph for \(b=2\)

Next, plot the function \(y=\cos(2x)\). When \(b=2\), the period of the graph is going to be half of the regular one, completing 2 cycles between 0 and \(2\pi\).

Step 3: Sketch the graph for \(b=3\)

Finally, plot the function \(y=\cos(3x)\). With \(b=3\), the period of the graph is going to be about one third, completing 3 cycles between 0 and \(2\pi\).

Key Concepts

Trigonometric Graph Transformations

Understanding the transformations of trigonometric graphs is essential when dealing with problems involving the plotting of these functions. Let's discuss how transformations apply to trigonometric functions, specifically through the multiplication of the argument of the cosine function by a constant, like in the given exercise with functions of the form y = cos(bx).

The parameter b in the trigonometric function affects the horizontal stretch or compression of the graph. When b is greater than 1, the graph of the function will be compressed because the function will go through its cycle more quickly. Conversely, if b is less than 1 but greater than 0, like 1/2, the graph is stretched out which means it will take longer to complete one cycle.

A transformed trigonometric graph maintains the same shape as the original graph; it is merely stretched or compressed along the x-axis. Imagine it as if the waveform is being squeezed or stretched horizontally while its amplitude remains consistent. These transformations require an understanding of the basic graphs of sine and cosine functions to properly predict the resultant graphs after applying the 'stretch' or 'compression' caused by the scaling factor b.

Cosine Function Period

The cosine function, y = cos(x), is one of the basic trigonometric functions, and its graph exhibits periodic behavior. The period of a function is the length of the smallest interval over which the function's values repeat. For the unmodified cosine function, the values repeat every 2π radians, which means its period is 2π.

When we introduce a coefficient, b, into the function as in y = cos(bx), we alter the original period. The new period is determined by dividing the standard period 2π by the absolute value of b: Period = 2π/|b|. For example, with y = cos(2x) the period is halved, meaning that the function will complete one full cycle over an interval of π instead of 2π. If b is less than 1, as with y = cos((1/2)x), the period will be doubled, resulting in a longer cycle. This modification affects how many complete cycles occur between 0 and 2π, directly impacting the graph's appearance across a specified domain.

Frequency of Trigonometric Graphs

Closely linked with the concept of the period is the frequency of trigonometric graphs. Frequency is defined as the number of cycles a periodic function completes over a specified interval. It is the inverse of the period: Frequency = 1/Period. In practical terms, it conveys how often the periodic event happens. If the cosine function has a standard period of 2π, the frequency is 1/(2π), meaning it completes one cycle for every 2π units traveled along the x-axis.

In our exercise, where b=2 in y = cos(2x), the graph completes 2 cycles between 0 and 2π, thus the frequency is 2/(2π) or 1/π. As b increases, the frequency also increases - the cycles become more frequent over the same interval. If b=3 in y = cos(3x), the frequency is 3/(2π), which indicates there are 3 complete cycles between 0 and 2π. As shown in these examples of transformed cosine functions, understanding the relationship between the coefficient b, the period, and the frequency allows us to accurately graph trigonometric functions and predict how many cycles will occur over any given interval.