Problem 21 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal stretches and compressions to graph at least two cycles of the given functions. $$f(x)=\sin (3 x)$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\csc (2 x)\) is a series of hyperbolas with vertical asymptotes at every multiple of π and passing through points at π/4 and 3π/4 for every cycle. The period of the function is π.

Step 1: Finding the Period

Start with the regular period of the \(\sin(x)\) and \(\csc(x)\), which is \(2\pi\). Now, take into account the horizontal stretches and compressions due to \(2x\). The new period, P, is \(\frac{2\pi}{|2|}=\pi\).

Step 2: Plotting Critical Points

For \(\sin( 2x)\) within one period from 0 to \(\pi\), the critical values occur at \(x=0\), \(x=\frac{\pi}{4}\), \(x=\frac{\pi}{2}\), \(x=\frac{3\pi}{4}\) and \(x=\pi\). The \(\csc\) function is undefined wherever \(\sin( 2x)\) is 0. So at \(x=0\) and \(x=\pi\), \(\csc (2 x)\) is undefined and there are vertical asymptotes. Draw these asymptotes on the graph.

Step 3: Plotting the Graph

Now, use the critical points of the \(\sin( 2x)\) function to plot the graph of \(\csc (2 x)\). At \(x=\frac{\pi}{4}\), \(\sin(2x)=1\) so \(\csc (2 x)=1\). At \(x=\frac{\pi}{2}\), \(\sin(2x)=0\) so \(\csc (2 x)\) is undefined. At \(x=\frac{3\pi}{4}\), \(\sin(2x)=-1\) so \(\csc (2 x)=-1\). Repeat this process for at least two cycles.

Key Concepts

Horizontal Stretches and Compressions

When graphing trigonometric functions, particularly transformations, understanding horizontal stretches and compressions is vital. This process entails modifying the standard function to either widen or narrow its graph along the horizontal axis. Trigonometric functions often undergo these changes, influenced by the coefficient in front of the variable within the function.
Consider the function \(\csc(2x)\). Here, the coefficient 2 causes a horizontal compression. To comprehend this, look at the function's input variable \(x\). If this variable is multiplied by a factor greater than 1, the graph compresses. Conversely, if it were a fraction (less than 1 but greater than 0), the graph would stretch out wider. To visualize horizontal compression, think of taking each point on the trigonometric function's graph and pulling it closer together towards the y-axis. This results in a graph where the features (like peaks and troughs) occur more frequently across the same interval than the original function would without compression.

Period of Trigonometric Functions

The period of a trigonometric function is the horizontal length of one complete cycle of the waveform. For basic trigonometric functions like \(\sin(x)\) and \(\csc(x)\), the standard period is \(2\pi\).
However, modifications involving stretches and compressions alter this period. For \(\csc(2x)\), the coefficient 2 compresses the period.

• Determine the new period by dividing the standard period, \(2\pi\), by the absolute value of the coefficient: \(\frac{2\pi}{2}=\pi\).
• This calculation shows that the function completes its cycle twice as quickly, so within \(\pi\), the entire waveform repeats.
• Thus, in a function like \(\csc(2x)\), for every \(\pi\) interval, you see a full cycle of the cosecant curve.

This adjustment in the period ultimately influences how you plot and interpret the function over a specified domain.

Vertical Asymptotes in Graphs

Vertical asymptotes occur where a trigonometric function, specifically one with reciprocal properties like \(\csc(x)\) or \(\sec(x)\), approaches infinity. These are crucial to sketching accurate graphs of such functions because they represent where the function is undefined.
In the context of \(\csc(2x)\):

• The sine component \(\sin(2x)\) becomes 0 at specific points, causing \(\csc(2x)\), the reciprocal, to become undefined.
• Identifying these points helps locate vertical asymptotes: for \(\csc(2x)\), at \(x=0\) and \(x=\pi\), there are vertical asymptotes.

To represent these on a graph:

• Draw vertical dashed lines at these undefined points. These lines remind us that the function doesn't exist there.
• This aids in understanding the behavior of the graph as it approaches these asymptotes, typically evoking increasing or decreasing without bound.

Vertical asymptotes not only signal where a graph splits but also guide us in correctly plotting and anticipating a trigonometric graph's shape.