Problem 22 - Precalculus: A Concise Course

Question

Sketch the graph of the function. Include two full periods. $$ y=3 \csc 4 x $$

Step-by-Step Solution

Answer

The graph of the function \( y = 3 \csc 4x \) is a graph with period \( \frac{\pi}{2} \), vertical asymptotes at points where \( \sin(4x) \) is zero, and tops and bottoms peaks at the values 3 and -3 respectively. The graph showcases two full periods.

Step 1: Determine the Period of the Function

The period of the function can be found by dividing \(2\pi\) by the coefficient of x present in the function. Here the coefficient of x is 4. Therefore, the period for this given cosecant function is \( \frac{2\pi}{4} = \frac{\pi}{2} \)

Step 2: Plot the Sine Function

Before sketching the function \( y = 3 \csc 4x \), it's helpful to first sketch the corresponding sine function, \( y = 3\sin 4x \), which has the same period and frequency. Draw a standard sine curve within one period.

Step 3: Plot Asymptotes for the Cosecant Function

The cosecant function has vertical asymptotes wherever the sine function is zero. This is because the cosecant function is undefined at these points. In the \(\sin(4x) \) function, the function value is zero at \(x=0\), \(x=\frac{\pi}{4}\), \(x=\frac{\pi}{2}\), etc. within the given period. Hence, set these x-values as asymptotes when graphing.

Step 4: Plot the Function

The curve of the cosecant function lies entirely above the x-axis where the sine function is positive, and below the x-axis where the sine function is negative. Draw these parts of the graph then extend beyond one period to include two full periods, repeating the pattern. The tops and bottoms peaks of the function hit at 3 and -3 respectively, which are got by multiplying the amplitude of the sine function (1) by 3.

Key Concepts

Period of Trigonometric Functions

Understanding the period of a trigonometric function is crucial to graphing it accurately. The period represents the length over which the function repeats its values. For the basic sine and cosine functions, this is commonly known as the time it takes for one complete cycle, which is typically \(2\pi\) radians.

However, when a coefficient is applied to the variable, such as the \(4x\) in the function \(y = 3 \csc 4x\), it will affect the period. To find the new period, simply divide the standard period (\(2\pi\) for sine and cosine) by the absolute value of the coefficient of the variable. In this case, the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

• This directly informs us how often the cosecant function will repeat its shaped pattern.
• When graphing trigonometric functions, it's essential to mark one full period on your graph to ensure you capture the essence of the function's behavior.

Meanwhile, other functions like tangent and cotangent have a standard period of \(\pi\), and the same rule for adjusting the period based on the coefficient applies.

Asymptotes in Trigonometric Graphs

Asymptotes play a pivotal role in graphing cosecant and secant functions, which are the reciprocals of sine and cosine. An asymptote is, in essence, an invisible line that the graph of the function will approach, but never actually touch or cross.

For cosecant, the function approaches infinity or negative infinity whenever the corresponding sine function equals zero since dividing by zero is undefined. In the function \(y = 3 \csc 4x\), you can identify the asymptotes by looking at the values of \(x\) where \(\sin(4x) = 0\). Here, the asymptotes occur at regular intervals of \(\frac{\pi}{4}\) within the identified period.

When graphing:

• Accurately draw vertical lines at these intervals to represent asymptotes.
• These lines guide you in sketching the curve, ensuring you avoid crossing these points as the function's value becomes infinitely large or small.

Remember, other trigonometric functions may also have asymptotes depending on their definition domains, which is essential to consider when plotting them.

Transformations of Trigonometric Functions

Transformations of trigonometric functions involve shifts, stretches, compressions, and reflections on the graph of the original function. They can dramatically change how the function appears when graphed. In the exercise given, the multiplication of the sine function by 3 to obtain \(y = 3 \csc 4x\) represents a vertical stretch by a factor of 3. This alteration affects the function's amplitude, causing the peaks of the cosecant function to hit at 3 and -3 instead of 1 and -1, respectively.

Other kinds of transformations include:

• Horizontal shifts, which are caused by adding or subtracting a constant to the \(x\) variable inside the trigonometric function.
• Vertical shifts, which result from adding or subtracting a constant outside the trigonometric function.
• Reflections, which occur when the function is multiplied by a negative factor, flipping it over the \(x\)-axis or \(y\)-axis depending on the function's form.

Keeping in mind, these transformations are systematic, they follow specific rules that, once understood, make graphing any trigonometric function a more methodical and approachable task.