Problem 76 - Precalculus

Question

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. $$ f(x)=-\frac{1}{4} \csc (x) $$

Step-by-Step Solution

Answer

Vertical shrink by 4, period is \(2\pi\), asymptotes at \(n\pi\).

Step 1: Understand the Function

The function given is \( f(x) = -\frac{1}{4} \csc(x) \). Recall that the cosecant function \( \csc(x) \) is the reciprocal of the sine function \( \sin(x) \), so it is undefined where \( \sin(x) = 0 \). The basic graph of \( \csc(x) \) has vertical asymptotes at integer multiples of \( \pi \) because that's where \( \sin(x) = 0 \).

Step 2: Determine the Stretching Factor

The function can be rewritten as \( f(x) = -\frac{1}{4} \cdot \csc(x) \). The leading coefficient \(-\frac{1}{4}\) represents a vertical stretching factor. The graph of \( \csc(x) \) is vertically shrunk by a factor of 4 and reflected across the x-axis.

Step 3: Determine the Period

The period of \( \csc(x) \) is the same as \( \sin(x) \), which is \( 2\pi \). The multiplier in the argument of \( \csc(x) \) is 1, hence the period remains \( 2\pi \).

Step 4: Identify the Asymptotes

Since \( \csc(x) \) is undefined where \( \sin(x) = 0 \), the vertical asymptotes will be at \( x = n\pi \), where \( n \) is an integer. This is because \( \sin(x) = 0 \) at these points.

Step 5: Sketch the Graph

Start by plotting the vertical asymptotes at \( x = 0, \pi, 2\pi, -\pi, -2\pi \). The basic shape of \( \csc(x) \) between \( (0, \pi) \) and \( (-\pi, 0) \) involves curves originating to \( +\infty \) towards the asymptotes and dipping down towards \( -\infty \) towards zero. Reflect these shapes across the x-axis and apply the vertical shrink; these reflections between asymptotes will occupy one period of \( \csc(x) \), while another follows for the remaining periods.

Key Concepts

Vertical Stretching Factor

The vertical stretching factor is a numerical value that modifies how a function's graph stretches or shrinks vertically. When you look at the function \( f(x) = -\frac{1}{4} \csc(x) \), the vertical stretching factor is \(-\frac{1}{4}\). This means each y-value that you would normally get from the standard \( \csc(x) \) function is first multiplied by \(-1/4\).
This change has two effects:

• First, it shrinks the height of the standard cosecant graph by a factor of 4. What used to go to, say, 4 units high now only goes to 1 unit.
• Second, the negative sign flips the graph upside down across the x-axis. So peaks become troughs and vice-versa.

When sketching the graph with this new vertical stretch factor, you should expect to see lower and inverted curves compared to the parent \(\csc(x)\) graph.

Period of a Function

The period of a function refers to the interval over which the function's graph repeats itself. For a cosecant function \(\csc(x)\), just like the sine function, this period is usually \(2\pi\).
In our function \( f(x) = -\frac{1}{4} \csc(x) \), the period remains at \(2\pi\) because there's no horizontal scaling factor altering the basic period. This is an important feature that tells us how frequently the patterns of the function recur.
When drawing your graph, you'll be sketching two full cycles of this pattern, making sure each cycle spans across an interval of \(2\pi\). Always remember the period is key to understanding where the function will repeat its behavior.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never actually touches. They occur in the cosecant function where the sine function is zero, as \(\csc(x) = 1/\sin(x)\), and cosine is undefined whenever \(\sin(x) = 0\).
In the function \( f(x) = -\frac{1}{4} \csc(x) \), vertical asymptotes will show up at integer multiples of \(\pi\), like \( x = 0, \pi, 2\pi, -\pi, \text{and} -2\pi\).

• These asymptotes form imaginary lines that mark the boundaries between the repeating curves of the function.
• They help us know where the graph dips toward infinity or negative infinity, behaving erratically as it approaches these lines.

By marking the asymptotes before sketching, you're able to correctly structure each period in your graph. They act like vertical guides to ensure the function is plotted correctly.