Problem 45
Question
Use a graphing utility to graph the function. Include two full periods. $$ y=-\csc (4 x-\pi) $$
Step-by-Step Solution
Verified Answer
The graph of the function \(y=-csc (4x-\pi)\) will have two 'n' shaped curves in each period, starting at x=\(\pi\), with asymptotes at integer multiples of \(\pi/4\), repeating every new period of \(\pi/4\). The graph will alternate between being above and below the x-axis due to the negative sign.
1Step 1: Understanding Cosecant Function
The cosecant function is the reciprocal of the sine function, and so never equals zero. It is undefined at integer multiples of \(\pi\). Its basic pattern is a 'U' shape that opens either up or down, repeating every \(\pi\) units. Its range is either above 1 or below -1.
2Step 2: Adjusting for Frequency and Phase
The coefficient of 4 affects the frequency of the function. It will make the graph oscillate four times as fast, shortening the period from \(\pi\) to \(\pi/4\). The phase shift of \(-\pi\) shifts the whole graph \(\pi\) units to the right. This means the 'U' shape will start at \(\pi\) and repeat every \(\pi/4\) units.
3Step 3: Drawing the Graph
Start with marking the asymptotes, which will be at multiples of the new period, \(\pi/4\), after the phase shift of \(\pi\). The 'U' shapes will be in the intervals between asymptotes. Remember the graph opens downward due to the negative sign, so draw 'n' shapes between asymptotes. Repeating this pattern, include two periods on each side of the y-axis. This will ensure two full periods are depicted.
Key Concepts
Cosecant FunctionPhase ShiftFrequencyGraph Analysis
Cosecant Function
The cosecant function is closely linked to the sine function, as it is its reciprocal. This means it is defined wherever the sine function is not zero. As a result, the cosecant function is undefined at integer multiples of \( \pi \).
When graphed, the cosecant function displays a distinct 'U' shape, which opens either upwards or downwards. This pattern is repeated every \( \pi \) units, forming a sequence of curves and vertical asymptotes. The range of the cosecant function is typically greater than 1 or less than -1. These characteristics are fundamental to understanding how to visualize the function on a graph.
When graphed, the cosecant function displays a distinct 'U' shape, which opens either upwards or downwards. This pattern is repeated every \( \pi \) units, forming a sequence of curves and vertical asymptotes. The range of the cosecant function is typically greater than 1 or less than -1. These characteristics are fundamental to understanding how to visualize the function on a graph.
Phase Shift
A phase shift in trigonometric functions occurs when the graph is shifted horizontally along the x-axis. For the function \( y = -\csc(4x - \pi) \), the phase shift is determined by solving the equation inside the function, \( 4x - \pi \).
To find the shift, set this equal to zero:
\[ 4x - \pi = 0 \]
Solve for \( x \) to get \( x = \frac{\pi}{4} \). This means the entire graph shifts \( \frac{\pi}{4} \) units to the right.
To find the shift, set this equal to zero:
\[ 4x - \pi = 0 \]
Solve for \( x \) to get \( x = \frac{\pi}{4} \). This means the entire graph shifts \( \frac{\pi}{4} \) units to the right.
- Start of the graph is shifted right.
- Ensure to adjust all relevant points accordingly.
Frequency
The frequency of a trigonometric function like cosecant determines how often the 'U' shapes occur within a given interval. In the function \( y = -\csc(4x - \pi) \), the coefficient 4 in front of \( x \) plays a significant role.
Frequency is directly related to the period. For sine and cosecant functions, the standard period is \( \pi \). To find the new period, divide the standard period by the absolute value of the coefficient. So, we have:
\[ \text{New Period} = \frac{\pi}{4} \]
This means the graph repeats itself four times within the typical length of one period. It creates closely packed 'U' shapes, showing how the graph's frequency has increased, resulting in more oscillations.
Frequency is directly related to the period. For sine and cosecant functions, the standard period is \( \pi \). To find the new period, divide the standard period by the absolute value of the coefficient. So, we have:
\[ \text{New Period} = \frac{\pi}{4} \]
This means the graph repeats itself four times within the typical length of one period. It creates closely packed 'U' shapes, showing how the graph's frequency has increased, resulting in more oscillations.
Graph Analysis
Graphing the cosecant function involves careful consideration of several components. Here's how to graph \( y = -\csc(4x - \pi) \):
- First, calculate the asymptotes. These are vertical lines occurring every \( \frac{\pi}{4} \) units after a shift of \( \pi \) units.
- Next, plot the 'U' shapes between these asymptotes. The negative sign indicates that these will open downwards, like an 'n'.
- Ensure to include two full periods to capture the graph's behavior accurately. Each period covers \( \frac{\pi}{4} \), which marks how the graph repeats its oscillation pattern.
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