Problem 49 Graph each function using transl... [FREE SOLUTION]

Chapter 4: Problem 49

Graph each function using translations. $$y=\csc \frac{x}{2}-1$$

Short Answer

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The graph of the given function first has a period of $2\pi$ as a result of the horizontal stretch, compared to the $\pi$ period of the base function, $\csc x$. The graph is not affected by x-values at multiples of $2\pi$ where $\sin x$ becomes 0. The graph then has every point moved down by 1 unit due to the vertical shift. This results in a transformed graph of the base function $\csc x$.

Step by step solution

Identify Transformations

First, identify the transformations on the function \[ y= \csc x \]. Here the function is transformed to \[ y = \csc \frac{x}{2} - 1 \]. This function includes a horizontal stretch by a factor of 2 and vertical shift down 1 unit.

Graph the Base Function

Next, graph the base function, which is \[ y = \csc x \]. Remember that $\csc x = 1 / \sin x$ and is undefined where $\sin x = 0$ (i.e., $x = n\pi$, where $n$ is any integer). The function has maximum and minimum points at every $ \pi $ in the positive and negative y-direction.

Apply the Horizontal Stretch

After graphing the base function, apply the horizontal stretch of factor 2. In effect, the period is being doubled, changing from $ \pi $ to $ 2\pi $. The graph will look horizontally 'stretched' or 'compressed' compared to before.

Apply the Vertical Shift

Finally, apply the vertical shift by moving each point down by 1 unit. This shifts entire graph down by 1 unit on y-axis. Afterwards, double check your graph to make sure all transformations have been accurately applied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Translations

Translations in graphing refer to shifting the entire graph of a function to a different position without altering its shape. In the context of trigonometric functions, this is a common transformation. Typically, you can translate a graph either vertically or horizontally.

Vertical translations involve shifting the graph up or down. This will modify the range of the function but will keep the relative positioning of points intact.
Horizontal translations, on the other hand, shift the graph left or right. This changes the phase of the function but does not affect its range.

When you look at the function $y= \csc \frac{x}{2} - 1$, the component $-1$ implies a vertical translation. No horizontal translation occurs unless there is a term added or subtracted within the function argument itself. These translations help in understanding and predicting changes in the graph's visual output.

Cosecant Function

The cosecant function, denoted as $ \csc x $, is the reciprocal of the sine function, essentially $ \csc x = \frac{1}{\sin x} $. Unlike sine, which has values ranging between -1 and 1, the cosecant function's outputs are either greater than or equal to 1, or less than or equal to -1. Because cosecant cannot be determined when sine equals zero (e.g., $x = n\pi$), the graph has asymptotes at these points.
Here are some key features of the cosecant function:

It has a periodic nature, repeating every $2\pi$ units.
Contains vertical asymptotes where the sine values become zero ($x = n\pi$).
The function produces arcs that go upwards and downwards to infinity, relative to maximum and minimum sinusoidal points it reciprocates.

Understanding these features is crucial when graphing the function initially before any transformations.

Horizontal Stretch

A horizontal stretch affects the period of a trigonometric function, essentially spreading the graph across a longer or shorter span of the x-axis. When you apply a horizontal stretch, you're changing the "speed" at which the function proceeds through its cycle. This is observed in the given function $y = \csc \frac{x}{2} - 1$.
In the equation $ \csc \frac{x}{2} $, the function is experiencing a horizontal stretch by a factor of 2 due to the division by 2 inside the function argument.

The normal period of $ \csc x $ is $2\pi$.
In $ \csc \frac{x}{2} $, the period doubles to $4\pi$. This results in the graph appearing stretched, with each repeated segment of the graph occupying more space on the x-axis compared to the unmodified function.

This stretching alters how often certain features like peaks and troughs occur.

Vertical Shift

Vertical shifts are straightforward transformations where the entire graph of a function is moved up or down along the y-axis. In the particular example of $y = \csc \frac{x}{2} - 1$, the $-1$ at the end causes a downward vertical shift.
Here鈥檚 how vertical shifts work:

A vertical shift downwards translates every point of the graph lower by a specific number of units; here, it is 1 unit down.
The principal use of a vertical shift is to adjust the baseline of the function graph. For cosecant, this implies the entire curve which would sit above and below +1 and -1 is now adjusted to levels -1 to -2 for minimums, and upwards arcs are similarly adjusted.

This means our graph's appearance relative to the y-axis centerline changes, a crucial element in portraying the alterations in real-world scenarios.

91影视

Graph each function using translations. $$y=\csc \frac{x}{2}-1$$

Short Answer

Step by step solution

Identify Transformations

Graph the Base Function

Apply the Horizontal Stretch

Apply the Vertical Shift

Key Concepts

Translations

Cosecant Function

Horizontal Stretch

Vertical Shift

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