Problem 78 Determine the period and sketch ... [FREE SOLUTION]

Chapter 5: Problem 78

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\frac{1}{2} \sec x$$

Short Answer

Expert verified

The period is $2\theta$, the range is $(-鈭�, -\frac{1}{2}] \text{ U } [\frac{1}{2}, 鈭�)$. Sketch from $\theta = -\frac{\theta}{2}$ to $\theta = \frac{\theta}{2}$.

Step by step solution

Understand the basic function

The given function is $y = \frac{1}{2} \text{sec} \theta$. The secant function, $\text{sec} \theta$, is the reciprocal of the cosine function. Therefore, $y = \frac{1}{2} \text{sec} \theta$ can be rewritten as $y = \frac{1}{2\text{cos} \theta}$.

Identify the period of the function

The period of the secant function, $\text{sec} \theta$, is $2\theta$. Hence, the period of $y = \frac{1}{2} \text{sec} \theta$ will be $2\theta$ as well.

Determine vertical asymptotes

Since the secant function is undefined where the cosine function is zero, $y = \frac{1}{2} \text{sec} \theta$ will have vertical asymptotes where $\theta = \frac{(2n+1)\theta}{2}$, where n is an integer.

Range of the function

The range of the secant function is $(-鈭�, -1] \text{ U } [1, 鈭�)$. Since we have $y = \frac{1}{2} \text{sec} \theta$, this scaling factor would shrink the usual secant values, resulting in $(-鈭�, -\frac{1}{2}] \text{ U } [\frac{1}{2}, 鈭�)$.

Sketch the graph

To sketch one cycle of the graph start from $\theta = -\frac{\theta}{2}$ to $\theta = \frac{\theta}{2}$, taking note of where the function is undefined (vertical asymptotes at $\theta = -\frac{\theta}{2}, 0, \frac{\theta}{2}$). Label the x-axis with intervals of $\frac{\theta}{2}$. Plot the points where $y = \frac{1}{2}$ and $y = -\frac{1}{2}$, then draw the typical U-shaped secant curves between these points, approaching the asymptotes.

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Key Concepts

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

vertical asymptotes

Vertical asymptotes occur where a function approaches infinity as the input approaches a certain value. For the secant function, these asymptotes happen where the cosine function is zero because the secant function is the reciprocal of the cosine function.

For the function $y = \frac{1}{2} \sec \theta$, we find vertical asymptotes where $ \cos \theta = 0 $. These occur at $\theta = \frac{(2n+1)\pi}{2}$ where $ n $ is an integer.

Thus, the asymptotes for $y = \frac{1}{2} \sec \theta$ exist at:

$\theta = -\frac{\pi}{2}$
$\theta = \frac{\pi}{2}$
$\theta = \frac{3\pi}{2}$
and so on.

Understanding these asymptotes is crucial for sketching the graph accurately.

function range

The range of a function is the set of all possible output values. For the secant function $ \sec \theta $, the range is $(-鈭�, -1] \cup [1, 鈭�) $.

When we scale the secant function by $ \frac{1}{2} $, the range shrinks proportionally. Thus, the range for $y = \frac{1}{2} \sec \theta$ changes to $(-鈭�, -\frac{1}{2}] \cup [\frac{1}{2}, 鈭�)$. This means the values will be either greater than or equal to $\frac{1}{2}$, or less than or equal to $-\frac{1}{2}$.

Specifically, the output values can't lie within $(-\frac{1}{2},\frac{1}{2})$. Understanding the range helps in predicting the behavior of the function and its graph.

graph sketching

When sketching the graph of $y = \frac{1}{2} \sec \theta$, follow these steps:

- Identify vertical asymptotes: From the earlier section, we know they occur at $\theta = \frac{(2n+1)\pi}{2}$.
- Determine the range: The function outputs values in the intervals $(-鈭�, -\frac{1}{2}] \cup [\frac{1}{2}, 鈭�)$.
- Plot key points: These are at the maximum and minimum values $y = \frac{1}{2}$ and $y = -\frac{1}{2}$.

Begin your sketch from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, noting asymptotes at the boundaries and zero. Plot the points at intervals of $\frac{\pi}{2}$. Draw the typical U-shapes of the secant curve, ensuring they approach but never touch the asymptotes.

This methodical approach ensures a clear and accurate graph of one cycle of the secant function.

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91影视

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\frac{1}{2} \sec x$$

Short Answer

Step by step solution

Understand the basic function

Identify the period of the function

Determine vertical asymptotes

Range of the function

Sketch the graph

Key Concepts

vertical asymptotes

function range

graph sketching

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