/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 89 Determine the period and range o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 89

Determine the period and range of each function. $$y=-3 \csc (2 x-\pi)-4$$

Short Answer

Expert verified
The period is \pi\, and the range is \left(-\infty, -7\right] \cup \left[-1, \infty\right).

Step by step solution

01

Understanding the Cosecant Function

Recognize that the given function is related to the cosecant function. The general form of a cosecant function is \(y = a \csc(bx - c) + d\), where the parameters affect the period and vertical shift.
02

Identifying the Coefficients

From the given function \(y = -3 \csc(2x - \pi) - 4\), identify the coefficients: \(a = -3\), \(b = 2\), \(c = \pi\), and \(d = -4\).
03

Determining the Period

The period of the cosecant function is given by \frac{2\pi}{b}\. Here, \(b = 2\). Thus, the period is \frac{2\pi}{2} = \pi.\
04

Determining the Range

The range of the cosecant function is affected by the vertical shift and amplitude. Generally, the range of \(a \csc(x)\) is \left(-\infty, -|a|\right] \cup \left[|a|, \infty\right).\ Here, \(a = -3\), so the initial range is \left(-\infty, -3\right] \cup \left[3, \infty\right).\ After applying the vertical shift \(d = -4\), shift each part of the range down by 4 units: \left(-\infty, -7\right] \cup \left[-1, \infty\right).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

cosecant function
The cosecant function, often denoted as \(\text{csc}(x)\), is one of the fundamental trigonometric functions. It is defined as the reciprocal of the sine function: $$\text{csc}(x) = \frac{1}{\text{sin}(x)}$$. This function is unique because it is undefined wherever the sine function is zero. This means \(\text{csc}(x)\) has vertical asymptotes at these points.

The general form of the cosecant function is: $$y = a \text{csc}(bx - c) + d$$. Each parameter plays a significant role in transforming the function:
  • \(a\): Amplitude, or the vertical stretch/compression.
  • \(b\): Affects the period of the function.
  • \(c\): Horizontal shift.
  • \(d\): Vertical shift.
Understanding these changes helps in graphing and analyzing the behavior of the function.
period of a function
The period of a trigonometric function is the interval after which the function repeats its values. For the general cosecant function \(y = a \text{csc}(bx - c) + d\), the period is influenced by the coefficient \(b\).

To find the period, use the formula: $$\text{Period} = \frac{2\text{蟺}}{b}$$. For instance, in the function \(y = -3 \text{csc}(2x - \text{蟺}) - 4\), we identify \(b = 2\). So, the period is calculated as:
\[\text{Period} = \frac{2\text{蟺}}{2} = \text{蟺}\].

This means every 蟺 units along the x-axis, the cosecant function will repeat its values, giving insights into its cyclical nature.
range of a function
The range of a trigonomic function refers to the set of all possible output values (y-values). For the cosecant function \(y = a \text{csc}(bx - c) + d\), the range primarily depends on the amplitude \(a\) and the vertical shift \(d\).

In its untransformed state, the range of \(\text{csc}(x)\) is 袄(袄迟别虫迟调-鈭� < y 鈮� -1} 鈭� \text{1 鈮� y < 鈭瀩\). When transformed, \(a\) modifies this by scaling it vertically, and \(d\) shifts it up or down. For example, in the function \(y = -3 \text{csc}(2x - \text{蟺}) - 4\):
  • \(a = -3\) which scales the range to \(-3\).
  • \(d = -4\) which shifts the entire range 4 units down.
Therefore, the transformed range is:
袄(袄迟别虫迟调-鈭� < y 鈮� -7} 鈭� \text{-1 鈮� y < 鈭瀩\).
vertical shift in trigonometric functions
The vertical shift in a trigonometric function moves the entire graph up or down along the y-axis. In the general form \(y = a \text{csc}(bx - c) + d\), the parameter \(d\) represents the vertical shift.

A positive value of \(d\) moves the graph up, while a negative value shifts it downward. For example, in the function \(y = -3 \text{csc}(2x - \text{蟺}) - 4\):
  • The \(d = -4\) indicates a vertical shift of 4 units downwards.
Vertical shifts are essential for determining the range and the overall positioning of the graph on the coordinate plane. They remain consistent with the function's periodic behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine the period and range of each function. $$y=4 \csc (3 x-\pi)+5$$

Convert the degree measure \(225^{\circ}\) to radians.

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\sec (3 x)$$

Solve each problem. Photographing the Earth From an altitude of \(161 \mathrm{mi}\), the Orbiting Geophysical Observatory (OGO-1) can photograph a path on the surface of the earth that is approximately \(2000 \mathrm{mi}\) wide. Find the central angle, to the nearest tenth of a degree, that intercepts an arc of \(2000 \mathrm{mi}\) on the surface of the earth (radius 3950 mi). (IMAGE CANNOT COPY)

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\csc (3 x+\pi)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.