/*! 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 121 Find the \(x\) -intercept(s) of ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 121

Find the \(x\) -intercept(s) of \(y=A \sin B x+A\)

Short Answer

Expert verified
The x-intercepts occur at \( x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \), where \( k \) is an integer.

Step by step solution

01

Set Y to Zero

To find the x-intercepts of the function, we need to set the equation equal to zero because the x-intercepts occur where the graph crosses the x-axis (i.e., where \( y = 0 \)). Thus, set \[ 0 = A \sin(Bx) + A \]
02

Solve for \\sin(Bx)

From the equation \( 0 = A \sin(Bx) + A \), subtract \( A \) from both sides to isolate the sine term: \[ -A = A \sin(Bx) \]. Then, divide both sides by \( A \) (assuming \( A eq 0 \)): \[ -1 = \sin(Bx) \].
03

Find Solutions for \\sin(Bx) = -1

The values of \( x \) for which \( \sin(Bx) = -1 \) occur at specific points: \[ Bx = \frac{3\pi}{2} + 2k\pi \], where \( k \) is an integer. These values are where the sine function repeats its pattern.
04

Solve for x

Solve for \( x \) by dividing both sides of the equation from Step 3 by \( B \): \[ x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \]. This provides the general form for the x-intercepts of the function.

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.

x-intercepts
X-intercepts are the points where a graph crosses the x-axis. This means that at these points, the function's value is zero. For any trigonometric equation like \( y = A \sin(Bx) + A \), identifying these points requires setting the equation equal to zero.
Step-by-step, you would:
  • Set the function equal to zero: \( 0 = A \sin(Bx) + A \)
  • Isolate \( \sin(Bx) \) by manipulating the equation: \( -A = A \sin(Bx) \)
  • Divide by \( A \) assuming \( A eq 0 \), resulting in: \( -1 = \sin(Bx) \)
This equation shows where the sinusoidal wave crosses the x-axis. It is crucial to understand that solving trigonometric functions for x-intercepts often involves finding angles corresponding to particular function values.
sine function
The sine function is a periodic and smooth curve that oscillates between -1 and 1. In any expression like \( y = A \sin(Bx) + A \), changes in the function's parameters directly affect its shape and position.
Here's a quick breakdown:
  • Amplitude: The coefficient \( A \) determines the amplitude (the wave's height from the center to its peak).
  • Frequency: The coefficient \( B \) affects the frequency, which is how often the wave oscillates in a given period.
  • Transformation: Adding the constant \( A \) to the sine function shifts the whole curve upwards.
Understanding these characteristics helps interpret the sine function's behavior over its domain and better solve related problems such as finding x-intercepts.
periodicity
Periodicity is a fundamental feature of the sine function, meaning it repeats itself after a certain interval called the period. For a function like \( y = A \sin(Bx) + A \), knowing the period is crucial for calculating where certain values, like the x-intercepts, occur repeatedly.
Here's what you need to know:
  • The period of \( \sin(Bx) \) is calculated as \( \frac{2\pi}{B} \). This tells us how long it takes for the sine wave to complete one full cycle.
  • Understanding periodicity lets you project where the function will cross the x-axis again, as observed in the general formula for x-intercepts: \( x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \).
  • This formula incorporates the idea of periodicity by including \( 2k\pi \), which accounts for the function's repeating nature.
Mastering periodicity is key to understanding and working with functions that involve trigonometric components, as it explains their repetitive nature.

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

What is the period of the function \(y=\tan x+\cot x ?\) Use a graphing calculator to graph \(Y_{1}=\tan x+\cot x\) and \(Y_{3}=2 \csc (2 x)\) in the same viewing window.

$$\int_{7 \pi / 6}^{5 \pi / 4} \sec ^{2} x d x$$

For Exercises \(87-90\), refer to the following: Set the calculator in parametric and radian modes and let $$ \begin{array}{l} X_{1}=\cos T \\ Y_{1}=\sin T \end{array} $$ (TABLE CANNOT COPY) Set the window so that \(0 \leq \mathrm{T} \leq 2 \pi,\) step \(=\frac{\pi}{15},-2 \leq \mathrm{X} \leq 2\) and \(-2 \leq Y \leq 2 .\) To approximate the sine or cosine of a T value, use the \([\text { TRACE }]\) key, type in the T value, and read the corresponding coordinates from the screen. Approximate \(\sin \left(\frac{2 \pi}{3}\right)\) to four decimal places.

In Exercises \(57-66,\) state the domain and range of the functions. $$y=2 \sec (5 x)$$

Determine whether each statement is true or false. Is cosecant an even or an odd function? Justify your answer.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision 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.