/*! 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 84 In Exercises 83-86, assume \(A\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 84

In Exercises 83-86, assume \(A\) and \(B\) are positive real numbers. $$ \text { Find the } y \text {-intercept of the function } y=A \sin (B x) \text {. } $$

Short Answer

Expert verified
The y-intercept is at the point (0, 0).

Step by step solution

01

Identify the y-intercept location

The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the input value, \(x\), is equal to 0.
02

Substitute x with 0 in the function

Substitute \(x = 0\) into the function equation \(y = A \sin(Bx)\). This results in \(y = A \sin(B \cdot 0)\).
03

Simplify the expression

Recall that the sine of 0 is 0, i.e., \(\sin(0) = 0\). Substitute this into the equation to get \(y = A \cdot 0 = 0\).
04

State the y-intercept

The y-coordinate of the y-intercept is 0. Therefore, the y-intercept of the function \(y = A \sin(Bx)\) is the point \((0, 0)\).

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.

Sine Function
The sine function is a fundamental part of trigonometry and plays a crucial role in various mathematical settings, including calculus and physics. The sine function is often represented as \( \sin(x) \) and maps angles, typically measured in radians, to a ratio of side lengths in a right-angled triangle.
The sine function has several key characteristics:
  • It is periodic, which means it repeats its values in regular intervals. For the sine function, this interval, or period, is \( 2\pi \).
  • It oscillates between values of -1 and 1. Hence, the range of the sine function is \([-1, 1]\).
  • It has a wave-like pattern, often referred to as a sine wave, which can model various real-world phenomena such as sound and light waves.
When using the sine function in equations, it applies not just to pure angles. For example, in the given function \(y = A \sin(Bx)\), the amplitude and frequency of the sine wave can be modified with the coefficients \(A\) and \(B\), respectively.
Real Numbers
Real numbers are a set that includes both rational and irrational numbers. They form the continuous number line used in mathematics. Here are some key points to remember about real numbers:
  • They include positive numbers, negative numbers, and zero.
  • Examples of real numbers are integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\).
  • In the context of the function \(y = A \sin(Bx)\), both \(A\) and \(B\) are specified as positive real numbers. This means they can take on any positive value on the continuous number line.
  • Real numbers are crucial because they allow us to perform calculus and analyze continuous functions and curves, like those of the sine wave.
Understanding real numbers is essential in exploring how different coefficients in a function influence its graph. Functions with real number coefficients provide a complete picture over a continuous interval.
Graph of Functions
The graph of a function visually represents the relationship between the input values (often represented as \(x\)) and the output values (often represented as \(y\)). The graph of a sine function, such as \(y = A \sin(Bx)\), is an excellent example of this relationship in action.
Several features define the graph of the sine function:
  • Y-intercept: This is where the graph crosses the y-axis, corresponding to \(x = 0\). For the function \(y = A \sin(Bx)\), as solved, the y-intercept is \((0,0)\).
  • Periodicity: The function repeats its values every \(2\pi\) for \(\sin(x)\). However, the coefficient \(B\) in \(y = A \sin(Bx)\) modifies this to \(\frac{2\pi}{B}\).
  • Amplitude: This is the maximum distance from the center line to the peak. In our function, the amplitude is \(A\).
  • Symmetry: Sine functions are symmetric about the origin, a feature known as odd symmetry, meaning \(\sin(-x) = -\sin(x)\).
By plotting these values, students can understand how variables affect the graph and develop a deeper understanding of functions and their graphs.

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

In Exercises 83-86, determine whether each statement is true or false. $$ A \sin \left(\omega t+\frac{\omega \pi}{2}\right)=A \cos (\omega t) $$

In Exercises 51-56, state the domain and range of the functions. $$ y=2 \sec (5 x) $$

For Exercises 65-68, refer to the following: Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of QualComp computers during 2008-2010 is approximated by the function $$ s(t)=0.098 \sin (0.79 t+2.37)+0.387 \quad 1 \leq t \leq 12 $$ where \(t\) represents time in quarters ( \(t=1\) represents the end of the first quarter of 2008), and s(t) represents computer sales (quarterly revenue) in millions of dollars. Business/Economics. Find the vertical shift. Interpret its meaning.

For Exercises 65-68, refer to the following: Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of QualComp computers during 2008-2010 is approximated by the function $$ s(t)=0.098 \sin (0.79 t+2.37)+0.387 \quad 1 \leq t \leq 12 $$ where \(t\) represents time in quarters ( \(t=1\) represents the end of the first quarter of 2008), and s(t) represents computer sales (quarterly revenue) in millions of dollars. Business/Economics. Find the phase shift. Interpret its meaning.

In Exercises 63-66, determine whether each statement is true or false. For what values of \(n\) do \(y=\tan x\) and \(y=\tan (x-n \pi)\) have the same graph?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

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.