/*! 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 51 The problems that follow review ... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

The problems that follow review material we covered in Section 4.3. Graph one complete cycle. $$ y=3 \sin \left(\pi x-\frac{\pi}{2}\right) $$

Short Answer

Expert verified
Graph starts at \( x = \frac{1}{2} \) and ends at \( x = \frac{5}{2} \), with peaks and troughs determined by the amplitude and phase shift.

Step by step solution

01

Identify Key Parameters for the Sine Function

The general form of a sine function is \( y = a \sin(bx + c) + d \). For \( y = 3 \sin\left(\pi x - \frac{\pi}{2}\right) \), the parameters are: amplitude \( a = 3 \), frequency factor \( b = \pi \), and phase shift \( c = -\frac{\pi}{2}\). The vertical shift \( d \) is 0.
02

Determine Amplitude, Period, and Phase Shift

The amplitude is \( |a| = 3 \). The period of the sine function is given by \( \frac{2\pi}{b} = \frac{2\pi}{\pi} = 2 \). The phase shift is calculated as \( -\frac{c}{b} = \frac{\frac{\pi}{2}}{\pi} = \frac{1}{2} \). This indicates a horizontal shift to the right by \( \frac{1}{2} \).
03

Calculate Key Points on the Cycle

A sine cycle starts at the phase shift, peaks at a quarter period (\( \frac{1}{2} \), repeats midway (\( 1 + \frac{1}{2} \)) and completes a full cycle at the period. Key points are at \( x = \frac{1}{2}, \frac{5}{4}, 1, \frac{3}{2}\).
04

Plot the Sine Function

For \( x = \frac{1}{2} \), \( y = 3 \sin\left( \pi \times \frac{1}{2} - \frac{\pi}{2} \right) = 3\sin(0) = 0 \). At \( x = \frac{5}{4} \), \( y = 3\sin\left( \frac{3\pi}{4} - \frac{\pi}{2} \right) = 3\sin\left( \frac{\pi}{4} \right) = \frac{3}{\sqrt{2}} \approx 2.12 \). At \( x=1 \), \( y=3\sin(\pi) = 0 \). At \( x = \frac{3}{2} \), \( y = 3\sin\left( \frac{3\pi}{2} - \frac{\pi}{2} \right) = -3 \).
05

Draw the Graph

Using the key points and understanding of sine function properties (start at phase shift, peak, cross midline, hit a trough, complete cycle), draw smooth wave from \( x = \frac{1}{2} \) to \( x = \frac{5}{2} \). The cycle includes points: (\( \frac{1}{2}, 0 \)), (\( \frac{5}{4}, 2.12 \)), (\( 1, 0 \)), (\( \frac{3}{2}, -3 \)).

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 91Ó°ÊÓ!

Key Concepts

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

Amplitude
The amplitude of a sine function is a measure of its vertical stretch or compression. It represents the maximum distance that the function's wave reaches from its central axis, usually the x-axis unless there's a vertical shift.
In mathematical terms, the amplitude is indicated by the coefficient "a" in the function's formula:
  • The general formula is: \[ y = a \sin(bx + c) + d \]
  • Here, the amplitude is \(|a|\), which means you only consider the positive magnitude.
For the function given in the exercise, \[ y = 3 \sin\left(\pi x - \frac{\pi}{2}\right), \]the amplitude is 3.
This means the sine wave will rise to a maximum of 3 units above the x-axis and descend to 3 units below it. Amplitude never reflects a negative value, as it’s purely about distance. If you imagine a wave, the amplitude determines its height. It affects how "tall" the peaks and how "deep" the troughs of the wave will feel when you graph it.
Period
The period of a sine function defines how long it takes for the wave to complete a full cycle before repeating. This characteristic is determined by the formula:
  • \[ \text{Period} = \frac{2\pi}{b} \]
  • Here, "b" is the frequency factor.
The period tells you the length along the x-axis needed for the wave to represent a single complete oscillation from start to finish, including moving through all phases such as rising, peaking, falling, and reaching its initial point again.
In our function \[ y = 3 \sin\left(\pi x-\frac{\pi}{2}\right), \]the frequency factor is \(b = \pi\). Therefore, the period of this sine wave is\[ \frac{2\pi}{\pi} = 2. \]
This means every 2 units along the x-axis, you will see the full cycle of the sine wave repeated.
Period is essential because it helps us to scale the x-axis properly when graphing, ensuring we capture the full form of the sine wave, complete with its highs and lows.
Phase Shift
The phase shift in a sine function describes how the graph is horizontally shifted from the regular sine wave's starting point. It determines where the cycle begins on the x-axis.
This shift is calculated through the formula:
  • \[ \text{Phase Shift} = -\frac{c}{b} \]
It involves the frequency factor "b" and the phase constant "c", the coefficient in front of the variable within the function arguments.
For the equation\[ y = 3 \sin\left(\pi x - \frac{\pi}{2}\right), \]"c" is \(-\frac{\pi}{2}\), and "b" is \(\pi\).
  • This results in a phase shift of \[ \frac{\frac{\pi}{2}}{\pi} = \frac{1}{2}. \]
This positive value indicates a shift to the right by \(\frac{1}{2}\) units on the x-axis, meaning the wave’s usual starting point is moved half a unit forward.
Understanding phase shift is crucial, as it helps position the sine wave accurately on a graph, ensuring you pinpoint the right moments for peak, trough, and intercepts.

One App. One Place for Learning.

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

Get started for free

Study anywhere. Anytime. Across all devices.