/*! 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 3 Use the techniques of shifting, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 3

Use the techniques of shifting, stretching, compressing, and reflecting to sketch at least one cycle of the graph of the given function. $$ y=2-\sin x $$

Short Answer

Expert verified
The graph is a vertically shifted and reflected sine wave with peaks at 3 and troughs at 1.

Step by step solution

01

Identify the Base Function

The given function is \( y = 2 - \sin x \). The base function here is \( \sin x \). Understanding how the base function behaves will help us apply transformations.
02

Recognize Transformations

In \( y = 2 - \sin x \), we identify two transformations: a vertical shift and a reflection. The base function \( \sin x \) is reflected across the x-axis, because of the negative sign before it, giving us \( -\sin x \). It is then shifted upwards by 2 units due to the +2.
03

Graph the Reflected Function

Start by graphing \( -\sin x \). The reflection across the x-axis means for any point \( (x, \sin x) \), we now have \( (x, -\sin x) \). Thus, the maximum amplitude becomes -1, and the minimum becomes 1.
04

Apply the Vertical Shift

Now, take the reflected function \( -\sin x \) and shift the entire graph upward by 2 units. This means the highest value of the sine wave is 2 + 1 = 3, and the lowest value is 2 - 1 = 1.
05

Sketch at Least One Cycle

The sine function \( \sin x \) has a period of \( 2\pi \). Therefore, one full cycle of \( -\sin x \), spanning from 0 to \( 2\pi \), will fit the transformations. Mark the key points: starting at \( (0, 2) \), peak at \( (\frac{\pi}{2}, 3) \), crosses the middle line at \( (\pi, 2) \), minimum at \( (\frac{3\pi}{2}, 1) \), and back to starting level at \( (2\pi, 2) \).

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.

Vertical Shift
When working with transformations of graphs, a vertical shift is a straightforward adjustment. It involves moving a graph up or down without altering its shape. For the function \( y = 2 - \sin x \), the "+2" in the equation signifies a vertical shift. Specifically, each point on the graph \( -\sin x \) is moved 2 units upwards.
This means, for every point \((x, -\sin x)\), it shifts to \((x, -\sin x + 2)\).
This affects the range of the function:
  • The highest value increases by 2 units.
  • The lowest value increases by 2 units.
For our function, the maximum of the sine wave becomes \( 3 \), and the minimum becomes \( 1 \). These changes are critical in sketching the correct graph.
Reflection
Reflections are another central aspect of graph transformations. For the function \( y = 2 - \sin x \), the reflection occurs due to the negative sign in front of \( \sin x \).
This reflects the graph over the x-axis. In simpler terms, the peaks and troughs of the wave are inverted. For any point \((x, \sin x)\), it becomes \((x, -\sin x)\).
  • The original peaks of \( \sin(x) \) at 1 become troughs at -1 for \( -\sin(x) \).
  • Conversely, the original troughs of \( \sin(x) \) at -1 are peaks at 1 for \( -\sin(x) \).
This reflection is an essential step before applying any shifts or further transformations.
Trigonometric Functions
Trigonometric functions, like sine and cosine, are foundational in mathematics, especially in understanding waves and oscillations. The base function in the exercise is \( \sin(x) \), a periodic function that oscillates between -1 and 1. This periodic nature (period \( 2\pi \)) means it repeats its pattern every \( 2\pi \) units.
  • Key characteristics include amplitude, period, and frequency.
  • \( \sin(x) \) has peaks at \( 1 \), troughs at \( -1 \), and crosses the x-axis periodically.
These are crucial in addressing how transformations will impact the function. For instance, knowing \( \sin(x) \)'s behavior helps us visualize transformations like reflection or vertical shifts.
Sinusoidal Graphs
Sinusoidal graphs are a type of periodic graph that visually represent sine and cosine functions. These graphs are characterized by their wave-like shape, with consistent amplitude and wavelength.
In the problem, we begin with \( y = \sin x \), a classic sine wave.
  • The function \( y=2-\sin x \) transforms this wave into a reflected and shifted version.
  • Such modifications result in different maximum and minimum points, but the wave nature remains.
Sketching sinusoidal graphs post-transformations involves identifying these new points and drawing the smooth periodic curve that connects them.

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

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=-2 \cos \left(2 x-\frac{\pi}{6}\right) $$

Find the period and the vertical asymptotes of the given function. Sketch at least one cycle of the graph. $$ y=-\sec x $$

Find the indicated value without the use of a calculator. $$ \sec \frac{29 \pi}{4} $$

Use a sum-to-product-formula in Theorem 4.7.2 to find the exact value of the expression. Do not use a calculator. $$ 2 \cos 195^{\circ}-2 \cos 105^{\circ} $$

Hours of Daylight The number \(H\) of daylight hours per day in various locations in the world can be modeled by a function of the form $$ H(t)=A \sin B(t-C)+D $$ where the variable \(t\) represents the number of days in a year corresponding to a specific calendar date (for example, February 1 corresponds to \(t=32\) In this problem we construct a model for Los Angeles, CA for the year 2017 (not a leap year) using data obtained from the U.S. Naval Observatory, Washington, D.C. (a) Find the amplitude \(A\) if 14.43 is the maximum number of daylight hours at the summer solstice and if 9.88 is the minimum number of daylight hours at the winter solstice. (b) Find \(B\) if the function \(H(t)\) is to have the period 365 days. (c) For Los Angeles in the year 2017 , we choose \(C\) \(=79 .\) Explain the significance of this number. [Hint: \(C\) has the same units as \(t\).] (d) Find \(D\) if the number of daylight hours at the vernal equinox for 2017 is 12.14 and occurs on March 20 (e) What does the model \(H(t)\) predict to be the number of daylight hours on January 1 ? On June 21 ? On August 1 ? On December 21 ? (f) Using a graphing utility to obtain the graph of \(H(t)\) on the interval [0,365] .
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

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.