Problem 50 Sketch the graph of the function... [FREE SOLUTION]

Chapter 4: Problem 50

Sketch the graph of the function. (Include two full periods.) $$ y=\sin (x-2 \pi) $$

Short Answer

Expert verified

After applying the shift, the function $y=\sin(x-2\pi)$ has key points at $(2\pi, 0), (\frac{5\pi}{2}, 1), (3\pi, 0), (\frac{7\pi}{2}, -1)$, and $(4\pi, 0)$. Draw a smooth curve connecting these points to complete the graph.

Step by step solution

Identify Key Points

Start with the basic sine function, $y=\sin(x)$, and find key points. The key points for a basic sine function in one period from 0 to $2\pi$ are: (0, 0), $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, and $(2\pi, 0)$.

Apply Shift

With a right shift of $2\pi$, the key points just found will also shift $2\pi$ units to the right. Thus, adjusting the points gives: $(2\pi, 0), (\frac{5\pi}{2}, 1), (3\pi, 0), (\frac{7\pi}{2}, -1)$, and $(4\pi, 0)$.

Draw the Graph

Plot these shifted key points on an x-y graph. After plotting, connect the points with a smooth curve and repeat to complete two full periods. Extend the pattern.

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, noted as $y = \sin(x)$, is one of the fundamental trigonometric functions. It reports the y-coordinates of a point as it varies along the unit circle. In simpler terms, it tells you how high or low a point is as it travels counterclockwise from the circle鈥檚 starting point. The sine function waves up and down, creating a smooth, continuous curve.
Sine functions are central to a broad range of applications like in sound waves, light waves, and alternating current in electricity.

Its maximum value is 1.
The minimum value is -1.
The sine wave starts at 0 and returns to 0 at the end of each cycle, at times known as "zero crossings".

You can visualize this as a hill and a valley that repeats, resembling the smooth undulations we often see in nature.

Periodic Functions

A function is periodic if its values repeat at regular intervals, known as the period. The sine function is a prime example of such a behavior. Its period is $2\pi$, meaning it completes a full cycle 鈥� up and down, back to the start 鈥� every $2\pi$ units horizontally along the x-axis.
Consider what this means:

Every $x$ value that increases by $2\pi$ will give you the same $y$ value.
This repetition is what gives the sine wave its characteristic wave pattern.

Periodic functions are important for understanding cycles in science and nature, where processes often repeat in a predictable manner over time.

Phase Shift

Phase shift in trigonometric functions refers to the horizontal movement of a graph along the x-axis. It describes how far the graph is "shifted" from the usual starting point.
The expression $y=\sin(x-2\pi)$ in the original exercise has a phase shift of $2\pi$ to the right. This means:

Each point on the sine wave moves $2\pi$ units to the right.
The wave begins its upward motion not at 0 on the x-axis, but instead effectively "starts" at $2\pi$.

Phase shifts don't alter the shape of the graph. They simply reposition it along the x-axis, maintaining its features but adjusting its starting point.

Key Points

Key points on a sine graph help you understand where the function is at its extremities and zero crossings. In one period of the sine function, which spans from 0 to $2\pi$, key points include:

Start at (0, 0) where the wave crosses the axis.
Reach its peak at $(\frac{\pi}{2}, 1)$.
Fall back to 0 at $(\pi, 0)$.
Drop to its lowest point at $(\frac{3\pi}{2}, -1)$.
Return to 0 at the end of the period, $(2\pi, 0)$.

In the given problem, these points shift because of the phase shift:

New start at $(2\pi, 0)$.
New peak at $(\frac{5\pi}{2}, 1)$.
Return to neutral at $(3\pi, 0)$.
New low at $(\frac{7\pi}{2}, -1)$.
And again at neutral at $(4\pi, 0)$.

These key points help you sketch the graph accurately, no matter the shifts.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Sketch the graph of the function. (Include two full periods.) $$ y=\sin (x-2 \pi) $$

Short Answer

Step by step solution

Identify Key Points

Apply Shift

Draw the Graph

Key Concepts

Sine Function

Periodic Functions

Phase Shift

Key Points

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.