Problem 73 Use a graphing utility to graph ... [FREE SOLUTION]

Chapter 4: Problem 73

Use a graphing utility to graph each function. $$y=\sin x-\cos \frac{x}{2}$$

Short Answer

Expert verified

The graph of $y=\sin x - \cos \dfrac{x}{2}$ would appear as a wave pattern similar to traditional sine and cosine functions, but with variations in amplitude and wavelength due to the subtraction of the altered cosine function from the sine function.

Step by step solution

Understanding the Functions

Firstly, understand that the given function is a composition of two functions, namely the sine function and the cosine function. The sine function is $\sin x$ where x represents the angle in the unit circle. Its graph is a wave that oscillates between -1 and 1. The cosine function, on the other hand, is presented as $\cos \dfrac{x}{2}$. This is a transformed cosine function, where the input angle x is divided by 2. This modification will result in the stretching of the cosine wave by a factor of 2 along the x-axis.

Graphing the Individual Functions

Using a graphing utility, graph each individual function separately. For $\sin x$, the resulting graph will be a wave that starts from the origin (0,0), peaking at (蟺/2, 1), going through (蟺, 0), and troughs at (3蟺/2, -1), and so on. For $\cos \dfrac{x}{2}$, it will start from (0,1), reach zero at (蟺, 0), go to trough at (2蟺, -1), and so on. But this will be stretched along the x-axis because its frequency is halved (x is divided by 2). Basically, it will take twice as long to complete a full cycle compared to a usual cosine function.

Combining the Functions

Now that we have graphed the individual functions, we need to combine them according to the given function, which is $y=\sin x - \cos \dfrac{x}{2}$. This means at each x-value, the y-value of the cosine function is subtracted from the y-value of the sine function to find the resultant y-value for the final function. Graph this combined function on the graphing utility as well.

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 $\sin x$ is one of the fundamental trigonometric functions, known for its smooth, wavelike oscillation. This function is periodic and repeats every $2\pi$. Its graph starts from the origin $(0,0)$, reaches its peak at $(\pi/2, 1)$, goes through $(\pi, 0)$, and reaches a trough at $(3\pi/2, -1)$, before returning to zero at $(2\pi, 0)$. \

The sine function oscillates between -1 and 1.
The function is symmetric about the origin, illustrating odd symmetry.

\Understanding the basic shape of $\sin x$ is essential to analyzing more complex trigonometric functions and transformations.

Cosine Function

The cosine function $\cos x$ is quite similar to the sine function but has a phase shift. It begins at its maximum value of 1 when $x = 0$. The cosine graph can be likened to a sine graph shifted $\pi/2$ units to the left. \

The $\cos x$ function is periodic with a period of $2\pi$.
It oscillates between -1 and 1, similar to the sine function.

\For the problem at hand, the cosine function is modified to $\cos \frac{x}{2}$. This transformation stretches the original cosine function along the x-axis, doubling its period to $4\pi$, which plays a critical role in the function's behavior.

Graphing Utility

A graphing utility is a powerful tool for visualizing mathematical functions. By inputting equations like $y = \sin x - \cos \frac{x}{2}$, you can see how transformations impact their graph. \

Graphing utilities help in understanding interaction between functions, such as addition and subtraction.
They allow for zooming in and out, providing detailed analysis at various scales.
Using a graphing utility can reveal patterns or intersections that might not be immediately obvious from equations alone.

\By breaking down the graphing into steps鈥攆irst separately and then together鈥攊t's easier to comprehend how these functions combine to form new patterns.

Function Transformation

Function transformation involves altering a function's formula to change its graph's shape or position. In this problem, $\cos \frac{x}{2}$ is a transformed version of $\cos x$. By dividing the input $x$ by 2, the cosine wave stretches, increasing its period from $2\pi$ to $4\pi$. \

Stretching or compressing graphs horizontally or vertically by transforming inputs or outputs.
Shifting graphs left/right or up/down by adjusting the constants in the equation.

\Such transformations are essential to understand as they help predict how different operations on functions affect their graphs. By analyzing the $y = \sin x - \cos \frac{x}{2}$ function, students gain insights into these transformations' effects, laying groundwork for more complex problems.

91影视

Use a graphing utility to graph each function. $$y=\sin x-\cos \frac{x}{2}$$

Short Answer

Step by step solution

Understanding the Functions

Graphing the Individual Functions

Combining the Functions

Key Concepts

Sine Function

Cosine Function

Graphing Utility

Function Transformation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Probability and Statistics

Pure Maths

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.