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Chapter 5: Problem 71

Graph each of the following from \(x=0\) to \(x=8\). \(y=\frac{1}{2} x+\sin \pi x\)

Short Answer

Expert verified
Graph the wave-like pattern formed by a line with a slope of 0.5, modulated by the sine function.

Step by step solution

01

Understand the Function

The function given is a combination of a linear function and a trigonometric function: \[ y = \frac{1}{2} x + \sin \pi x \]The first term \( \frac{1}{2} x \) represents a straight line with a slope of \( \frac{1}{2} \). The second term \( \sin \pi x \) is a sine wave with a period of 2, since the sine wave \( \sin kx \) has a period \( \frac{2\pi}{k} \). Here, since \( k = \pi \), the period is \( \frac{2\pi}{\pi} = 2 \).
02

Calculate Key Points

We need several points to accurately graph the function. Let鈥檚 calculate key points by evaluating the function at integer values of \( x \) from 0 to 8. For example:- At \( x = 0 \), \( y = \frac{1}{2}(0) + \sin(\pi \cdot 0) = 0 \)- At \( x = 1 \), \( y = \frac{1}{2}(1) + \sin(\pi \cdot 1) = \frac{1}{2} + 0 = \frac{1}{2} \)- At \( x = 2 \), \( y = \frac{1}{2}(2) + \sin(\pi \cdot 2) = 1 + 0 = 1 \)- At \( x = 3 \), \( y = \frac{1}{2}(3) + \sin(\pi \cdot 3) = \frac{3}{2} + 0 = \frac{3}{2} \)- Continue for \( x = 4 \) to \( x = 8 \), noticing the repeated sine pattern.
03

Plot the Points

With the calculated values, plot the key points on the graph:- Plot (0, 0), (1, 0.5), (2, 1), (3, 1.5), (4, 2), (5, 2.5), (6, 3), (7, 3.5), (8, 4) Overlay the sine wave shape influenced by \( \sin \pi x \).
04

Sketch the Graph

Connect the plotted points smoothly: - The linear component creates a steadily increasing slope. - The sine component introduces a wavy pattern repeating every 2 units. - At integer multiples of 2, the sine function returns to 0, resulting in peaks or valleys where the linear and wave components intersect.

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Key Concepts

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

Linear Functions
A linear function is one of the simplest types of mathematical models in algebra. It depicts a constant rate of change and can be represented by the equation: \[ y = mx + b \]where:
  • m is the slope of the line, which indicates the steepness and direction.
  • m tells us how much y changes for every unit increase in x.
  • b is the y-intercept, the value where the line crosses the y-axis.
In the given exercise, our linear part is expressed as \frac{1}{2}x, indicating a slope of \frac{1}{2} and a y-intercept of 0. This means for every 2 units moved horizontally, the value of y increases by 1 unit vertically.
This creates a steadily rising line when plotted on a graph.
Sine Wave
The sine wave is a smooth, periodic oscillation that appears frequently in mathematics, physics, and engineering. It is described by the function:\[ y = \sin( heta) \]where:
  • \theta is the angle in radians.
  • The sine function represents periodic motion, like a wave repeating its pattern at regular intervals.
In the problem, the term \sin(\pi x) forms the sine wave part. Its distinctive quality is that it introduces oscillations to other functions it combines with.
This oscillation repeats its pattern within the given period, adding a wavy feature to our original function. The peaks and valleys of this wave add to the linear portion, creating a combined effect that introduces periodic rises and falls in the final graph.
Period of Trigonometric Functions
The period of a trigonometric function, like our sine wave, is the interval after which the function repeats its values. For the basic sine function \[ y = \sin(kx) \]the period is determined by the formula:\[ \frac{2\pi}{k} \]In our given function, k = \pi which means the period is \[ \frac{2\pi}{\pi} = 2 \].
  • This implies that every 2 units along the x-axis, the sine wave starts its pattern of growth and decay again.
  • Such periodicity contributes to making our graph repeat its wavy motion every 2 units.
This regular repeatability makes predicting parts of the graph based on known points possible and is crucial in understanding the behavior of many natural phenomena related to waves and oscillations.

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Most popular questions from this chapter

Prove that each of the following identities is true: $$ \frac{1+\cos x}{1-\cos x}=(\csc x+\cot x)^{2} $$

Prove that each of the following identities is true: $$ \sec ^{2} \theta-\tan ^{2} \theta=1 $$

Prove that each of the following identities is true: $$ \cos ^{2} x\left(1+\tan ^{2} x\right)=1 $$

Prove each identity. $$ \sec \theta-\cos (-\theta)=\frac{\sin ^{2} \theta}{\cos \theta} $$

Prove that each of the following identities is true: $$ \frac{\cos x}{1+\sin x}+\frac{1+\sin x}{\cos x}=2 \sec x $$
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