Problem 19

Question

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=2 \sin (x+\pi / 4) $$

Step-by-Step Solution

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Answer

Graph a sinusoid with amplitude 2, period $2\pi$, left-shifted by $\pi/4$.

1Step 1: Identify the Parent Function

The parent function here is the sine function, which is given by $y=\sin x$. The sine function has a standard waveform with an amplitude of 1, period of $2\pi$, and is symmetric about the origin.

2Step 2: Identify Transformations

The given function, $ y=2 \sin(x+\pi/4) $, has several transformations applied to the parent function: an amplitude change, a horizontal shift, and a phase shift. The amplitude is 2, indicating a vertical stretch. The expression $x+\pi/4$ reflects a horizontal shift or phase shift by $-\pi/4$ to the left.

3Step 3: Determine the Amplitude

In $y=2\sin(x+\pi/4)$, the amplitude is determined by the coefficient of the sine function, $2$. This means the graph of the sine function will vertically stretch by a factor of 2. The highest point of the sine curve will now be 2, and the lowest point will be -2.

4Step 4: Determine the Period

For the sine function, the standard period is $2\pi$, and no horizontal scaling is applied, so the period remains $2\pi$.

5Step 5: Calculate Key Points

Determine key points using the horizontal shift. For $y=\sin x$, key points are at $0, \pi/2, \pi, 3\pi/2, $ and $2\pi$. Shift these points left by $\pi/4$:$-\pi/4, \pi/4, 3\pi/4, 5\pi/4, 7\pi/4$. Evaluate these points with $2\sin$:$-\pi/4 \to 0, \pi/4 \to 2, 3\pi/4 \to 0, 5\pi/4 \to -2, 7\pi/4 \to 0$.

6Step 6: Sketch the Graph

Plot the calculated key points. Starting at $-\pi/4,0$, rise to $\pi/4,2$, descend to $3\pi/4,0$, fall to $5\pi/4,-2$, and return to $7\pi/4, 0$. Smoothly connect these points with a sinusoidal wave that extends infinitely in both directions due to the periodicity of the sine function.

Key Concepts

Sine FunctionAmplitude and PeriodPhase ShiftWaveform Transformations

Sine Function

The sine function ($y = \sin x$) is a fundamental concept in trigonometry. It represents a periodic wave that oscillates between -1 and 1. When graphed, the sine function creates a wavelike pattern. This pattern repeats every $2\pi$ units along the x-axis, making it have a periodicity of $2\pi$.

The graph of $y = \sin x$ starts at the origin (0,0).
It rises to a crest at $\frac{\pi}{2}$, descends back to the x-axis at $\pi$, falls to a trough at $\frac{3\pi}{2}$, and returns to the x-axis at $2\pi$

This basic cycle is known as the "sine wave" and is crucial for understanding how various transformations affect trigonometric functions.

Amplitude and Period

Amplitude and period are two key characteristics of any trigonometric function. In the given function $y = 2 \sin(x+\pi/4)$, the amplitude is the absolute value of the coefficient in front of the sine function. Here, it's $2$.

Amplitude affects how tall the waves are, stretching them vertically.
For this function, the waves reach as high as $2$ and as low as $-2$.

The period is the complete cycle length of the wave along the x-axis. Normally, for $\sin x$, the period is $2\pi$. In $y = 2\sin(x+\pi/4)$, since there is no horizontal scaling factor other than $x$, the period remains $2\pi$.
This means that every $2\pi$, the wave pattern repeats itself, creating consistent cycles along the graph.

Phase Shift

Phase shift refers to horizontal shifts on the graph of a trigonometric function. These shifts occur because of an added component inside the function's argument, such as $+\pi/4$ in $y = 2 \sin(x+\pi/4)$.

Phase shifts move the entire graph of the function left or right along the x-axis.
In this function, the addition of $\pi/4$ means moving the graph to the left by $\pi/4$ units.

This leftward movement modifies the starting point of each cycle. Instead of beginning at 0, the wave starts at $-\pi/4$, making the graph's key points shift left accordingly. Understanding how to manipulate the phase shift allows precise alignment of sine and cosine waves to meet specific criteria.

Waveform Transformations

Waveform transformations provide further insights on how various alterations affect trigonometric graphs. In addressing $y = 2 \sin(x+\pi/4)$, there are two primary transformations to consider: amplitude changes and phase shifts.

The amplitude transformation is determined by the factor 2, which stretches the graph vertically.
The phase shift (caused by adding $+\pi/4$

Transformations like these reflect on the many applications of trigonometric functions, from sound waves to electromagnetic signals. By mastering these transformations, students can visualize how complex signals may transform while retaining key characteristics, helping solve real-world physics and engineering problems.

Problem 18

Problem 19

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