Question

Graph (1) \(\mathrm{y}=\sin \mathrm{x}(2) \mathrm{y}=4 \sin \mathrm{x}(3) \mathrm{y}=\sin 4 \mathrm{x}\) (4) \(\mathrm{y}=\sin [\mathrm{x}+(\pi / 4)](5) \mathrm{y}=\mathrm{A} \sin (\mathrm{Bx}+\mathrm{C})+\mathrm{D}\) where \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) are real constants.

Short answer

To graph the given sine functions, first understand the properties of the basic sine function: range between -1 and 1 with a period of \(2\pi\). Then, apply appropriate transformations for amplitude change, period change, phase shift, and vertical shift based on the function. To summarize: 1. For \(y = 4\sin(x)\), the amplitude is 4, oscillating between -4 and 4. 2. For \(y = \sin(4x)\), the period is \(\frac{\pi}{2}\), making the function oscillate more rapidly. 3. For \(y = \sin(x + \pi/4)\), there's a phase shift of \(-\pi/4\), shifting the graph left by \(\pi/4\) units. 4. For \(y = A\sin(Bx + C) + D\), apply all transformations: Amplitude: |A|, Period: \(\frac{2\pi}{B}\), Phase Shift: \(-\frac{C}{B}\), Vertical Shift: D, resulting in \(y = A \cdot sin(\frac{2 \pi}{\text{period}}(x -\frac{C}{B})) + D\).

Step-by-step solution

Understand the basic sine function's graph

The basic sine function, \(y=\sin(x)\), has the following properties: 1. Range: The output values for the sine function lie in the interval \([-1, 1]\), i.e., \(y\) always lies between -1 and 1. 2. Period: The sine function repeats itself after every \(2\pi\) intervals, which means that the sine function has a period of \(2\pi\). 3. It oscillates between -1 and 1 and crosses the x-axis at \(x = n\pi\), where \(n\) is an integer. Now that we have understood the basic sine function's graph properties, let's move on to graphing the given functions.

Graph y = 4sin(x)

The function \(y = 4\sin(x)\) is a simple amplitude change from the basic sine function. The amplitude has been multiplied by 4. To graph this function, follow these steps: 1. The function will now oscillate between -4 and 4, instead of -1 and 1. 2. The period remains \(2\pi\) and it still crosses the x-axis at \(x = n\pi\). This function will look like an expanded version of the basic sine function in the vertical direction.

Graph y = sin(4x)

The function \(y = \sin(4x)\) is a simple period change from the basic sine function. The period changes according to the following rule: \(\text{new period} = \frac{\text{old period}}{k}\), where \(k\) is the coefficient of \(x\). To graph this function: 1. The amplitude remains 1, so the function oscillates between -1 and 1. 2. The new period will be \(\frac{2\pi}{4}=\frac{\pi}{2}\). 3. As the period has reduced, the function will appear to oscillate more rapidly.

Graph y = sin(x + \pi/4)

The function \(y = \sin(x + \pi/4)\) is a simple phase shift from the basic sine function. The phase shift is the change in the \(x\) direction according to the value inside the brackets. To graph this function: 1. The amplitude remains 1, so the function oscillates between -1 and 1. 2. The period remains \(2\pi\). 3. It has a phase shift of \(-\pi/4\), which means the entire graph shifts \(\pi/4\) units left.

Graph y = A sin(Bx + C) + D

The function \(y = A\sin(Bx + C) + D\) is a combination of all transformations (amplitude change, period change, phase shift, and vertical shift). To graph this function: 1. Amplitude is given by the absolute value of A. The function oscillates between \(-A\) and \(A\). 2. The period will be \(\frac{2\pi}{B}\). 3. It has a phase shift of \(-\frac{C}{B}\), which means the entire graph shifts \(\frac{C}{B}\) units left if C > 0 or right if C < 0 4. The vertical shift is given by D, moving the graph up (D > 0) or down (D < 0) by D units. Summarizing the transformations, the given functions will have the following form: \(y = A \cdot sin(\frac{2 \pi}{\text{period}}(x -\frac{C}{B})) + D\)

Key concepts

Amplitude Change

When you encounter the sine function in the form of \(y = A \sin(x)\), the 'A' represents the amplitude. The amplitude tells you how far the peaks and valleys of the graph extend from the central axis. For the basic sine function, \(y = \sin(x)\), the amplitude is 1. This means the graph will peak at 1 and valley at -1. But when the amplitude changes, as in \(y = 4\sin(x)\), it multiplies the standard amplitude by 4.

With this change:

• The sine wave will stretch vertically, now oscillating between 4 and -4.
• The overall shape of the sine wave remains the same, only the height changes.

By understanding amplitude, you can easily predict how tall the graph of a sine function will be, which is crucial in applications ranging from physics to engineering.

Period Change

In sine functions, the period is the distance over which the wave repeats itself. The basic sine function, \(y = \sin(x)\), has a period of \(2\pi\), which means it completes one full wave cycle every \(2\pi\) units along the x-axis.

When you see a sine function like \(y = \sin(4x)\), the number multiplying x, in this case 4, affects the period.

• The new period is calculated by dividing the original period by this coefficient: \(\frac{2\pi}{4} = \frac{\pi}{2}\).
• This change means the wave completes its cycle faster, leading to more waves in the same length on the x-axis.

This knowledge can help in understanding how functions model real-world scenarios, such as sound waves, which might need to be adjusted to fit specific frequencies.

Phase Shift

Phase shifts refer to horizontal movements of the graph along the x-axis. When you see a sine function like \(y = \sin(x + \frac{\pi}{4})\), it involves a phase shift. The term \(+\frac{\pi}{4}\) inside the sine function indicates a shift.

Here's how it works:

• Plus inside the parentheses \(x + C\) shifts the graph to the left, so \(+\frac{\pi}{4}\) shifts it \(\frac{\pi}{4}\) units left.
• If the function was \(y = \sin(x - \frac{\pi}{4})\), it would shift \(\frac{\pi}{4}\) units to the right.

Understanding phase shifts is crucial when aligning waves in engineering, or predicting tides in oceanography.

Vertical Shift

Vertical shifts move the graph up or down along the y-axis. In a sine function of the form \(y = A\sin(Bx + C) + D\), the 'D' represents the vertical shift.

Let's break down this effect:

• If \(D\) is positive, the graph shifts upwards by \(D\) units.
• If \(D\) is negative, the graph shifts downwards by \(-D\) units.

This transformation does not affect the amplitude or period, only the vertical position. This concept helps in real-world applications where the baseline (or zero point) of a wave needs to be adjusted, such as in electricity for alternating current waveforms.