Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed. $$ y=2 \sin 2 x $$

Short answer

Amplitude: 2; Period: \( \pi \); No shifts.

Step-by-step solution

Identify the Amplitude

The amplitude of a sine function in the form \( y = a \sin(bx + c) + d \) is given by \(|a|\). In our function \( y = 2 \sin(2x) \), \( a = 2 \). Therefore, the amplitude is 2.

Determine the Period

The period of the sine function is calculated by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the function. For \( y = 2 \sin(2x) \), \( b = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \).

Identify the Horizontal Shift

The horizontal shift, also known as the phase shift, is calculated from the expression \( bx + c = 0 \). In our function, \( c = 0 \), so there is no horizontal shift.

Identify the Vertical Shift

The vertical shift is determined by the value of \( d \) in the function format \( y = a \sin(bx + c) + d \). In our function, \( d = 0 \), indicating there is no vertical shift.

Graph the Function

Plot the sine function using the identified characteristics. For \( y = 2\sin(2x) \), the amplitude is 2, the period is \( \pi \), and there are no shifts. The function completes a full cycle between 0 and \( \pi \), peaking at \( x = \frac{\pi}{4} \) and troughing at \( x = \frac{3\pi}{4} \). Repeat this pattern to map the graph over \(-5 \leq x \leq 5\).

Key concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often written as \( \sin(x) \). This function models wave-like patterns which repeat at regular intervals. In its basic form \( y = \sin(x) \), the graph oscillates between -1 and 1, creating a smooth, continuous wave.

Key characteristics of the sine function include its smooth periodic nature and its even symmetry about the origin. In practical terms, this function is often used to model phenomena such as sound waves, light waves, and alternating electrical currents due to its repetitive pattern.

The sine function is the base for various modifications involving transformations that alter its amplitude, period, phase, and vertical displacement, allowing it to be adapted to fit different periodic behaviors in advanced mathematical and engineering problems.

Amplitude

Amplitude refers to the height of the peaks and the depth of the troughs from the central axis of the sine wave. For a function in the format \( y = a \sin(bx + c) + d \), the amplitude is given by the absolute value of \( a \). This provides a measure of how 'tall' or 'short' the wave appears compared to its resting position.

In our example \( y = 2 \sin(2x) \), the amplitude is calculated as \(|2|\), which equals 2. This means the wave reaches a maximum height of 2 and a minimum height of -2. The sine wave oscillates between these bounds with the central axis as the reference.

Understanding amplitude is crucial for interpreting the physical significance of trigonometric models, especially in fields like acoustics where it directly correlates to sound volume. A larger amplitude corresponds to more intense wave properties, whether in light brightness or sound loudness.

Period

The period of a trigonometric function describes how long it takes for the sine wave to complete one full cycle and start repeating. Expressed mathematically, the period \( T \) for the function \( y = a \sin(bx + c) + d \) is determined by \( \frac{2\pi}{b} \).

In the provided function \( y = 2 \sin(2x) \), since \( b = 2 \), the period is given by \( \frac{2\pi}{2} = \pi \). Thus, this curve completes one full oscillation every \( \pi \) units of the x-axis.

A clear understanding of the period is pivotal when graphing transformations or solving problems where timing or frequency of waves is crucial, such as electrical engineering, signal processing, or physics. The ability to determine and adjust the period allows for precise control over how often a wave pattern repeats, affecting everything from sound pitch in music to time cycles in clocks.

Graphing Transformations

Graphing transformations refer to the modifications applied to the basic sine function to adjust its graphical representation. These transformations can alter amplitude, period, phase shift (horizontal shift), and vertical shift.

To graph the function \( y = 2\sin(2x) \), we apply these transformations:

• Amplitude Change: The graph stretches vertically by a factor of 2, doubling the height of the wave peaks and the depth of the troughs.
• Period Adjustment: The period of the graph shrinks to \( \pi \), compressing the wave horizontally to complete a full cycle twice as quickly compared to the standard sine wave.
• Horizontal and Vertical Shifts: Since there are no additional constants added within the function or outside, there are no horizontal (phase) or vertical shifts in this particular graph.

Graphing these transformations involves plotting characteristic points such as maxima, minima, and intercepts over the interval \(-5 \leq x \leq 5\), ensuring the modifications are consistently applied across the entire range. Understanding these principles enhances the ability to interpret and depict various waveforms suited to real-world applications.