Question

Determine the phase shift and the vertical displacement with respect to \(y=\sin x\) for each function. Sketch a graph of each function. a) \(y=\sin \left(x-50^{\circ}\right)+3\) b) \(y=\sin (x+\pi)\) c) \(y=\sin \left(x+\frac{2 \pi}{3}\right)+5\) d) \(y=2 \sin \left(x+50^{\circ}\right)-10\) e) \(y=-3 \sin \left(6 x+30^{\circ}\right)-3\) f) \(y=3 \sin \frac{1}{2}\left(x-\frac{\pi}{4}\right)-10\)

Short answer

a) 50° right, 3 up. b) π left, no vertical shift. c) -2π/3 left, 5 up. d) -50° left, 10 down. e) -5° left, 3 down. f) π/2 right, 10 down.

Step-by-step solution

Identify Components of the Function

Each function can be written in the form of \( y = A \, \sin(B(x - C)) + D \), where \(A\) is the amplitude, \(B\) affects the period, \(C\) accounts for phase shift, and \(D\) is the vertical displacement. Identify \(C\) and \(D\) for each function.

Function a) Analysis

Given \( y = \, \sin(x - 50^{\circ}) + 3 \):- Phase Shift (\(C\)) = \( 50^{\circ} \) to the right - Vertical Displacement (\(D\)) = 3 units upward.

Function b) Analysis

Given \( y = \, \sin(x + \pi) \):- Phase Shift (\(C\)) = \( -\pi \) to the left - Vertical Displacement (\(D\)) = 0 (no vertical displacement).

Function c) Analysis

Given \( y = \, \sin(x + \frac{2\pi}{3}) + 5 \):- Phase Shift (\(C\)) = \( -\frac{2\pi}{3} \) to the left - Vertical Displacement (\(D\)) = 5 units upward.

Function d) Analysis

Given \( y = 2 \, \sin(x + 50^{\circ}) - 10 \):- Phase Shift (\(C\)) = \( -50^{\circ} \) to the left - Vertical Displacement (\(D\)) = 10 units downward.

Function e) Analysis

Given \( y = -3 \, \sin(6x + 30^{\circ}) - 3 \):- Phase Shift (\(C\)) = \( -30^{\circ} \div 6 = -5^{\circ} \) to the left - Vertical Displacement (\(D\)) = 3 units downward.

Function f) Analysis

Given \( y = 3 \, \sin\left( \frac{1}{2}(x - \frac{\pi}{4}) \right) - 10 \):- Phase Shift (\(C\)) = \( \frac{\pi}{4} \times 2 = \frac{\pi}{2} \) to the right - Vertical Displacement (\(D\)) = 10 units downward.

Sketch the Graphs

Using the identified phase shifts and vertical displacements, sketch each function:- Function a) will be a sine wave shifted 50 degrees to the right and 3 units upward.- Function b) will be a sine wave shifted \(-\pi\) to the left.- Function c) will be a sine wave shifted \(-\frac{2\pi}{3}\) to the left and 5 units upward.- Function d) will be a sine wave with twice the amplitude, shifted \(-50^{\circ}\) to the left and 10 units downward.- Function e) will be a sine wave inverted and multiplied by 3, with period affected by 6, shifted \(-5^{\circ}\) to the left and 3 units downward.- Function f) will be a sine wave with triple amplitude, its period affected by \( \frac{1}{2} \) factor, shifted to the right by \(\frac{\pi}{2}\) and 10 units downward.

Key concepts

trigonometric transformations

Trigonometric transformations change the position, shape, or size of trigonometric functions like sine and cosine. These changes can include shifting the graph horizontally or vertically, stretching or compressing it, or reflecting it. When analyzing these transformations for sine functions, it's important to recognize how each parameter affects the graph. Transformations help tailor these functions to different real-world contexts, making them incredibly useful in fields ranging from engineering to physics.

sine function graph

A sine function graph, typically of the form \( y = \sin(x) \), is a smooth, continuous wave that starts at the origin (0,0). It oscillates between -1 and 1, with a repetitive pattern called a 'period'. When dealing with transformations, the basic shape of the sine graph remains the same, but its properties like position, height, and width can change. Understanding the basic sine function graph helps in visualizing how transformations affect its shape and position.

phase shift

Phase shift refers to the horizontal movement of the sine function. It's determined by the value of 'C' in the function \(y = \sin(B(x - C)) + D\). If 'C' is positive, the graph shifts to the right; if negative, it shifts to the left. For example, in the function \(y = \sin(x - 50^\text{°}) + 3\), there's a phase shift of 50 degrees to the right. This means the wave starts 50 degrees later than \(y = \sin(x)\). Each phase shift changes when the cycle of the sine wave begins.

vertical displacement

Vertical displacement moves the sine wave up or down and is given by the value 'D' in the function \( y = \sin(B(x - C)) + D \). If 'D' is positive, the graph shifts upward; if negative, it shifts downward. For instance, in the function \( y = \sin(x - 50^\text{°}) + 3 \), there's a vertical displacement of 3 units upward. This changes the central line of the sine waves from y=0 to y=3, meaning the wave oscillates around this new line.

amplitude

Amplitude, denoted by 'A' in the function \( y = A \sin(B(x - C)) + D \), is the height of the wave from its central line to its peak. It tells you how tall or short the wave is. The amplitude is always a positive number. For example, in \( y = 2 \sin(x + 50^\text{°}) - 10 \), the amplitude is 2. This means the wave will reach 2 units above and 2 units below its central line. A larger amplitude makes the wave taller, and a smaller amplitude makes it shorter.

period

The period of a sine function is the distance (or length) of one complete cycle of the wave. It's determined by the coefficient 'B' in the function \( y = \sin(B(x - C)) + D \) and is given by \( \frac{2\pi}{B} \). For instance, if \( B = 1 \), the period remains \( 2\pi \). A smaller value of 'B' stretches the wave horizontally, making the period longer, while a larger 'B' compresses it, making the period shorter. Understanding the period helps in predicting how often the sine wave repeats itself.