Question

Determine the amplitude of each function. Then, use the language of transformations to describe how each graph is related to the graph of \(y=\sin x\) a) \(y=3 \sin x\) b) \(y=-5 \sin x\) c) \(y=0.15 \sin x\) d) \(y=-\frac{2}{3} \sin x\)

Short answer

a) Amplitude: 3, vertical stretch by 3. b) Amplitude: 5, vertical stretch by 5, reflection across x-axis. c) Amplitude: 0.15, vertical compression by 0.15. d) Amplitude: 0.67, vertical compression by 0.67, reflection across x-axis.

Step-by-step solution

Identify the base function

Each function given is a transformation of the base function, which is the sine function, described by the equation: \[ y = \sin x \].

Understand amplitude

The amplitude of a sine function is determined by the coefficient in front of the sine term. For \( y = a \sin x \), the amplitude is the absolute value of \( a \).

Determine amplitude of \( y = 3 \sin x \)

The function is \( y = 3 \sin x \). Here, \( a = 3 \). Therefore, the amplitude is:\[ |3| = 3 \]

Describe transformation for \( y = 3 \sin x \)

Since the coefficient is positive, the graph of \( y = 3 \sin x \) involves a vertical stretch by a factor of 3 compared to \( y = \sin x \).

Determine amplitude of \( y = -5 \sin x \)

The function is \( y = -5 \sin x \). Here, \( a = -5 \). Therefore, the amplitude is:\[ |-5| = 5 \]

Describe transformation for \( y = -5 \sin x \)

Since the coefficient is negative, the graph of \( y = -5 \sin x \) involves a vertical stretch by a factor of 5 and a reflection across the x-axis compared to \( y = \sin x \).

Determine amplitude of \( y = 0.15 \sin x \)

The function is \( y = 0.15 \sin x \). Here, \( a = 0.15 \). Therefore, the amplitude is:\[ |0.15| = 0.15 \]

Describe transformation for \( y = 0.15 \sin x \)

Since the coefficient is positive, the graph of \( y = 0.15 \sin x \) involves a vertical compression by a factor of 0.15 compared to \( y = \sin x \).

Determine amplitude of \( y = -\frac{2}{3} \sin x \)

The function is \( y = -\frac{2}{3} \sin x \). Here, \( a = -\frac{2}{3} \). Therefore, the amplitude is:\[ | -\frac{2}{3} | = \frac{2}{3} \]

Describe transformation for \( y = -\frac{2}{3} \sin x \)

Since the coefficient is negative, the graph of \( y = -\frac{2}{3} \sin x \) involves a vertical compression by a factor of \(\frac{2}{3}\) and a reflection across the x-axis compared to \( y = \sin x \).

Key concepts

Transformations of Sine Functions

Sine functions are often transformed to model different kinds of oscillatory behavior. The base function is the simple sine function, described by the equation: \( y = \sin x \).Transformations can alter its amplitude, period, and horizontal shifts.In this article, we will focus on how the amplitude and other specific transformations modify the basic sine graph.When you have a function like \(y = a \sin x \,\), the amplitude is determined by the coefficient 'a' in front of the sine term.Transformations affect the shape and orientation of the sine graph in various ways:

• A positive 'a' will retain the original direction of the sine waves.
• A negative 'a' will flip the sine waves across the x-axis.
• A larger absolute value of 'a' can stretch or compress the graph vertically.

Vertical Stretch

A vertical stretch occurs when the absolute value of the coefficient 'a' in \(y = a \sin x \) is greater than 1.For example, in the function \(y=3 \sin x \,\) the coefficient 'a' is 3.The amplitude of the sine function is always the absolute value of 'a', so here, the amplitude is |3| = 3.This transforms the base sine function by stretching it vertically by a factor of 3.This means each peak and trough of the sine wave is three times higher and lower, respectively, compared to the base graph \(y= \sin x \).Vertical stretching makes the sine waves taller but does not alter their frequency or horizontal position.

Vertical Compression

Vertical compression happens when the absolute value of the coefficient 'a' in \( y = a \sin x \) is less than 1 but greater than 0.For example, consider the function \(y = 0.15 \sin x \).Here, the coefficient 'a' is 0.15, and the amplitude is |0.15| = 0.15.This compresses the graph vertically by a factor of 0.15.Therefore, each peak and trough is only 15% of the height and depth of the base sine graph \(y= \sin x \).Vertical compression makes the sine waves shorter, squashing the vertical height of the oscillations without changing the wave's frequency or horizontal direction.

Reflection Across the X-Axis

Reflection across the x-axis happens when the coefficient 'a' in \( y = a \sin x \) is negative.For instance, in the function \(y= -5 \sin x \), the coefficient 'a' is -5.The amplitude here is | -5 | = 5.This transformation not only involves a vertical stretch by a factor of 5 but also a reflection across the x-axis since 'a' is negative.As a result, the peaks of the sine wave become troughs, and the troughs become peaks, creating a mirrored image of the base sine graph \(y= \sin x \).This type of reflection transforms the graph by flipping it upside down, reversing the direction of the sine waves while retaining the same frequency and period.