Problem 116 - Precalculus

Question

In Exercises \(115-118, A\) and \(B\) are positive real numbers. Find the \(y\) -intercept of the function \(y=A \sin B x\)

Step-by-Step Solution

Answer

The y-intercept is at \( y = 0 \).

Step 1: Understand the Function Structure

The function given is in the form of a sine function, specifically, \( y = A \sin(Bx) \). In general, a sine function \( \sin(x) \) has an amplitude \( A \) and a frequency modification factor \( B \), which affect its vertical stretching and horizontal compression/stretching.

Step 2: Define the y-intercept

The y-intercept of a function is the point where the graph of the function intersects the y-axis. This occurs when \( x = 0 \). So, we need to evaluate the function \( y = A \sin(Bx) \) at \( x = 0 \).

Step 3: Substitute to Find the y-intercept

Substitute \( x = 0 \) into the function \( y = A \sin(Bx) \). This gives:\[y = A \sin(B \times 0) = A \sin(0)\]We know from trigonometry that \( \sin(0) = 0 \).

Step 4: Calculate the y-intercept

Since \( \sin(0) = 0 \), substituting yields:\[y = A \times 0 = 0\]Thus, the y-intercept of the function is at \( y = 0 \).

Key Concepts

Understanding the Sine Function

The sine function is one of the basic types of trigonometric functions. It is denoted as \( \sin(x) \) and it represents a wave-like pattern that repeats every \( 2\pi \) radians. Because of this periodic nature, sine functions are often used to model cyclical behaviors such as sound waves, temperature fluctuations, and tides. The simplest form of the sine function is \( y = \sin(x) \), which oscillates between -1 and 1.When we introduce coefficients to this function, as in \( y = A \sin(Bx) \), it modifies the shape and frequency of the wave:

• A is the amplitude, which affects the height of the wave's peaks and troughs.
• B modifies the frequency, affecting how tightly the waves are packed.

These modifications allow the sine function to be adjusted to fit various real-world phenomena more accurately.

Amplitude and Its Effects

Amplitude is a crucial feature of the sine function that determines the vertical stretch of the wave. It is represented by the coefficient \( A \) in the function \( y = A \sin(Bx) \).

• The amplitude indicates the maximum distance from the midline of the wave to its peak or trough. If \( A = 2 \), for example, the wave peaks at 2 and troughs at -2.
• When the amplitude is greater than 1, the wave stretches upwards and downwards from the x-axis.
• If \( A \) is less than 1, the wave compresses, reducing the height of its peaks and troughs. If \( A = 0.5 \), the wave peaks at 0.5 and troughs at -0.5.

In contexts like physics and engineering, modifying amplitude can help represent phenomena such as the intensity of sound or light waves.

Exploring Trigonometric Functions

Trigonometric functions cover an array of mathematical functions including sine, cosine, and tangent, which are essential in trigonometry. These functions relate the angles of triangles to the lengths of their sides and are crucial in analyzing periodic patterns.

• Sine (\( \sin \)): As discussed, represents the y-coordinate of a point on the unit circle at any given angle.
• Cosine (\( \cos \)): Similar to sine but represents the x-coordinate of a point on the unit circle.
• Tangent (\( \tan \)): Expresses the ratio of the sine and cosine values. It helps determine the steepness of the line connecting the point on the unit circle to the origin.

These functions do more than simply relate to triangle measurements; they model oscillations, rotations, and waves in various scientific fields. Understanding these functions and their characteristics enables students to solve complex problems related to cycles and periodic events.