Problem 121 - Precalculus

Question

Find the \(x\) -intercept(s) of \(y=A \sin B x+A\)

Step-by-Step Solution

Answer

The x-intercepts occur at \( x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \), where \( k \) is an integer.

Step 1: Set Y to Zero

To find the x-intercepts of the function, we need to set the equation equal to zero because the x-intercepts occur where the graph crosses the x-axis (i.e., where \( y = 0 \)). Thus, set \[ 0 = A \sin(Bx) + A \]

Step 2: Solve for \\sin(Bx)

From the equation \( 0 = A \sin(Bx) + A \), subtract \( A \) from both sides to isolate the sine term: \[ -A = A \sin(Bx) \]. Then, divide both sides by \( A \) (assuming \( A eq 0 \)): \[ -1 = \sin(Bx) \].

Step 3: Find Solutions for \\sin(Bx) = -1

The values of \( x \) for which \( \sin(Bx) = -1 \) occur at specific points: \[ Bx = \frac{3\pi}{2} + 2k\pi \], where \( k \) is an integer. These values are where the sine function repeats its pattern.

Step 4: Solve for x

Solve for \( x \) by dividing both sides of the equation from Step 3 by \( B \): \[ x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \]. This provides the general form for the x-intercepts of the function.

Key Concepts

x-intercepts

X-intercepts are the points where a graph crosses the x-axis. This means that at these points, the function's value is zero. For any trigonometric equation like \( y = A \sin(Bx) + A \), identifying these points requires setting the equation equal to zero.
Step-by-step, you would:

• Set the function equal to zero: \( 0 = A \sin(Bx) + A \)
• Isolate \( \sin(Bx) \) by manipulating the equation: \( -A = A \sin(Bx) \)
• Divide by \( A \) assuming \( A eq 0 \), resulting in: \( -1 = \sin(Bx) \)

This equation shows where the sinusoidal wave crosses the x-axis. It is crucial to understand that solving trigonometric functions for x-intercepts often involves finding angles corresponding to particular function values.

sine function

The sine function is a periodic and smooth curve that oscillates between -1 and 1. In any expression like \( y = A \sin(Bx) + A \), changes in the function's parameters directly affect its shape and position.
Here's a quick breakdown:

• Amplitude: The coefficient \( A \) determines the amplitude (the wave's height from the center to its peak).
• Frequency: The coefficient \( B \) affects the frequency, which is how often the wave oscillates in a given period.
• Transformation: Adding the constant \( A \) to the sine function shifts the whole curve upwards.

Understanding these characteristics helps interpret the sine function's behavior over its domain and better solve related problems such as finding x-intercepts.

periodicity

Periodicity is a fundamental feature of the sine function, meaning it repeats itself after a certain interval called the period. For a function like \( y = A \sin(Bx) + A \), knowing the period is crucial for calculating where certain values, like the x-intercepts, occur repeatedly.
Here's what you need to know:

• The period of \( \sin(Bx) \) is calculated as \( \frac{2\pi}{B} \). This tells us how long it takes for the sine wave to complete one full cycle.
• Understanding periodicity lets you project where the function will cross the x-axis again, as observed in the general formula for x-intercepts: \( x = \frac{3\pi}{2B} + \frac{2k\pi}{B} \).
• This formula incorporates the idea of periodicity by including \( 2k\pi \), which accounts for the function's repeating nature.

Mastering periodicity is key to understanding and working with functions that involve trigonometric components, as it explains their repetitive nature.