Problem 24 - Precalculus Mathematics for Calculus

Question

Find the amplitude and period of the function, and sketch its graph. $$y=4 \sin (-2 x)$$

Step-by-Step Solution

Answer

Amplitude is 4, and the period is \(\pi\).

Step 1: Identify the form of the sine function

The given function is \(y = 4 \sin(-2x)\). This function is a variation of the general sine function \(y = A \sin(Bx)\), where \(A\) is the amplitude, and \(B\) affects the period of the function.

Step 2: Determine the amplitude

The amplitude of a sine function \(y = A \sin(Bx)\) is given by the absolute value of \(A\). Here, \(A = 4\), so the amplitude is \(|4| = 4\).

Step 3: Determine the period

The period of a sine function \(y = A \sin(Bx)\) is calculated as \(\frac{2\pi}{|B|}\). In this case, \(B = -2\), so the period is \(\frac{2\pi}{|-2|} = \frac{2\pi}{2} = \pi\).

Step 4: Sketch the function

To sketch the function \(y = 4 \sin(-2x)\), note that there is a reflection across the y-axis due to the negative in front of 2x. The amplitude is 4, causing vertical stretching, and the period is \(\pi\), meaning one complete oscillation occurs from \(0\) to \(\pi\). Start plotting at the origin with a downward slope due to the negative sign, reaching -4 at \(\pi/2\), back to 0 at \(\pi\), reaching 4 at \(3\pi/2\), and returning to 0 at \(2\pi\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions essential for understanding oscillatory movements. It is periodic, meaning it repeats values in regular intervals. Mathematically, the sine function is expressed as \( y = \sin(x) \), representing the y-coordinate of a point on the unit circle as x varies. The basic sine curve oscillates above and below the x-axis.

The sine function starts at 0, climbs to its peak at \( y = 1 \), descends back to 0, reaches its lowest point at \( y = -1 \), and returns to 0 to complete one full cycle. This cyclical behavior is crucial to understanding how to transform and manipulate sine functions for different applications.

Trigonometric Graph Sketching

Sketching trigonometric functions can be a delightful task once you understand the pattern of the sine and cosine curves. The sine function graph is a smooth curve, called a sinusoid, due to its periodic and wave-like nature.

To sketch \( y = \sin(x) \), start at the origin, then guide your way along the x-axis: up to its peak at \( x = \pi/2 \), down through the origin again at \( x = \pi \), to the trough at \( x = 3\pi/2 \), and back to the origin at \( x = 2\pi \). This creates one complete wave.

• Identify amplitude and period.
• Note key points and plot them sequentially.
• Consider any horizontal or vertical translations.
• Check for reflections across axes.

Visual aids, like graphs, help to grasp these concepts more fully and allow you to see the impact of changes in expression parameters.

Function Amplitude

Amplitude in trigonometric functions is one of the key components that determine how much the graph stretches above and below its equilibrium line. For the general form \( y = A \sin(Bx) \), the amplitude is given by the absolute value of \( A \).

The amplitude indicates the 'tallness' of the wave. It measures the distance from the wave’s midline to its peak or trough. In the function \( y = 4 \sin(-2x) \), the amplitude is \( |4| = 4 \), which means the sine wave reaches up to 4 units above and dips down 4 units below the center line at y = 0. This stretching effect increases the "height" of the oscillation without altering how frequently it repeats.

Function Period

The period of a trigonometric function describes how often the function repeats its value. For the sine function, the period is determined by the coefficient \( B \) in its argument \( y = A \sin(Bx) \).

To calculate the period, use the formula \( \frac{2\pi}{|B|} \). This represents the length of one complete cycle on the graph. For \( y = 4 \sin(-2x) \), \( B = -2 \). Although \( B \) is negative, only its magnitude affects the period, resulting in \( \frac{2\pi}{|-2|} = \pi \).

• The period length determines how quickly the function completes an oscillation.
• Graphs can have their periods compressed or stretched by this factor.

Understanding the period is vital for predicting the function's behavior over extended inputs, which is particularly useful in mathematical modeling and real-world applications like signal processing.