Problem 125 - Precalculus

Question

Use a graphing calculator to graph \(Y_{1}=5 \sin x\) and \(Y_{2}=\sin (5 x) .\) Is the following statement true based on what you see? \(y=\sin c x\) has the same graph as \(y=c \sin x\)

Step-by-Step Solution

Answer

The statement is false; the graphs differ in amplitude and frequency.

Step 1: Understanding the Functions

First, we need to understand what each function represents. The function \( Y_1 = 5 \sin x \) means that the amplitude of the sine wave has been multiplied by 5, making it five times taller than \( \sin x \). The function \( Y_2 = \sin(5x) \) modifies the frequency of the basic sine wave, making it oscillate five times faster in the same interval.

Step 2: Graphing \( Y_1 = 5 \sin x \)

Use your graphing calculator to plot \( Y_1 = 5 \sin x \). You should see a sine wave with peaks at \( y = 5 \) and troughs at \( y = -5 \). This indicates the increased amplitude as expected.

Step 3: Graphing \( Y_2 = \sin(5x) \)

Now, plot \( Y_2 = \sin(5x) \). This graph should display a sine wave with standard amplitude \( y = 1 \) and \( y = -1 \), but with five complete cycles occurring within the interval where \( \sin x \) would have just one. This shows the increased frequency.

Step 4: Comparing the Graphs

Now compare the two graphs. The graph of \( Y_1 = 5 \sin x \) is taller but has the same frequency as \( \sin x \). In contrast, \( Y_2 = \sin(5x) \) has the same height as \( \sin x \), but the frequency is higher. The graphs of \( y = c \sin x \) and \( y = \sin(cx) \) look different due to changes in amplitude and frequency, respectively.

Step 5: Conclusion on the Statement

Based on the graphs, the statement "\( y = \sin cx \) has the same graph as \( y = c \sin x \)" is false. Changing the constant outside the sine function adjusts amplitude, whereas changing the constant inside the sine function adjusts frequency.

Key Concepts

Amplitude

Amplitude is a key feature of trigonometric functions, especially with the sine function. It represents the maximum value that the sine wave reaches. In simpler terms, the amplitude determines how tall the wave is. For example, in the function \( Y_1 = 5 \sin x \), the amplitude is 5. This means that the wave will reach heights of 5 and depths of -5.
The original sine function \( \sin x \) has an amplitude of 1, where the wave peaks at 1 and dips at -1. By multiplying it by 5, we stretch the wave 5 times taller. This has a dramatic visual effect on the graph, showcasing the power of amplitude in altering the appearance of trigonometric graphs.

Frequency

Frequency in the context of trigonometric functions refers to how often the wave cycles repeat over a given interval. The standard sine function, \( \sin x \), completes one full cycle from 0 to \( 2\pi \).
In the function \( Y_2 = \sin(5x) \), the frequency is increased by a factor of 5. This adjustment causes the wave to complete 5 full cycles within the same interval compared to the standard sine function's single cycle. By increasing the number inside the sine function, you directly increase the frequency of the wave, altering how rapidly it oscillates.

• Higher frequency means more cycles in a given interval.
• Frequency changes affect the wave's rapidity, but not its height.

This demonstrates the distinct roles of amplitude and frequency on sine waves.

Graphing Calculator

A graphing calculator is an essential tool for visualizing complex mathematical functions like trigonometric equations. It allows you to input functions and instantly see their graphical representation on the screen.
When working with the functions \( Y_1 = 5 \sin x \) and \( Y_2 = \sin(5x) \):

• You can easily observe how changes in amplitude and frequency affect the shape of the graphs.
• Graphing calculators help you compare different functions, such as seeing the distinct differences in height and number of cycles between \( Y_1 \) and \( Y_2 \).

These visual insights provide a deeper understanding of abstract mathematical concepts. The use of such calculators can make complex ideas more approachable and understandable for students.

Sine Function

The sine function is one of the fundamental trigonometric functions that describe the wave-like oscillations present in various real-world phenomena. The basic form \( \sin x \) oscillates between -1 and 1, with a smooth, periodic movement repeating every \( 2\pi \) radians.
Key characteristics of the sine function include:

• Amplitude: Determines the wave peak height.
• Frequency: Controls how quickly successive peaks occur.

By adjusting these properties, as seen with \( Y_1 = 5 \sin x \) and \( Y_2 = \sin(5x) \), you can create waves with varying attributes.
The versatility of the sine function allows it to model many situations, from sound waves to alternating current in electrical circuits. Understanding its behavior and how it can be manipulated becomes crucial for students engaging with such mathematical concepts.