Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

Trigonometry

4. Graphing Trigonometric Functions

Graphs of the Sine and Cosine Functions

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2:51 minutes

Problem 1

Textbook Question

In Exercises 1–6, determine the amplitude and period of each function. Then graph one period of the function.y = 3 sin 4x

Verified step by step guidance

Identify the general form of the sine function: \( y = a \sin(bx) \), where \( a \) is the amplitude and \( \frac{2\pi}{b} \) is the period.

For the given function \( y = 3 \sin 4x \), compare it with the general form to find \( a = 3 \) and \( b = 4 \).

Determine the amplitude: The amplitude is the absolute value of \( a \), so the amplitude is \( |3| = 3 \).

Calculate the period: Use the formula \( \frac{2\pi}{b} \) to find the period. Substitute \( b = 4 \) to get \( \frac{2\pi}{4} = \frac{\pi}{2} \).

Graph one period of the function: Start at \( x = 0 \), and plot the sine wave over the interval \( [0, \frac{\pi}{2}] \), marking key points at \( x = 0, \frac{\pi}{8}, \frac{\pi}{4}, \frac{3\pi}{8}, \frac{\pi}{2} \) with corresponding \( y \)-values based on the amplitude.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis or equilibrium position. In the context of sine functions, it is determined by the coefficient in front of the sine term. For the function y = 3 sin 4x, the amplitude is 3, indicating that the graph will oscillate between 3 and -3.

Period

The period of a trigonometric function is the length of one complete cycle of the wave. For sine functions, the period can be calculated using the formula P = 2π / |b|, where b is the coefficient of x. In the function y = 3 sin 4x, the period is P = 2π / 4 = π, meaning the function completes one full cycle over the interval from 0 to π.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the function's values over a specified interval. For y = 3 sin 4x, one period can be graphed from 0 to π, showing the wave's oscillation between its amplitude limits. Understanding the amplitude and period is crucial for accurately representing the function's behavior on a graph.