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Graphing Sine Function

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Graphing Sine Function

Study Guide

Key Definition

The sine function is defined as $y = \sin(x)$. Its graph is a continuous wave that oscillates between $-1$ and $1$.

Important Notes

The period of $y = \sin(x)$ is $2\pi$.
The amplitude of $y = a\sin(bx)$ is $|a|$.
The sine function is an odd function, meaning it has symmetry about the origin.
The x-intercepts of $y = \sin(x)$ occur at $n\pi$, where $n$ is an integer.
The y-intercept of $y = \sin(x)$ is at $(0,0)$.

Mathematical Notation

$\sin(x)$ is the sine function

$\pi$ equals approximately 3.14159

$2\pi$ represents the period of the sine function

$|a|$ denotes the amplitude

$n\pi$ indicates the x-intercepts where $n$ is an integer

Remember to use proper notation when solving problems

Why It Works

The sine wave represents periodic oscillations in nature and engineering, modeled by $y = \sin(x)$.

Remember

The sine function is periodic with a period of $2\pi$ and its graph is symmetric around the origin.

Quick Reference

Period (parent function):$2\pi$

Amplitude:$|a|$

Domain:$(-\infty, \infty)$

Range (parent function):$[-1, 1]$

Understanding Graphing Sine Function

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The sine function $y = \sin(x)$ has a wave-like pattern.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the period of the sine function $y = \sin(x)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a Ferris wheel with radius 1 and its center at height 0 that completes one full rotation every $2\pi$ minutes. What is the height of a seat at $\pi$ minutes?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Predict the behavior of the sine function $y = a\sin(bx)$ as $a$ and $b$ are changed.

Challenge Quiz

Single Choice Quiz

Advanced

For the function $y = 3\sin(2x)$, what is the amplitude?

Recap

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Review key concepts and takeaways