Become an expert at graphing sine and cosine functions with this ultimate guide from Mario's Math Tutoring! This video takes you through 10 examples, starting with the basics and gradually moving to more challenging transformations. If you're studying trigonometry or pre-calculus, this tutorial is essential for mastering sinusoidal graphs. Learn how to derive the basic sine and cosine graphs from the unit circle, then dive deep into transformations including: Amplitude: Vertical stretching and shrinking, and reflections. Period: How the 'B' value affects the cycle length (2π/B). Phase Shift: Horizontal shifting (left and right). Vertical Shift: Moving the graph up and down. We'll cover factoring out the 'B' value for clarity and even demonstrate a powerful table method for complex transformations. By the end, you'll be confidently graphing any sine or cosine function! Timestamps: 0:00 Introduction to Graphing Sine and Cosine Functions 0:40 Deriving the Basic Sine Graph from the Unit Circle 2:00 Deriving the Basic Cosine Graph from the Unit Circle 3:00 Key Differences Between Sine and Cosine Starting Points 3:45 Understanding General Form (y = A sin B(x-H) + K) 4:10 Amplitude (Vertical Stretch/Shrink & Reflection) 4:45 Period Formula (2π/B) 5:15 Phase Shift (Horizontal Shift H) 5:35 Vertical Shift (K) 6:00 Example 3: Graphing y = 2sin(4x) (Amplitude & Period) 8:25 Example 4: Graphing y = -cos(1/2 x) (Reflection & Period) 10:30 Tip for Drawing Smooth Curves 10:50 Example 5: Graphing y = sin(x - π/2) + 2 (Phase & Vertical Shift) 13:50 Example 6: Graphing y = 3cos(x) - 1 (Amplitude & Vertical Shift) 16:00 Example 7: Graphing y = 1/2 sin(2x + π/2) + 1 (Factoring B, All Shifts) 19:50 Example 8: Graphing y = -2sin(π/2 x - 3) - 2 (Complex Transformations) 23:45 Example 9: Graphing y = 4cos(1/3 x + π) + 2 (Table Method for Transformations) 29:55 Example 10: Graphing y = -3sin(π/8 x - 5) - 2 (Advanced Table Method) ➡️JOIN the channel as a CHANNEL MEMBER at the "ADDITIONAL VIDEOS" level to get access to my math video courses(Algebra 1, Algebra 2/College Algebra, Geometry, PreCalculus), midterm & final exam reviews, ACT and SAT prep videos and more! (Over 390+ videos)