How to Graph Sine and Cosine with Transformations

Learn to sketch graphs of sine and cosine functions with various transformations in this comprehensive video lesson. Explore periodic functions, the unit circle, and circular functions before delving into graphing sine and cosine waves. Master techniques for vertical and horizontal stretches, compressions, and shifts, including phase shifts. Practice with multiple examples, from basic sine graphs to complex transformations like y = 3 cos (3x + 2π/3) + 2. Gain a thorough understanding of amplitude, period, and key x-values for trigonometric functions, enhancing your ability to analyze and graph these essential mathematical concepts.

Introduction.
What are periodic functions?.
Revisiting the unit circle.
Sine and Cosine are periodic functions.
Revisiting sine and cosine values from the unit circle.
Revisiting the circular functions.
Values for y = sin x.
Values for y = cos x.
Basic points on a sine graph.
Graphing y = sin x over 1 period.
Graphing the sine wave, y = sin x.
Creating points from negative angle measures.
A few facts about the sine function.
Showing that sine is an odd function.
Obtaining the five key x-values for sine.
Graphing y = a sin x problem 1.
Graphing y = a sin x problem 2.
Graphing y = a sin x problem 3, reflection across x-axis.
Showing the range for y = a sin x.
Finding the amplitude.
Finding the period.
Considering a horizontal stretch or horizontal compression.
Graph of f(x) = a sin bx.
Graphing f(x) = 3 sin 2x.
Graphing f(x) = 3 sin x/3.
What is a phase shift?.
Graphing y = 4 sin (2x - 2π/3).
Vertical shift.
Graphing y = 4 sin (3x + 3π/4) - 1.
Graphing y = cos x.
Graphing y = 3 cos (3x + 2π/3) + 2.