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Graphing Sine and Cosine Functions with Transformations

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Summary

Learn how to graph sine and cosine functions, including transformations such as amplitude changes, phase shifts, and vertical shifts.

Highlights

Introduction to Sine and Cosine Graphs

00:00

The video starts by explaining the general formula for graphing sine and cosine functions, including the roles of amplitude, period, horizontal shift, and vertical shift.

Basic Sine Graph

01:15

Explanation on how to plot the basic sine graph using the unit circle, and the significance of key points like 0, π/2, π, 3π/2, and 2π.

Basic Cosine Graph

03:45

Demonstrates the plotting process for the basic cosine graph and comparison with the sine graph.

Example: y = 2 sin(1/2 x)

05:30

Provides a step-by-step guide on graphing a transformed sine function with vertical stretch and period changes.

Example: y = -cos(2(x - π/4))

09:15

Illustrates graphing a transformed cosine function with reflection, period adjustment, and horizontal shift.

Example: y = 3 sin(x) - 3

14:00

Describes graphing a sine function with a vertical shift and amplitude change.

Example: y = cos(π/4(x + 2)) + 1

17:30

Explains the procedure to graph a cosine function with phase shift, vertical shift, and period alteration.

Complex Example: y = 1/2 sin(3x + π) - 2

22:00

Analyzes and graphs a more complex sine function involving all transformations, including vertical shrink, phase shift, and vertical shift.

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