Sine and Cosine Functions Overview

Sine and Cosine Functions Overview

• The sine function, f(x) = sin(x), has a period of 2π.
• The cosine function, f(x) = cos(x), serves as the basis for cosine graphs.

Graph Characteristics

• The graph f(x) = sin(x) + 4 passes through (0, 4) and has a minimum at (3π/2, 3).
• f(x) = -4sin(x) has a domain of all real numbers with an amplitude of 4 and y-intercept at (0, 0).

Transformation Properties

• The transformation resulting in f(x) = -2sin(x) + 3 involves reflecting across the x-axis, stretching vertically by a factor of 2, and translating vertically by 3 units.
• The function y = 1/2sin(x) - 2 represents a vertical compression and downward translation.

Range and Maximum Values

• The range of f(x) = 5cos(x) + 1 spans from -4 to 6, indicating the highest and lowest values of the function.
• The maximum value of f(x) = sin(x) is 1, illustrating the peak of the sine wave.
• The range of y = -5sin(x) is from -5 to 5, showing the effects of amplitude and reflection.

Specific Function Graphs

• y = 4cos(x) + 3 features a vertical shift upward by 3 units from the basic cosine function.

X-intercepts and Important Points

• The graph of f(x) = sin(x) can illustrate various key points, including the x-intercept and other transformations required for graphing.