Algebra 2B - Unit 4: Trigonometric Functions

Graphs of Sine Functions

• The midline of the function can be determined, e.g., y = 2 represents the midpoint of a sine graph.
• Amplitude measures the height from the midline to the peak or trough. Values are represented by specific numbers, e.g., 3, 2, or 4.
• Frequency determines how many cycles fit into a specified interval. Notable frequencies include values like 4 or 2 for various functions.
• The period indicates the length of one complete cycle in a trigonometric function, such as π or 4π.
• Function rules like 2 sin(3x) + 4 and sin(2x) - 3 express transformations in sine graphs.

Graphs of Cosine Functions

• Cosine graphs can be characterized by their midline, amplitude, period, and frequency.
• The amplitude can be found in functions like 3 or 4 cos(2x) + 6 and can also show different transformations.
• Period examples include π/2 and 2π/3, indicating distances between repeating features in these graphs.
• Cosine function rules are seen in expressions such as y = 4 cos(7x) - 1.

Graphs of Tangent Functions

• Tangent functions demonstrate unique properties, including a period of π/2.
• Vertical asymptotes occur at multiples of π/4, specifically noted as x = pi/4, 3pi/4, and so forth.
• Function representations include y = tan(2x) and y = tan(3x), showcasing different transformations and their resulting graphs.

Modeling with Trigonometric Functions

• Certain sine functions may model multiple cycles over given intervals; for example, f(x) = sin(2x) completes 2 cycles over (0, 2π].
• Cosine functions can be tailored to specific attributes, such as f(x) = 2 cos(x) + 1 for a midline or f(x) = 7 cos(x) + 4 for amplitude.
• The height of objects modeled by cosine functions, like a Ferris wheel, can be represented through functions such as f(x) = -106 cos(4x) + 106.
• Cycling frequency in trigonometric functions translates to practical questions like the number of rotations, e.g., a yo-yo represented by f(x) = 2 cos(8πx) + 2 completes 4 rotations in 11 seconds.

Transformations of Trigonometric Functions

• Transformations involve shifting, stretching, or compressing the original functions. For instance, g(x) = f(x + π/4) indicates a leftward phase shift.
• When f(x) = sin(x) is translated down by 4 units, it follows the transformation g(x) = f(x) - 4.
• Various transformations can also involve reflections, such as g(x) = f(−x), which showcases an over-the-y-axis reflection.
• The transformation g(x) = f(2x) demonstrates a horizontal compression by a factor of 2.

Summary of Function Rules

• Specific function rules are highlighted, including transformation rules for adjusted functions like g(x) = -3 cos(2x) + 2, representing reflections and translations in the graphs.
• Study the specific attributes and transformations required for each sine, cosine, and tangent function, as each contributes uniquely to the understanding of trigonometric behavior in different scenarios.