Trig and geometry Ultimate Study Guide

Sinusoidal Function

A function that can be described by sine or cosine, typically having the form y = a sin(b(x - c)) + d or y = a cos(b(x - c)) + d.

Amplitude

The maximum value of a sinusoidal function, calculated as the absolute value of parameter a: Amplitude = |a|.

Period

The length of one complete cycle of a sinusoidal function, determined by the parameter b: Period = 2π/b.

Phase Shift

The horizontal shift of a sinusoidal graph, determined by the parameter c in y = a sin(b(x - c)) + d.

Vertical Translation

The upward or downward shift of a sinusoidal graph, determined by the parameter d in the equation.

Vertical Stretch

When |a| > 1, the graph undergoes a vertical stretch by a factor of a.

Vertical Shrink

When 0 < |a| < 1, the graph undergoes a vertical shrink (compression) by a factor of a.

Horizontal Stretch

When 0 < b < 1, the graph undergoes a horizontal stretch by a factor of 1/b.

Horizontal Shrink

When b > 1, the graph undergoes a horizontal shrink by a factor of 1/b.

Reflection Across x-axis

When a is negative (-a), the graph is reflected across the x-axis.

Midline

The horizontal line that represents the average value of the function, given by y = d.

Transformation Analysis

A systematic examination of the effects of parameters a, b, c, and d on the graph of sinusoidal functions.

Reflection

A transformation that flips the graph over a specific axis, in this case achieved by negative coefficients.

Coefficient

A numerical factor in front of a variable, which affects the amplitude or rate of change of a function.

Resulting Equation

An equation constructed based on given characteristics of a sinusoidal function, involving parameters a, b, c, and d.