Graphing Trigonometric Functions: A Complete Guide

Overview

Covers graphing sine, cosine, tangent, secant, cosecant, and cotangent functions.
Introduction to transformations: shifting, stretching, compressing.
Reference to the unit circle for graph origins.

Equation: y = sin(x)
Key Points:
- 0 radians: sin value = 0
- π/2 or 90 degrees: sin value = 1
- π or 180 degrees: sin value = 0
- 3π/2 or 270 degrees: sin value = -1
- 2π or 360 degrees: sin value = 0
Shape: Repeats starting at the midline, maxima at 1, minima at -1.
Amplitude (2sin(x)):
- Vertical stretch of 2: maxima at 2, minima at -2.
Period (sin(2x)):
- Period = 2π/B
- Calculation: B = 2 → Period = π
- Graph divided into 4 parts.
Transformation (sin(x + π) - 2):
- Right shift by π, down shift by 2.

Equation: y = cos(x)
Key Points:
- 0 radians: cos value = 1
- π/2 or 90 degrees: cos value = 0
- π or 180 degrees: cos value = -1
- 3π/2 or 270 degrees: cos value = 0
- 2π or 360 degrees: cos value = 1
Shape: Starts at maximum, goes to midline, minimum, then repeats.
Reflections and Periods:
- Negative reflection flips graph over x-axis.
- Half B value changes period (cos(1/2x)): Period = 4π.
Transformation (cos(x - π/2) + 1):
- Right shift by π/2, up shift by 1.

Multiple transformations (3sin(1/2x - π) - 2):
- Period = 4π, right shift by π, down shift by 2, amplitude stretch to 3.
Secant and Cosecant Graphs:
- Based on cosine and sine graphs respectively.
- Vertical asymptotes at x-intercepts of parent function.
- Reflections and stretches applied as described.

Basics:
- Tangent is y/x from the unit circle.
- Period = π/B
- Key points and asymptotes from -π/2 to π/2.
Transformations:
- Coefficient affects vertical stretch/shrink.
- Period = π/1/2 = 2π
- Examples with phase shift and vertical shift.

Basics:
- Cotangent is x/y from the unit circle.
- Period = π/B
- Key points and asymptotes from 0 to π.
Transformations:
- Reflections, phases, and vertical shifts discussed.