Trigonometric functions can be transformed to model various real-world phenomena. By adjusting parameters, we can shift, stretch, or reflect these functions to accurately represent cyclic patterns in nature and engineering.

Understanding these transformations allows us to adapt basic sine, cosine, and tangent functions to fit specific scenarios. We'll explore how changing amplitude, period, phase, and vertical shifts impacts the graph and equation of trig functions.

Transformations of Trigonometric Functions

Transformations of trigonometric functions

• Vertical translations move graph up or down $f(x) = \sin(x) + c$ shifts up by $c$ units
• Horizontal translations shift left or right $f(x) = \sin(x - h)$ moves right by $h$ units
• Reflections flip graph over axis $f(x) = -\sin(x)$ reflects over x-axis $f(x) = \sin(-x)$ reflects over y-axis
• Dilations stretch or compress graph vertically $f(x) = a\sin(x)$ $|a| > 1$ stretches $0 < |a| < 1$ compresses
• Horizontal stretch/compression $f(x) = \sin(bx)$ $0 < |b| < 1$ stretches $|b| > 1$ compresses

Effects of parameter changes

• Amplitude changes affect graph height $f(x) = a\sin(x)$ $|a|$ is amplitude
• Period changes alter horizontal stretch/compression $f(x) = \sin(bx)$ period is $\frac{2\pi}{|b|}$
• Phase shifts create horizontal translations $f(x) = \sin(x - c)$ shifts right by $c$ units
• Vertical shifts move graph up/down $f(x) = \sin(x) + d$ shifts up by $d$ units
• Key features affected include max/min values x-intercepts midline equation

Equations from transformed graphs

• Identify parent function (sine cosine tangent)
• Determine amplitude from graph height
• Calculate period and frequency using horizontal stretch
• Identify phase and vertical shifts from graph position
• Combine transformations into equation
• General form $f(x) = a\sin(b(x - c)) + d$ or $f(x) = a\cos(b(x - c)) + d$

Graphing transformed functions

• Begin with parent function graph
• Apply transformations in order:
  1. Horizontal compression/stretch
  2. Phase shift
  3. Reflection
  4. Vertical stretch/compression
  5. Vertical shift
• Plot key points (max/min x-intercepts inflection points)
• Sketch curve through key points
• Label period amplitude midline

Real-world applications of transformations

• Identify periodic phenomena (tides seasons sound waves alternating current)
• Model situation using trigonometric function
• Choose appropriate function (sine cosine tangent)
• Determine amplitude from max/min values
• Calculate period from cyclic behavior
• Identify phase and vertical shifts
• Interpret parameters in context (amplitude as max deviation period as cycle time)
• Use model for predictions or analysis (find function values solve equations)
• Recognize model limitations in real-world scenarios