Function Transformations

General Study Advice & Fundamentals

  • Preparation: Maintain handwritten paper notes and step-by-step graphical solutions for online assignments to prepare for exams.

  • Linear Equations: Both variables raised to first power (y=mx+by=mx+b). Graph using slope and known points.

  • Table of Values: Substitute arbitrary xx-values into a function to calculate yy-values when function shape is forgotten.

Parent Functions

  • Constant: y=cy=c (Horizontal line)

  • Identity: y=xy=x (Line through origin, slope 11)

  • Absolute Value: y=∣x∣y=∣x∣ (V-shape, vertex at (0,0)(0,0), range [0,∞)[0,∞))

  • Quadratic: y=x2y=x2 (U-shaped parabola, vertex at (0,0)(0,0))

  • Cubic: y=x3y=x3 (S-shaped curve through origin)

  • Square Root: y=xy=x​ (Sideways half-parabola, domain/range [0,∞)[0,∞))

  • Cube Root: y=x3y=3x​ (Origin-symmetric curve)

  • Reciprocal: y=1xy=x1​ (Asymptotes at x=0x=0 and y=0y=0)

  • Reciprocal Square: y=1x2y=x21​ (Branches above x-axis, asymptotes at x=0x=0 and y=0y=0)

Transformations of Functions

  • General Formula: y=a⋅f(b(x+c))+dy=a⋅f(b(x+c))+d

  • Vertical Stretch/Compress (aa): Vertical stretch if ∣a∣>1∣a∣>1; vertical compression if 0<∣a∣<10<∣a∣<1.

  • Vertical Reflection (aa): Reflected across x-axis if a<0a<0.

  • Horizontal Stretch/Compress (bb): Horizontal compression if b>1b>1; horizontal stretch if 0<b<10<b<1.

  • Horizontal Reflection (bb): Reflected across y-axis if b<0b<0.

  • Vertical Shift (dd): Shift up if +d+d; shift down if −d−d.

  • Horizontal Shift (cc): Shift left if +c+c; shift right if −c−c (opposite sign rule).

Piecewise Functions

  • Definition: Functions defined by multiple sub-functions applying to distinct domain intervals.

  • Endpoints: Solid circle for included boundaries (≤≤, ≥≥); open circle for excluded boundaries (<<, >>).

Composition of Functions

  • Notation: (f∘g)(x)=f(g(x))(f∘g)(x)=f(g(x))

  • Evaluation: Solve inside function g(x)g(x) first, then substitute that result into outside function f(x)f(x).