Master Form & Parameter Roles

• Master form: \(y = a \cdot f\big(b(x + c)\big) + d\)
• a: vertical stretch/compression by \(|a|\); reflect in \(x\)-axis if \(a < 0\)
• b: horizontal compression/stretch; scale factor = \(1/|b|\); reflect in y-axis if \(b < 0\)
• c: horizontal shift; \(+c\) moves left \(c\) units, \(-c\) moves right \(c\) units
• d: vertical shift; \(+d\) up, \(-d\) down
• Mapping rule: If \((x, y)\) lies on \(y = f(x)\), then on \(y = a f(b(x + c)) + d\) the corresponding point is:
  • \(x' = \frac{x}{b} - c\)
  • \(y' = a y + d\)
• Order of effects: apply “inside” (\(b, c\)) first (horizontal), then “outside” (\(a, d\)) (vertical).

Quick Mapping: What Each Transformation Does

• Vertical stretch by \(k\) (\(k>1\)): \(y = k f(x) \rightarrow (x, k y)\)
• Vertical compression by k (0 < k < 1):
  Equation: \( y = k f(x) \)
  Transformation: \( (x, ky) \)
• Vertical reflection: \(y = -f(x) \rightarrow (x, -y)\)
• Horizontal compression by \(k\) (\(k>1\)): \(y = f(kx) \rightarrow (\frac{x}{k}, y)\)
• Horizontal stretch by k (0 < k < 1):
  Equation: \( y = f(kx) \)
  Transformation: \( \left( \frac{x}{k}, y \right) \)
• Horizontal reflection: \(y = f(-x) \rightarrow (-x, y)\)
• Horizontal shift right by \(c\): \(y = f(x - c) \rightarrow (x + c, y)\)
• Horizontal shift left by \(c\): \(y = f(x + c) \rightarrow (x - c, y)\)
• Vertical shift up by \(d\): \(y = f(x) + d \rightarrow (x, y + d)\)
• Vertical shift down by \(d\): \(y = f(x) - d \rightarrow (x, y - d)\)

Domain, Range, Intercepts, Asymptotes

• Domain: apply the inverse horizontal map to parent domain endpoints: \(x' = \frac{x}{b} - c\)
• Range: apply vertical map to parent range: \(y' = a y + d\)
• x-intercepts: solve \(a f(b(x + c)) + d = 0\) (do not just map x-intercepts unless \(d = 0\))
• y-intercept: set \(x = 0\) \(\rightarrow\) \(y = a f(b c) + d\)
• Vertical asymptotes \(x = L\) map to \(x = \frac{L}{b} - c\)
• Horizontal asymptotes \(y = L\) map to \(y = a L + d\)

Absolute Value, Reciprocal, and Inverse

• \(y = |f(x)|\): Points with \(y \geq 0\) stay the same; points with \(y < 0\) reflect across the x-axis to \((x, -y)\). Domain unchanged; range becomes \([0, \infty)\) if parent is symmetric about x-axis.
• \(y = f(|x|)\): Keep the right half (\(x \geq 0\)) of \(y = f(x)\); mirror it to the left. Domain becomes \(( -\infty, \infty )\) if \(f\) is defined for \(x \geq 0\); range = range of \(f\) on \(x \geq 0\).
• \(y = \frac{1}{f(x)}\): Map \((x, y)\) to \((x, 1/y)\); zeros of \(f\) become vertical asymptotes; vertical asymptotes of \(f\) remain vertical asymptotes. Where \(|y|\) is large, \(1/y\) is near 0; where \(y\) is near 0, \(1/y\) is large in magnitude. Domain excludes \(x\) where \(f(x) = 0\); range excludes 0 if \(f\) has no infinities.
• \(y = f^{-1}(x)\) (inverse): Reflect across \(y = x\): \((x, y) \rightarrow (y, x)\). Domain and range swap; function must be one-to-one (restrict domain if needed).

Trigonometric Transformations (sin, cos, tan)

• Standard forms:
  • \(y = a \sin(b(x + c)) + d\)
  • \(y = a \cos(b(x + c)) + d\)
  • \(y = a \tan(b(x + c)) + d\)
• Common features (sin/cos):
  • Amplitude: \(|a|\)
  • Midline: \(y = d\)
  • Period (sin/cos): \(\frac{2\pi}{|b|}\) (radians), \(\frac{360^\circ}{|b|}\) (degrees)
  • Phase shift: \(-c\) (left if \(c > 0\), right if \(c < 0\))
  • Range: \([d - |a|,\ d + |a|]\); domain: all real \(x\)
• Zeros (anchors):
  • sin: \(b(x + c) = n\pi \rightarrow x = -c + \frac{n\pi}{b}\)
  • cos: \(b(x + c) = \frac{\pi}{2} + n\pi \rightarrow x = -c + \frac{\pi/2 + n\pi}{b}\)
• Tangent specifics:
  • Period = \(\frac{\pi}{|b|}\)
  • Vertical asymptotes where \(b(x + c) = \frac{\pi}{2} + n\pi \rightarrow x = -c + \frac{\pi/2 + n\pi}{b}\)
  • Zeros where \(b(x + c) = n\pi\)
  • Domain: all real x except VA lines; range: all real \(y\)

Strategy for Sketching/Analysis (Exam Checklist)

• Rewrite into master form \(y = a f(b(x + c)) + d\)
• Extract a, b, c, d and record: horizontal scale 1/|b|, shifts, reflections, vertical scale |a|
• Choose key parent points (intercepts, turning points, zeros/asymptotes), then map each using: \(x' = \frac{x}{b} - c,\quad y' = a y + d\)
• For trig: compute amplitude, period, phase shift, midline; mark one full period with zeros/extrema/asymptotes
• For absolute value, reciprocal, inverse: note new domain/range, locate asymptotes (if any), and map representative points
• Compute intercepts explicitly when d ≠ 0 (do not assume they map from the parent)
• Label asymptotes, midline, and key points on the final graph; use exact values (π fractions, radicals) until the final step