Why Study Transformations?
Transformations let you predict, sketch, and analyze any function by understanding how its graph changes with shifts, stretches, compressions, and reflections. This is a core skill for solving equations, graphing, and modeling real-world problems.
Master Form & Mapping Rule
- General form: \(y = a f(b(x + c)) + d\)
- Order of effects: Inside \((b(x + c))\) = horizontal (stretch/compress, reflect, shift); outside \((a, d)\) = vertical (stretch/reflect/shift)
- Coordinate mapping: If \((x, y)\) is on \(y = f(x)\), then on \(y = a f(b(x + c)) + d\):
\((x_\text{new}, y_\text{new}) = (\frac{x}{b} - c,\ a y + d)\)
What Each Transformation Does
| Transformation | Equation | Mapped Point |
|---|---|---|
| Vertical stretch by \(k\) | \(y = k f(x)\) | \((x,\ ky)\) |
| Vertical compression by \(k\) (\(0 < k < 1\)) | \(y = k f(x)\) | \((x,\ ky)\) |
| Vertical reflection | \(y = -f(x)\) | \((x,\ -y)\) |
| Horizontal compression by \(k\) (\(k > 1\)) | \(y = f(kx)\) | \((\frac{x}{k},\ y)\) |
| Horizontal stretch by \(k\) (\(0 < k < 1\)) | \(y = f(kx)\) | \((\frac{x}{k},\ y)\) |
| Horizontal reflection | \(y = f(-x)\) | \((-x,\ y)\) |
| Horizontal translation right by \(c\) | \(y = f(x - c)\) | \((x + c,\ y)\) |
| Horizontal translation left by \(c\) | \(y = f(x + c)\) | \((x - c,\ y)\) |
| Vertical translation up by \(d\) | \(y = f(x) + d\) | \((x,\ y + d)\) |
| Vertical translation down by \(d\) | \(y = f(x) - d\) | \((x,\ y - d)\) |
Sign Notes & Order of Operations
- \(b\) inside means horizontal scaling by \(1/b\). Larger \(b =\) compression.
- \(+c\) inside \((x + c)\) shifts left by \(c\); \((x − c)\) shifts right by \(c\).
- Negative \(a =\) reflect over \(x\)-axis and scale by \(|a|\).
- Apply horizontal (inside) changes first, then vertical (outside) changes.
Absolute Value, Reciprocal, and Inverse Graphs
- \(y = |f(x)|\): Reflect any part below x-axis upward. \((x, y)\) with \(y \geq 0\) stays, \(y < 0\) maps to \((x, -y)\).
- \(y = f(|x|)\): Right side (\(x \geq 0\)) stays, left side is a mirror of the right.
- \(y = \frac{1}{f(x)}\): \((x, y) \rightarrow (x, 1/y)\), zeros become vertical asymptotes.
- \(y = f^{-1}(x)\): \((x, y) \rightarrow (y, x)\), reflect across \(y = x\). Domain/range swap.
Trigonometric Graphs: Sine, Cosine, Tangent
- Standard form: \(y = a \sin(b(x + c)) + d\) or \(y = a \cos(b(x + c)) + d\)
- Amplitude = \(|a|\), Midline = \(y = d\), Period = \(\frac{2\pi}{|b|}\) (or \(\frac{360^\circ}{|b|}\)), Phase shift = \(-c\) (right if \(c < 0\), left if \(c > 0\))
- Tangent: \(y = a \tan(b(x + c)) + d\), period = \(\frac{\pi}{|b|}\), vertical asymptotes where \(b(x + c) = \frac{\pi}{2} + n\pi\)
- For sin: zeros at \(b(x + c) = n\pi \rightarrow x = -c + \frac{n\pi}{b}\)
- For cos: zeros at \(b(x + c) = \frac{\pi}{2} + n\pi \rightarrow x = -c + \frac{\pi/2 + n\pi}{b}\)
- For tan: zeros at \(b(x + c) = n\pi\)
- sin/cos range: \([d - |a|,\ d + |a|]\), domain all real numbers
- tan: range all real, domain excludes vertical asymptotes
Exam-Ready Checklist
- Write all transformed equations in the master form \(y = a f(b(x + c)) + d\)
- Use the mapping \((\frac{x}{b} - c,\ a y + d)\) on key points
- Check sign conventions: does +c shift left or right?
- For trig: compute amplitude, period, phase shift, midline, and label one full period
- For absolute/reciprocal/inverse: identify domain/range/asymptotes, map sample points
- For "exact" values use surds or π fractions; justify by mapping points
Short Worked Examples (Exam Style)
- Given \(y = 2 \sin(3(x + \frac{\pi}{6})) - 1\): amplitude = 2, period = \(\frac{2\pi}{3}\), midline \(y = -1\), phase shift left \(\frac{\pi}{6}\)
- If \(f(x) = x^2\) has \((1,1)\), then \(g(x) = -2 f(3(x + 2)) + 5\) maps to \((\frac{1}{3} - 2,\ -2 \cdot 1 + 5) = (-\frac{5}{3}, 3)\)
- If \(y = |f(x)|\) and \(f\) has \((2, -3)\), then g has \((2, 3)\)