Function Transformations - Formulas & Rules
IB Mathematics Analysis & Approaches (SL & HL)
📐 Translations (Shifts)
Definition:
A translation moves the entire graph without changing its shape, size, or orientation.
Vertical Translation:
\[y = f(x) + k\]
• If \(k > 0\): Shifts graph UP by \(k\) units
• If \(k < 0\): Shifts graph DOWN by \(|k|\) units
• Translation vector: \(\begin{pmatrix} 0 \\ k \end{pmatrix}\)
Horizontal Translation:
\[y = f(x - h)\]
• If \(h > 0\): Shifts graph RIGHT by \(h\) units
• If \(h < 0\): Shifts graph LEFT by \(|h|\) units
• Translation vector: \(\begin{pmatrix} h \\ 0 \end{pmatrix}\)
• ⚠️ CAUTION: \(f(x - 3)\) moves RIGHT, \(f(x + 3)\) moves LEFT
🪞 Reflections
Definition:
A reflection flips the graph over a line (axis), creating a mirror image.
Reflection in the x-axis:
\[y = -f(x)\]
• All y-values change sign
• Points \((x, y)\) become \((x, -y)\)
• Graph is flipped vertically (upside down)
Reflection in the y-axis:
\[y = f(-x)\]
• All x-values change sign
• Points \((x, y)\) become \((-x, y)\)
• Graph is flipped horizontally (left-right)
Reflection in the line \(y = x\):
\[y = f^{-1}(x)\]
• Gives the inverse function
• Points \((x, y)\) become \((y, x)\)
• x and y coordinates swap
↕️ Vertical Stretches & Compressions
General Form:
\[y = a \cdot f(x)\]
where \(a\) is the vertical scale factor (affects y-values)
Vertical Stretch:
When \(|a| > 1\): Vertical stretch by scale factor \(|a|\)
• All y-values are multiplied by \(a\)
• Graph becomes taller/steeper
• Points \((x, y)\) become \((x, ay)\)
• Example: \(y = 3f(x)\) stretches vertically by factor of 3
Vertical Compression:
When \(0 < |a| < 1\): Vertical compression by scale factor \(|a|\)
• All y-values are multiplied by \(a\)
• Graph becomes flatter/shorter
• Example: \(y = \frac{1}{2}f(x)\) compresses vertically by factor of \(\frac{1}{2}\)
↔️ Horizontal Stretches & Compressions
General Form:
\[y = f(bx)\]
where \(b\) affects x-values (⚠️ scale factor is \(\frac{1}{b}\))
Horizontal Compression:
When \(|b| > 1\): Horizontal compression by scale factor \(\frac{1}{|b|}\)
• All x-values are divided by \(b\)
• Graph becomes narrower
• Points \((x, y)\) become \(\left(\frac{x}{b}, y\right)\)
• Example: \(y = f(2x)\) compresses horizontally by factor of \(\frac{1}{2}\)
Horizontal Stretch:
When \(0 < |b| < 1\): Horizontal stretch by scale factor \(\frac{1}{|b|}\)
• All x-values are divided by \(b\)
• Graph becomes wider
• Example: \(y = f\left(\frac{x}{2}\right)\) stretches horizontally by factor of 2
🔄 General Transformation Form
Combined Transformation:
\[y = a \cdot f(b(x - h)) + k\]
Parameter Effects:
\(a\): Vertical stretch/compression by factor \(|a|\) (and reflection if \(a < 0\))
\(b\): Horizontal compression by factor \(\frac{1}{|b|}\) (and reflection if \(b < 0\))
\(h\): Horizontal translation (shift right by \(h\) units)
\(k\): Vertical translation (shift up by \(k\) units)
🔢 Order of Transformations
Recommended Order (INSIDE to OUTSIDE):
1. Horizontal stretch/compression: Apply \(b\) (inside)
2. Horizontal translation: Apply \(h\) (inside)
3. Vertical stretch/compression: Apply \(a\) (outside)
4. Vertical translation: Apply \(k\) (outside)
5. Reflections: Apply throughout
Important Note:
When multiple transformations are applied, work from the inside of the function outward. Transformations inside the function parentheses affect x-values (horizontal), while those outside affect y-values (vertical).
📏 Effect on Asymptotes
Vertical Asymptotes:
• Affected by horizontal transformations only
• If original asymptote is \(x = c\), after \(f(b(x-h))\):
• New asymptote: \(x = \frac{c}{b} + h\)
Horizontal Asymptotes:
• Affected by vertical transformations only
• If original asymptote is \(y = d\), after \(a \cdot f(x) + k\):
• New asymptote: \(y = ad + k\)
⚡ Quick Reference Table
| Transformation | Equation | Effect |
|---|---|---|
| Vertical shift up | \(y = f(x) + k\) | Move up \(k\) units |
| Vertical shift down | \(y = f(x) - k\) | Move down \(k\) units |
| Horizontal shift right | \(y = f(x - h)\) | Move right \(h\) units |
| Horizontal shift left | \(y = f(x + h)\) | Move left \(h\) units |
| Reflect in x-axis | \(y = -f(x)\) | Flip vertically |
| Reflect in y-axis | \(y = f(-x)\) | Flip horizontally |
| Vertical stretch | \(y = af(x)\), \(a > 1\) | Stretch by factor \(a\) |
| Vertical compression | \(y = af(x)\), \(0 < a < 1\) | Compress by factor \(a\) |
| Horizontal compression | \(y = f(bx)\), \(b > 1\) | Compress by factor \(\frac{1}{b}\) |
| Horizontal stretch | \(y = f(bx)\), \(0 < b < 1\) | Stretch by factor \(\frac{1}{b}\) |
⚠️ Common Mistakes to Avoid
❌ Mistake 1: Thinking \(f(x + 3)\) shifts right (it shifts LEFT)
✅ Remember: \(f(x - h)\) shifts right, \(f(x + h)\) shifts left
❌ Mistake 2: Confusing horizontal scale factor with the coefficient
✅ Remember: For \(f(bx)\), the scale factor is \(\frac{1}{b}\), not \(b\)
❌ Mistake 3: Forgetting to transform asymptotes
✅ Remember: Asymptotes move with the graph based on the transformation
❌ Mistake 4: Applying transformations in the wrong order
✅ Remember: Work from inside to outside: stretch → translate
💡 Exam Tip: Always identify which transformations are applied by looking at the position of coefficients and constants. INSIDE the function = horizontal effects (on x), OUTSIDE the function = vertical effects (on y). Use your GDC to verify your transformed graphs. Remember the sign conventions for translations!