\[y = f(x) + k\]

• If \(k > 0\): Shifts graph UP by \(k\) units
• If \(k < 0\): Shifts graph DOWN by \(|k|\) units
• Translation vector: \(\begin{pmatrix} 0 \\ k \end{pmatrix}\)

\[y = f(x - h)\]

• 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

\[y = -f(x)\]

• All y-values change sign
• Points \((x, y)\) become \((x, -y)\)
• Graph is flipped vertically (upside down)

\[y = f(-x)\]

• All x-values change sign
• Points \((x, y)\) become \((-x, y)\)
• Graph is flipped horizontally (left-right)

\[y = f^{-1}(x)\]

• Gives the inverse function
• Points \((x, y)\) become \((y, x)\)
• x and y coordinates swap

\[y = a \cdot f(x)\]

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

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 \(\frac{1}{2}\)

\[y = f(bx)\]

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 \(\frac{1}{2}\)

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

\[y = a \cdot f(b(x - h)) + k\]

\(a\): 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)

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

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).

• Affected by horizontal transformations only
• If original asymptote is \(x = c\), after \(f(b(x-h))\):
• New asymptote: \(x = \frac{c}{b} + h\)

• Affected by vertical transformations only
• If original asymptote is \(y = d\), after \(a \cdot f(x) + k\):
• New asymptote: \(y = ad + k\)

❌ 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