IB Mathematics AI – Topic 2

General Form:

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

Effect:

• If k > 0: Shift UP by k units
• If k < 0: Shift DOWN by |k| units

Examples:

• \(y = x^2 + 3\): Parabola shifted up 3 units
• \(y = \sin(x) - 2\): Sine curve shifted down 2 units
• \(y = e^x + 5\): Exponential shifted up 5 units

Key Point:

Adding/subtracting OUTSIDE the function affects the OUTPUT (y-values)

⚠️ Common Pitfalls & Tips:

• The constant is ADDED to the entire function
• Positive k = up, negative k = down (intuitive)
• All points on the graph move by the same amount
• Asymptotes also shift vertically

General Form:

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

Effect (OPPOSITE to sign!):

• If h > 0 (minus inside): Shift RIGHT by h units
• If h < 0 (plus inside): Shift LEFT by |h| units

Examples:

• \(y = (x - 4)^2\): Parabola shifted RIGHT 4 units
• \(y = (x + 2)^2\): Parabola shifted LEFT 2 units
• \(y = \sin(x - \pi/2)\): Sine shifted RIGHT by \(\pi/2\)

Key Point:

Replacing x with (x - h) affects the INPUT (x-values)

Memory trick: The sign is opposite to what you expect!

⚠️ Common Pitfalls & Tips:

• CRITICAL: Sign is opposite! \(f(x-3)\) moves RIGHT, not left
• The transformation is INSIDE the function brackets
• Think: "What value of x gives the original function?"
• Vertical asymptotes also shift horizontally

Solution:

Step 1: Identify transformations

Compare \(g(x) = (x + 3)^2 - 5\) with \(f(x) = x^2\)

Horizontal transformation: (x + 3)

This is \(f(x - (-3))\), so h = -3

Direction: LEFT by 3 units (because of +3)

Vertical transformation: - 5

This adds -5 to the function

Direction: DOWN by 5 units

Complete description:

1. Translate LEFT 3 units (horizontal)
2. Translate DOWN 5 units (vertical)

Vertex movement:

Original vertex: (0, 0)

New vertex: (-3, -5)

1. Reflection in the x-axis:

\[ y = -f(x) \]

Effect: Flips the graph upside down

Every y-value becomes its opposite: (x, y) → (x, -y)

Example: \(y = -x^2\) is an upside-down parabola

2. Reflection in the y-axis:

\[ y = f(-x) \]

Effect: Flips the graph left to right

Every x-value becomes its opposite: (x, y) → (-x, y)

Example: \(y = e^{-x}\) is \(e^x\) reflected in y-axis

Key Differences:

• Negative OUTSIDE: \(-f(x)\) reflects in x-axis
• Negative INSIDE: \(f(-x)\) reflects in y-axis

⚠️ Common Pitfalls & Tips:

• Don't confuse \(-f(x)\) with \(f(-x)\) – location of negative matters!
• x-axis reflection: negative outside (affects output)
• y-axis reflection: negative inside (affects input)
• For even functions, \(f(-x) = f(x)\), so no visible change

General Form:

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

Effect:

• If |a| > 1: Vertical STRETCH by factor a (graph taller)
• If 0 < |a| < 1: Vertical COMPRESSION by factor a (graph shorter)
• If a < 0: Stretch/compress AND reflect in x-axis

Examples:

• \(y = 3x^2\): Stretch by factor 3 (narrower parabola)
• \(y = \frac{1}{2}\sin(x)\): Compression by factor 1/2 (amplitude = 0.5)
• \(y = -2x^2\): Stretch by 2 AND flip

Key Point:

Multiplying OUTSIDE affects y-values: all y-coordinates multiplied by a

⚠️ Common Pitfalls & Tips:

• Stretch factor |a| > 1 makes graph TALLER (seems narrower for parabolas)
• x-intercepts stay the same (y = 0 multiplied by anything is still 0)
• y-intercept changes by factor a

General Form:

\[ y = f(bx) \]

Effect (RECIPROCAL!):

• If |b| > 1: Horizontal COMPRESSION by factor 1/b
• If 0 < |b| < 1: Horizontal STRETCH by factor 1/b
• If b < 0: Stretch/compress AND reflect in y-axis

Examples:

• \(y = (2x)^2\): Compression by factor 1/2 (narrower)
• \(y = \sin(\frac{x}{2})\): Stretch by factor 2 (period doubles)
• \(y = f(3x)\): Compression by factor 1/3

Key Point:

Factor is RECIPROCAL: \(f(bx)\) stretches by factor \(\frac{1}{b}\)

⚠️ Common Pitfalls & Tips:

• CRITICAL: Effect is reciprocal! \(f(2x)\) compresses by 1/2
• y-intercepts stay the same (x = 0)
• x-intercepts are divided by b
• For sinusoidal: period becomes \(\frac{2\pi}{b}\)

General Form:

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

Order of Transformations:

1. Horizontal stretch/compression: Factor 1/b
2. Horizontal translation: h units right (if positive)
3. Vertical stretch/compression: Factor a
4. Vertical translation: k units up (if positive)

Memory Aid - "HSVT" Order:

• Horizontal Stretch
• Horizontal Translation (Shift)
• Vertical Stretch (Height)
• Vertical Translation

Alternative: Think INSIDE-OUT

Work from inside the function outward:

1. Start with innermost x-transformations (horizontal)
2. Then apply outer transformations (vertical)

⚠️ Common Pitfalls & Tips:

• Order matters! Apply transformations systematically
• Horizontal transformations come from INSIDE the function
• Vertical transformations come from OUTSIDE
• Reflections can be done with stretches/compressions
• Always work from inside-out when analyzing

Solution:

Rewrite in standard form:

\[ g(x) = -2(x - (-1))^2 + 3 \]

Compare with: \(y = a \cdot f(b(x - h)) + k\)

Step 1: Identify parameters

a = -2 (vertical stretch AND reflection)

b = 1 (no horizontal stretch)

h = -1 (horizontal translation)

k = 3 (vertical translation)

Step 2: Describe each transformation

1. Horizontal translation: LEFT 1 unit (x + 1)
2. Vertical stretch: Factor 2 (makes it narrower)
3. Reflection: In x-axis (negative sign makes it upside-down)
4. Vertical translation: UP 3 units

Vertex movement:

Original: (0, 0)

After horizontal shift: (-1, 0)

After vertical stretch & reflection: (-1, 0) [vertex on x-axis stays]

After vertical shift: (-1, 3)

New vertex: (-1, 3)

INSIDE vs OUTSIDE

• INSIDE: Affects x (horizontal)
• OUTSIDE: Affects y (vertical)
• Inside changes often opposite direction

Critical Signs

• \(f(x - h)\): RIGHT if h > 0
• \(f(x) + k\): UP if k > 0
• Horizontal is opposite!

Stretch Factors

• Vertical: \(a \cdot f(x)\) by factor a
• Horizontal: \(f(bx)\) by factor 1/b
• Horizontal is reciprocal!

Order

• Horizontal stretch first
• Then horizontal shift
• Then vertical stretch
• Finally vertical shift