IB Mathematics AI – Topic 2

General Form:

\[ y = mx + c \]

Common Applications:

• Cost functions: Total cost = fixed cost + (variable cost × quantity)
• Distance-time: Distance = speed × time (constant speed)
• Temperature conversion: F = 1.8C + 32
• Depreciation: Straight-line depreciation

⚠️ Tips:

• m represents rate of change per unit
• c is the initial value (when x = 0)
• Linear models work best for short-term predictions

General Forms:

Standard form:

\[ y = ax^2 + bx + c \]

Vertex form:

\[ y = a(x - h)^2 + k \]

where (h, k) is the vertex

Key Features:

• Vertex: Maximum or minimum point
• Axis of symmetry: \(x = -\frac{b}{2a}\)
• Direction: Opens up if a > 0, down if a < 0

Applications:

• Projectile motion: Height vs. time
• Area optimization: Maximum enclosed area
• Profit maximization: Revenue - Cost
• Satellite dishes: Parabolic reflectors

⚠️ Tips:

• Use vertex form when you know maximum/minimum
• Use GDC to find vertex and roots
• Check domain restrictions for real-world problems

Solution:

(a) Maximum height:

Maximum occurs at vertex. Time at vertex:

\[ t = -\frac{b}{2a} = -\frac{20}{2(-5)} = -\frac{20}{-10} = 2 \text{ seconds} \]

Maximum height:

\[ h(2) = -5(2)^2 + 20(2) + 2 = -20 + 40 + 2 = 22 \text{ meters} \]

(b) When ball hits ground (h = 0):

Solve: \(-5t^2 + 20t + 2 = 0\)

Using quadratic formula or GDC:

\[ t = \frac{-20 \pm \sqrt{20^2 - 4(-5)(2)}}{2(-5)} = \frac{-20 \pm \sqrt{440}}{-10} \]

\[ t \approx 4.10 \text{ seconds (taking positive value)} \]

Answers: (a) 22 meters, (b) 4.10 seconds

Direct Variation:

y varies directly as x (or y is directly proportional to x)

\[ y = kx \quad \text{or} \quad y \propto x \]

where k is the constant of proportionality

Properties: When x doubles, y doubles; when x = 0, y = 0

Inverse Variation:

y varies inversely as x (or y is inversely proportional to x)

\[ y = \frac{k}{x} \quad \text{or} \quad y \propto \frac{1}{x} \]

Properties: When x doubles, y halves; xy = k (constant)

Applications:

• Direct: Distance = speed × time (constant speed)
• Direct: Hooke's Law: Force ∝ extension
• Inverse: Speed = distance/time (constant distance)
• Inverse: Boyle's Law: Pressure ∝ 1/volume

⚠️ Tips:

• Direct variation: straight line through origin
• Inverse variation: hyperbola, never touches axes
• Always find k first using given values

General Form:

\[ y = Ab^x \quad \text{or} \quad y = Ae^{kx} \]

where A is initial value, b is base (or e with rate k)

Growth (b > 1 or k > 0):

• Population growth: \(P(t) = P_0e^{rt}\)
• Compound interest: \(A = P(1+r)^t\)
• Bacterial growth

Decay (0 < b < 1 or k < 0):

• Radioactive decay: \(N(t) = N_0e^{-\lambda t}\)
• Drug concentration in body
• Temperature cooling (Newton's Law)

Half-life and Doubling Time:

Half-life: Time for quantity to reduce to half

\[ t_{1/2} = \frac{\ln 2}{k} \]

Doubling time: Time for quantity to double

\[ t_d = \frac{\ln 2}{r} \]

⚠️ Tips:

• Exponential growth increases rapidly
• Use natural log (ln) to solve for time
• Check if model uses e or another base

General Form:

\[ y = a + b\ln(x) \quad \text{or} \quad y = a + b\log(x) \]

Properties:

• Grows slowly (slower than linear for large x)
• Only defined for x > 0
• Has vertical asymptote at x = 0

Applications:

• pH scale: Acidity measurement
• Richter scale: Earthquake magnitude
• Decibels: Sound intensity
• Learning curves: Skill acquisition

General Form:

\[ y = A\sin(B(x - C)) + D \quad \text{or} \quad y = A\cos(B(x - C)) + D \]

Parameters:

• A: Amplitude (half the distance between max and min)
• B: Affects period: \(P = \frac{2\pi}{B}\)
• C: Horizontal shift (phase shift)
• D: Vertical shift (midline)

Applications:

• Tides: Sea level vs. time
• Temperature: Daily/seasonal variation
• Sound waves: Oscillations
• Ferris wheel: Height vs. time
• Daylight hours: Throughout the year

⚠️ Tips:

• Amplitude = (max - min)/2
• Midline D = (max + min)/2
• Use GDC regression for sinusoidal fitting

Solution:

Step 1: Find amplitude A

\[ A = \frac{\text{max} - \text{min}}{2} = \frac{28 - 12}{2} = 8 \]

Step 2: Find midline D

\[ D = \frac{\text{max} + \text{min}}{2} = \frac{28 + 12}{2} = 20 \]

Step 3: Find period and B

Period = 24 hours (daily cycle)

\[ B = \frac{2\pi}{24} = \frac{\pi}{12} \]

Step 4: Find phase shift C

Maximum at t = 14 (2pm), so use cosine:

\[ T(t) = 8\cos\left(\frac{\pi}{12}(t - 14)\right) + 20 \]

Answer: \(T(t) = 8\cos\left(\frac{\pi}{12}(t - 14)\right) + 20\)

General Form:

\[ y = \frac{L}{1 + Ae^{-kx}} \]

where:

• L: Carrying capacity (limiting value)
• A: Constant determining initial value
• k: Growth rate

Properties:

• S-shaped curve (sigmoid)
• Starts with exponential growth
• Slows as it approaches carrying capacity L
• Horizontal asymptote at y = L

Applications:

• Population growth: Limited by resources
• Disease spread: Limited by total population
• Learning curves: Skill mastery plateaus
• Technology adoption: Market saturation

⚠️ Tips:

• More realistic than exponential for long-term growth
• Use GDC for logistic regression
• Carrying capacity is the maximum sustainable value

Definition: A function defined by different formulas on different parts of its domain.

General Form:

\[ f(x) = \begin{cases} f_1(x) & \text{if } x < a \\ f_2(x) & \text{if } a \leq x < b \\ f_3(x) & \text{if } x \geq b \end{cases} \]

Applications:

• Tax brackets: Different rates for income ranges
• Shipping costs: Rates by weight/distance
• Parking fees: First hour, additional hours
• Utility bills: Tiered pricing

⚠️ Tips:

• Check which piece applies for given x value
• Pay attention to < vs ≤ at boundaries
• Graph may have jumps (discontinuities)
• Evaluate carefully at boundary points

When to Use Each Model

• Linear: Constant rate of change
• Quadratic: Has max/min point
• Exponential: Rapid growth/decay
• Sinusoidal: Periodic patterns

Key Characteristics

• Direct: y = kx
• Inverse: y = k/x
• Logistic: S-curve with limit
• Piecewise: Different rules