Course Cheatsheet
Section 04: Advanced Functions
Polynomial Functions
General Form
Polynomial of degree n: \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\)
Where:
- \(a_n \neq 0\) (leading coefficient)
- \(n\) is the degree (highest power)
- Domain: All real numbers
Polynomial Classification by Degree
| Degree | Name | General Form | Example |
|---|---|---|---|
| 0 | Constant | \(f(x) = a\) | \(f(x) = 5\) |
| 1 | Linear | \(f(x) = ax + b\) | \(f(x) = 2x + 3\) |
| 2 | Quadratic | \(f(x) = ax^2 + bx + c\) | \(f(x) = x^2 - 4\) |
| 3 | Cubic | \(f(x) = ax^3 + bx^2 + cx + d\) | \(f(x) = 2x^3 - x\) |
| 4 | Quartic | \(f(x) = ax^4 + ...\) | \(f(x) = x^4 - 5x^2 + 4\) |
End Behavior Rules
Determined by degree and leading coefficient:
| Degree | Leading Coefficient | Left End | Right End | Shape |
|---|---|---|---|---|
| Even | Positive (\(a > 0\)) | ↑ | ↑ | Both ends up |
| Even | Negative (\(a < 0\)) | ↓ | ↓ | Both ends down |
| Odd | Positive (\(a > 0\)) | ↓ | ↑ | Left down, right up |
| Odd | Negative (\(a < 0\)) | ↑ | ↓ | Left up, right down |
Key Properties
Number of Zeros (x-intercepts):
- Maximum possible: equal to degree
- Can be less if roots are repeated or complex
Number of Turning Points:
- Maximum: degree - 1
- A turning point is where function changes from increasing to decreasing (or vice versa)
Multiplicity of Zeros:
- Odd multiplicity: Graph crosses x-axis
- Even multiplicity: Graph touches x-axis but doesn’t cross
Power Functions & Roots
Power Functions
General form: \(f(x) = kx^p\)
Where \(k\) is coefficient and \(p\) is the power
Common Power Functions:
| Function | Domain | Range | Key Features |
|---|---|---|---|
| \(f(x) = x^2\) | All real | \([0, \infty)\) | Parabola, even function |
| \(f(x) = x^3\) | All real | All real | Cubic, odd function |
| \(f(x) = x^{1/2} = \sqrt{x}\) | \([0, \infty)\) | \([0, \infty)\) | Square root |
| \(f(x) = x^{-1} = \frac{1}{x}\) | \(x \neq 0\) | \(y \neq 0\) | Reciprocal/Hyperbola |
Root Functions
Square Root: \(f(x) = \sqrt{x}\)
- Domain: \(x \geq 0\)
- Range: \([0, \infty)\)
- Always increasing
Cube Root: \(f(x) = \sqrt[3]{x}\)
- Domain: All real numbers
- Range: All real numbers
- Always increasing
- Passes through origin
General Root: \(f(x) = \sqrt[n]{x}\)
- Even n: Domain restricted to \(x \geq 0\)
- Odd n: Domain is all real numbers
Rational Exponents
Conversion Rules:
- \(x^{1/n} = \sqrt[n]{x}\)
- \(x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\)
- \(x^{-n} = \frac{1}{x^n}\)
Properties:
- \((x^a)^b = x^{ab}\)
- \(x^a \cdot x^b = x^{a+b}\)
- \(\frac{x^a}{x^b} = x^{a-b}\)
Exponential Functions
Definition and Properties
Exponential function: \(f(x) = a \cdot b^x\) where \(b > 0, b \neq 1\)
Key Components:
- \(a\): Initial value (when \(x = 0\))
- \(b\): Base (growth/decay factor)
- \(x\): Exponent (time, periods)
Growth vs Decay
| Type | Base | Behavior | Example |
|---|---|---|---|
| Exponential Growth | \(b > 1\) | Increases rapidly | \(f(x) = 2^x\) |
| Exponential Decay | \(0 < b < 1\) | Decreases rapidly | \(f(x) = (0.5)^x\) |
Standard Form: \(f(t) = P_0 \cdot (1 + r)^t\)
- \(P_0\): Initial amount
- \(r\): Growth rate (positive) or decay rate (negative)
- \(t\): Time
The Natural Exponential: \(e\)
Euler’s number: \(e \approx 2.71828\)
Continuous Growth: \(f(t) = Pe^{rt}\)
- Most natural model for continuous processes
- \(r > 0\): continuous growth
- \(r < 0\): continuous decay
Applications:
- Compound interest: \(A = Pe^{rt}\)
- Population growth
- Radioactive decay
- Bacterial growth
Exponential Properties
Essential Rules:
- \(b^x \cdot b^y = b^{x+y}\)
- \(\frac{b^x}{b^y} = b^{x-y}\)
- \((b^x)^y = b^{xy}\)
- \(b^0 = 1\)
- \(b^{-x} = \frac{1}{b^x}\)
Doubling and Half-Life
Doubling Time: Time for quantity to double - If \(P(t) = P_0 \cdot 2^{t/T_d}\), then \(T_d\) is doubling time
Half-Life: Time for quantity to reduce by half - If \(P(t) = P_0 \cdot (0.5)^{t/T_h}\), then \(T_h\) is half-life
Logarithmic Functions
Definition
Logarithm is the inverse of exponential: \[\log_b(x) = y \iff b^y = x\]
Key Points:
- \(\log_b(b) = 1\)
- \(\log_b(1) = 0\)
- \(\log_b(b^x) = x\)
- \(b^{\log_b(x)} = x\)
Common Logarithms
Common Log (base 10): \(\log(x)\) or \(\log_{10}(x)\) - Used in science, pH scale, Richter scale
Natural Log (base e): \(\ln(x)\) or \(\log_e(x)\) - Used in calculus, continuous growth - \(\ln(e) = 1\)
Logarithm Properties
Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
Power Rule: \(\log_b(x^n) = n\log_b(x)\)
Change of Base: \(\log_b(x) = \frac{\log_a(x)}{\log_a(b)} = \frac{\ln(x)}{\ln(b)}\)
Domain and Range
For \(f(x) = \log_b(x)\):
- Domain: \((0, \infty)\) (only positive numbers)
- Range: All real numbers
- Vertical asymptote at \(x = 0\)
- Passes through \((1, 0)\) and \((b, 1)\)
Solving Logarithmic Equations
Strategy 1: Combine logs \[\log(x) + \log(x-3) = 1\] \[\log(x(x-3)) = 1\] \[x(x-3) = 10^1\]
Strategy 2: Convert to exponential \[\log_2(x) = 5\] \[x = 2^5 = 32\]
Solutions must satisfy \(x > 0\) for all logarithmic arguments
Trigonometric Functions
The Unit Circle
Unit circle definition:
- Circle with radius 1 centered at origin
- Point on circle: \((\cos\theta, \sin\theta)\)
Key Angles and Values:
| Angle | Degrees | Radians | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|---|---|
| 0° | 0° | 0 | 0 | 1 | 0 |
| 30° | 30° | \(\pi/6\) | 1/2 | \(\sqrt{3}/2\) | \(1/\sqrt{3}\) |
| 45° | 45° | \(\pi/4\) | \(\sqrt{2}/2\) | \(\sqrt{2}/2\) | 1 |
| 60° | 60° | \(\pi/3\) | \(\sqrt{3}/2\) | 1/2 | \(\sqrt{3}\) |
| 90° | 90° | \(\pi/2\) | 1 | 0 | undefined |
Degrees and Radians
Conversion:
- \(180° = \pi\) radians
- Degrees to radians: multiply by \(\frac{\pi}{180}\)
- Radians to degrees: multiply by \(\frac{180}{\pi}\)
Why radians?
- Arc length: \(s = r\theta\) (when \(\theta\) in radians)
- Natural for calculus
- Simplifies many formulas
Basic Trigonometric Functions
Sine Function: \(f(x) = \sin(x)\)
- Domain: All real numbers
- Range: \([-1, 1]\)
- Period: \(2\pi\)
- Starts at origin
Cosine Function: \(f(x) = \cos(x)\)
- Domain: All real numbers
- Range: \([-1, 1]\)
- Period: \(2\pi\)
- Starts at maximum (1)
Tangent Function: \(f(x) = \tan(x) = \frac{\sin(x)}{\cos(x)}\)
- Domain: All real except \(x = \frac{\pi}{2} + n\pi\)
- Range: All real numbers
- Period: \(\pi\)
- Vertical asymptotes where \(\cos(x) = 0\)
Key Trigonometric Identities
Reciprocal Identities:
- \(\csc(x) = \frac{1}{\sin(x)}\)
- \(\sec(x) = \frac{1}{\cos(x)}\)
- \(\cot(x) = \frac{1}{\tan(x)}\)
Even/Odd Properties:
- \(\cos(-x) = \cos(x)\) (even)
- \(\sin(-x) = -\sin(x)\) (odd)
Transformations of Trigonometric Functions
General form: \(f(x) = A\sin(B(x - C)) + D\)
| Parameter | Effect | Name |
|---|---|---|
| \(A\) | Vertical stretch by \(\|A\|\) | Amplitude |
| \(B\) | Horizontal compression by \(\|B\|\) | Frequency factor |
| \(C\) | Horizontal shift right by \(C\) | Phase shift |
| \(D\) | Vertical shift up by \(D\) | Midline |
Period: \(\frac{2\pi}{|B|}\) (for sine and cosine)
Example: \(f(x) = 3\sin(2x - \pi) + 1\)
- Amplitude: 3
- Period: \(\frac{2\pi}{2} = \pi\)
- Phase shift: \(\frac{\pi}{2}\) right
- Midline: \(y = 1\)
Business Applications
Seasonal Patterns:
- Sales cycles
- Temperature variations
- Demand fluctuations
Example: Seasonal Revenue
\[R(t) = 50 + 20\sin\left(\frac{2\pi}{12}(t - 3)\right)\] - Average revenue: €50k - Seasonal variation: ±€20k - Period: 12 months - Peak: Month 6 (June)
Function Transformations (Universal)
The Transformation Framework
Standard form: \(g(x) = a \cdot f(b(x - h)) + k\)
Order of transformations:
- Horizontal shift: \(h\) (inside function)
- Horizontal stretch/compress: \(b\) (inside function)
- Vertical stretch/compress: \(a\) (outside function)
- Vertical shift: \(k\) (outside function)
Universal Transformation Rules
| Transformation | Formula | Effect |
|---|---|---|
| Shift up | \(f(x) + k\) | Move up \(k\) units |
| Shift down | \(f(x) - k\) | Move down \(k\) units |
| Shift right | \(f(x - h)\) | Move right \(h\) units |
| Shift left | \(f(x + h)\) | Move left \(h\) units |
| Vertical stretch | \(a \cdot f(x)\), \(\|a\| > 1\) | Stretch by factor \(\|a\|\) |
| Vertical compress | \(a \cdot f(x)\), \(0 < \|a\| < 1\) | Compress by factor \(\|a\|\) |
| Horizontal stretch | \(f(bx)\), \(0 < \|b\| < 1\) | Stretch by factor \(1/\|b\|\) |
| Horizontal compress | \(f(bx)\), \(\|b\| > 1\) | Compress by factor \(1/\|b\|\) |
| Reflect over x-axis | \(-f(x)\) | Flip upside down |
| Reflect over y-axis | \(f(-x)\) | Flip left-right |
- Inside changes (affect \(x\)): Work opposite to intuition
- Outside changes (affect \(y\)): Work as expected
The 4-Step Method for Transformations
- Identify base function
- Find key points (intercepts, max/min, asymptotes)
- Track transformations systematically
- Verify with test point
Rational Functions
Definition and Structure
Rational function: \(f(x) = \frac{P(x)}{Q(x)}\) where \(P\) and \(Q\) are polynomials
Key components:
- Numerator \(P(x)\): Determines zeros (x-intercepts)
- Denominator \(Q(x)\): Determines vertical asymptotes
- Degree comparison: Determines horizontal/oblique asymptotes
Domain
Domain: All real numbers except where \(Q(x) = 0\)
Steps to find domain: 1. Set denominator equal to zero 2. Solve for \(x\) 3. Exclude these values from domain
Asymptotes and Holes
Vertical Asymptotes:
- Occur where denominator = 0 (after cancellation)
- Graph approaches \(\pm\infty\)
- To find: Solve \(Q(x) = 0\) after simplifying
Horizontal Asymptotes (comparing degrees):
| Condition | Horizontal Asymptote |
|---|---|
| deg(\(P\)) < deg(\(Q\)) | \(y = 0\) |
| deg(\(P\)) = deg(\(Q\)) | \(y = \frac{a_n}{b_n}\) (ratio of leading coefficients) |
| deg(\(P\)) > deg(\(Q\)) | No horizontal asymptote |
Oblique (Slant) Asymptotes:
- When deg(\(P\)) = deg(\(Q\)) + 1
- Find by polynomial long division
- Graph approaches this line as \(x \to \pm\infty\)
Holes (Removable Discontinuities):
- Occur when factor cancels from numerator and denominator
- To find: Factor completely, cancel common factors
- Point where hole occurs: \((a, f(a))\) where factor is \((x-a)\)
Systematic Analysis Process
Always follow this order:
- Factor completely (both numerator and denominator)
- Cancel common factors → These become holes
- Find vertical asymptotes (remaining zeros of denominator)
- Find horizontal/oblique asymptotes (degree comparison)
- Find x-intercepts (zeros of simplified numerator)
- Find y-intercept (evaluate at \(x = 0\) if defined)
Example: \(f(x) = \frac{x^2 - 4}{x^2 - x - 2}\)
- Factor: \(f(x) = \frac{(x-2)(x+2)}{(x-2)(x+1)}\)
- Cancel: \(f(x) = \frac{x+2}{x+1}\), \(x \neq 2\) (hole at \(x = 2\))
- Vertical asymptote: \(x = -1\)
- Horizontal asymptote: \(y = 1\) (equal degrees, ratio 1/1)
- x-intercept: \(x = -2\)
- y-intercept: \(f(0) = 2\)
- Hole at: \((2, \frac{4}{3})\)
Business Applications
Average Cost Functions: \[AC(x) = \frac{C(x)}{x} = \frac{FC + VC \cdot x}{x} = \frac{FC}{x} + VC\]
- Vertical asymptote at \(x = 0\)
- Horizontal asymptote at \(y = VC\) (variable cost per unit)
- As production increases, average cost approaches variable cost
Business Applications Summary
Exponential Growth/Decay Models
Investment Growth:
\[A(t) = P(1 + r)^t \quad \text{or} \quad A(t) = Pe^{rt}\]
Population Models:
- Growth: \(P(t) = P_0e^{kt}\) where \(k > 0\)
- Decay: \(P(t) = P_0e^{-kt}\) where \(k > 0\)
Depreciation:
- Declining balance: \(V(t) = V_0(1 - r)^t\)
Periodic Business Patterns
Revenue Seasonality:
\[R(t) = R_0 + A\sin\left(\frac{2\pi}{12}(t - \phi)\right)\]
Where:
- \(R_0\): Average revenue
- \(A\): Seasonal amplitude
- \(\phi\): Phase shift (timing of peak)
Optimization with Functions
Revenue Maximization:
- For quadratic revenue: vertex formula
- For rational functions: calculus (Section 05)
Cost Minimization:
- Average cost minimum
- Production efficiency
Problem-Solving Strategies
General Approach
- Identify function type (polynomial, exponential, rational, etc.)
- Check domain restrictions (denominators, square roots, logs)
- Apply appropriate techniques
- Verify results make practical sense
- Consider constraints (production limits, budget)
Common Mistakes to Avoid
- Logarithms: Forgetting domain restrictions (\(x > 0\))
- Rational functions: Not factoring completely before finding asymptotes
- Transformations: Confusing inside vs outside changes
- Trigonometry: Mixing degrees and radians
- Exponential: Confusing growth rate with growth factor
Quick Reference: When to Use Each Function
| Business Situation | Function Type | Example |
|---|---|---|
| Constant growth/decay rate | Exponential | \(P(t) = P_0 \cdot b^t\) |
| Seasonal patterns | Trigonometric | \(S(t) = A\sin(Bt) + C\) |
| Average cost | Rational | \(AC(x) = \frac{C(x)}{x}\) |
| Revenue optimization | Quadratic | \(R(x) = px = p(a - bp)\) |
| Multi-step processes | Composition | \((f \circ g)(x)\) |
| Reverse relationships | Inverse | \(f^{-1}(x)\) |