Course Cheatsheet
Section 04: Advanced Functions
Polynomial Functions
General Form
\(P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\) where \(a_n \neq 0\)
| Degree | Name | Example |
|---|---|---|
| 0 | Constant | \(f(x) = 5\) |
| 1 | Linear | \(f(x) = 2x + 3\) |
| 2 | Quadratic | \(f(x) = x^2 - 4\) |
| 3 | Cubic | \(f(x) = 2x^3 - x\) |
| 4 | Quartic | \(f(x) = x^4 - 5x^2 + 4\) |
Domain: always all real numbers.
Key Properties
- Maximum zeros: equal to degree \(n\)
- Maximum turning points: \(n - 1\)
- Fundamental Theorem of Algebra: a degree-\(n\) polynomial has exactly \(n\) zeros (counting multiplicities and complex zeros)
End behavior depends on only two things: degree (even/odd) and sign of leading coefficient.
| Degree | Leading Coeff. | Left End | Right End |
|---|---|---|---|
| Even | Positive | \(\uparrow\) | \(\uparrow\) |
| Even | Negative | \(\downarrow\) | \(\downarrow\) |
| Odd | Positive | \(\downarrow\) | \(\uparrow\) |
| Odd | Negative | \(\uparrow\) | \(\downarrow\) |
Think: even degree = both ends go the same direction; odd degree = ends go opposite directions. The sign of \(a_n\) determines which way.
Zeros and Multiplicity
If \((x - c)^m\) is a factor of \(P(x)\), then \(c\) is a zero with multiplicity \(m\).
- Odd multiplicity: graph crosses x-axis at \(c\)
- Even multiplicity: graph touches x-axis and bounces off at \(c\)
- Higher multiplicity means flatter near the zero
Example: \(P(x) = -2(x+3)(x-1)^2(x-4)\)
- Zeros: \(x = -3\) (mult. 1, crosses), \(x = 1\) (mult. 2, touches), \(x = 4\) (mult. 1, crosses)
- Degree: \(1 + 2 + 1 = 4\) (even), leading coefficient \(-2\) (negative)
- End behavior: both ends down
Sketching from Factored Form
- Identify zeros and their multiplicities
- Determine end behavior (degree + leading coefficient)
- Find y-intercept: evaluate \(P(0)\)
- Plot key points and connect smoothly
- Between consecutive zeros, the polynomial does not cross the x-axis – use test points to determine if the graph is above or below
Rational Root Theorem
For a polynomial with integer coefficients, any rational zero \(\frac{p}{q}\) must satisfy:
- \(p\) divides the constant term \(a_0\)
- \(q\) divides the leading coefficient \(a_n\)
Test candidates systematically, starting with simple values (\(\pm 1, \pm 2, \ldots\)).
If \(P(x)\) is continuous on \([a, b]\) and \(P(a)\) and \(P(b)\) have opposite signs, then there is at least one zero \(c\) in \((a, b)\).
Business use: if profit is negative at low production and positive at high production, IVT guarantees a break-even point exists somewhere in between.
Note: IVT proves existence but does not give the exact location or count of zeros.
Power Functions and Roots
Power Functions: \(f(x) = kx^p\)
| Exponent \(p\) | Example | Domain | Key Behavior |
|---|---|---|---|
| Positive integer | \(x^2, x^3\) | All real | Polynomial building blocks |
| Negative integer | \(x^{-1} = \frac{1}{x}\) | \(x \neq 0\) | Hyperbolic, asymptotic |
| Fraction (even denom.) | \(x^{1/2} = \sqrt{x}\) | \(x \geq 0\) | Restricted domain |
| Fraction (odd denom.) | \(x^{1/3} = \sqrt[3]{x}\) | All real | Full domain |
Root Function Domains
- Even roots (\(\sqrt{x}, \sqrt[4]{x}\)): domain \(x \geq 0\), range \(y \geq 0\)
- Odd roots (\(\sqrt[3]{x}, \sqrt[5]{x}\)): domain all real, range all real
Rational Exponent Rules
- \(x^{m/n} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m\)
- \((x^a)^b = x^{ab}\)
- \(x^a \cdot x^b = x^{a+b}\)
- \(\frac{x^a}{x^b} = x^{a-b}\)
- \(x^{-n} = \frac{1}{x^n}\)
Growth Rate Comparison
For large \(x > 1\), higher exponent = faster growth:
\[x^{1/3} < x^{1/2} < x < x^{3/2} < x^2 < x^3\]
Business insight: exponent \(< 1\) (e.g., \(x^{0.7}\)) models economies of scale – cost grows slower than production. Exponent \(> 1\) models diseconomies at scale.
Exponential Functions
Definition
\(f(x) = a \cdot b^x\) where \(b > 0, b \neq 1\)
- \(a\): initial value (\(f(0) = a\))
- \(b > 1\): exponential growth
- \(0 < b < 1\): exponential decay
- Domain: all real numbers; Range: \((0, \infty)\) when \(a > 0\)
Growth and Decay Models
| Model | Formula | Use Case |
|---|---|---|
| Discrete growth | \(A(t) = A_0 \cdot (1+r)^t\) | Annual compounding |
| Discrete decay | \(A(t) = A_0 \cdot (1-r)^t\) | Depreciation |
| Continuous growth | \(A(t) = A_0 \cdot e^{kt}\), \(k > 0\) | Continuous processes |
| Continuous decay | \(A(t) = A_0 \cdot e^{-kt}\), \(k > 0\) | Radioactive decay |
| Doubling time | \(A(t) = A_0 \cdot 2^{t/T_d}\) | Population doubling |
| Half-life | \(A(t) = A_0 \cdot (0.5)^{t/T_h}\) | Medication decay |
Relationship: \(b = e^k\) or equivalently \(k = \ln(b)\).
The Natural Exponential \(e\)
\(e \approx 2.71828\), defined by \(e = \lim_{n \to \infty}(1 + 1/n)^n\)
Special property: the rate of change of \(e^x\) equals \(e^x\) itself (derivative equals the function).
Discrete: growth happens at fixed intervals (annual, monthly). Use \(A = P(1 + r/n)^{nt}\).
Continuous: growth happens constantly. Use \(A = Pe^{rt}\).
The two are related: any discrete model \(A_0 \cdot b^t\) can be written as \(A_0 \cdot e^{t \ln b}\), and vice versa. In practice, continuous compounding yields only slightly more than daily compounding.
Compound Interest
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
- \(P\): principal, \(r\): annual rate, \(n\): compounding frequency, \(t\): years
- Continuous compounding: \(A = Pe^{rt}\) (limit as \(n \to \infty\))
- Effective Annual Rate: \(\text{EAR} = (1 + r/n)^n - 1\)
Rule of 70
Doubling time \(\approx \frac{70}{r}\) where \(r\) is the percentage growth rate.
- 7% growth: doubles in \(\approx 10\) periods
- 2% inflation: prices double in \(\approx 35\) years
Exponential Properties
- \(b^x \cdot b^y = b^{x+y}\)
- \((b^x)^y = b^{xy}\) (not \(b^{x+y}\)!)
- \(b^0 = 1\) for any \(b \neq 0\)
- \(b^{-x} = 1/b^x\)
Exponential functions always overtake polynomial functions eventually, no matter the degree. For example, \(2^x\) eventually surpasses \(x^{100}\). This is why exponential growth is so powerful (and dangerous) in business contexts – early growth looks manageable, but it accelerates dramatically.
Logistic Growth
When exponential growth has limits:
\[P(t) = \frac{L}{1 + Ae^{-kt}}\]
- \(L\): carrying capacity (maximum)
- Three phases: slow start, rapid growth, saturation
- Inflection point at \(P = L/2\) (maximum growth rate)
- Models: product adoption, market penetration, population ecology
Logarithmic Functions
Definition
\(\log_b(x) = y \iff b^y = x\) (the logarithm is the inverse of the exponential)
Special values: \(\log_b(1) = 0\), \(\log_b(b) = 1\), \(\log_b(b^n) = n\), \(b^{\log_b(x)} = x\)
Common types: \(\log(x) = \log_{10}(x)\), \(\ln(x) = \log_e(x)\)
Logarithm Properties
| Rule | Formula |
|---|---|
| Product | \(\log_b(xy) = \log_b(x) + \log_b(y)\) |
| Quotient | \(\log_b(x/y) = \log_b(x) - \log_b(y)\) |
| Power | \(\log_b(x^n) = n \cdot \log_b(x)\) |
| Change of Base | \(\log_b(x) = \frac{\ln(x)}{\ln(b)}\) |
Domain: \((0, \infty)\) only. Range: all real numbers. Vertical asymptote at \(x = 0\).
The most frequent error: confusing the product rule with addition.
- Wrong: \(\log(a) + \log(b) = \log(a + b)\)
- Correct: \(\log(a) + \log(b) = \log(a \cdot b)\)
Also remember: \(\log(a - b) \neq \log(a) - \log(b)\). The quotient rule applies to \(\log(a/b)\), not subtraction inside the argument.
Always check that solutions keep all log arguments positive!
Solving Logarithmic Equations
- Combine logs: use product/quotient rules to get a single log
- Convert to exponential: \(\log_b(\text{expr}) = c \implies \text{expr} = b^c\)
- Check domain: all arguments of log must be positive
Example: \(\log_2(x) + \log_2(x-2) = 3\)
\[\log_2(x(x-2)) = 3 \implies x^2 - 2x = 8 \implies x = 4 \text{ (reject } x = -2\text{)}\]
Trigonometric Functions
Degrees and Radians
- \(180° = \pi\) radians
- Degrees to radians: multiply by \(\frac{\pi}{180}\)
- Radians to degrees: multiply by \(\frac{180}{\pi}\)
- Arc length: \(s = r\theta\) (with \(\theta\) in radians)
Always check which unit your problem uses. In mathematical formulas and calculus, radians are standard. Your calculator has a mode setting – make sure it matches!
Key conversions: \(90° = \pi/2\), \(180° = \pi\), \(360° = 2\pi\).
If a formula says \(\sin(2x)\) with no degree symbol, \(x\) is in radians.
Unit Circle Key Values
| Angle | Radians | \(\sin\) | \(\cos\) | \(\tan\) |
|---|---|---|---|---|
| \(0°\) | \(0\) | \(0\) | \(1\) | \(0\) |
| \(30°\) | \(\pi/6\) | \(1/2\) | \(\sqrt{3}/2\) | \(1/\sqrt{3}\) |
| \(45°\) | \(\pi/4\) | \(\sqrt{2}/2\) | \(\sqrt{2}/2\) | \(1\) |
| \(60°\) | \(\pi/3\) | \(\sqrt{3}/2\) | \(1/2\) | \(\sqrt{3}\) |
| \(90°\) | \(\pi/2\) | \(1\) | \(0\) | undefined |
Sine, Cosine, Tangent
| Property | \(\sin(x)\) | \(\cos(x)\) | \(\tan(x)\) |
|---|---|---|---|
| Domain | All real | All real | \(x \neq \pi/2 + n\pi\) |
| Range | \([-1, 1]\) | \([-1, 1]\) | All real |
| Period | \(2\pi\) | \(2\pi\) | \(\pi\) |
| Starts at | \(\sin(0) = 0\) | \(\cos(0) = 1\) | \(\tan(0) = 0\) |
Fundamental identity: \(\sin^2(x) + \cos^2(x) = 1\) (always, for all \(x\))
Relationship: \(\cos(x) = \sin(x + \pi/2)\) and \(\tan(x) = \sin(x)/\cos(x)\)
Transformations: \(y = A\sin(B(x - C)) + D\)
| Parameter | Name | Effect |
|---|---|---|
| \(A\) | Amplitude | Vertical stretch by \(|A|\); if \(A < 0\), reflect |
| \(B\) | Frequency | Period \(= 2\pi / |B|\) |
| \(C\) | Phase shift | Horizontal shift right by \(C\) |
| \(D\) | Vertical shift | Midline at \(y = D\) |
Example: \(y = 3\sin(2x) - 1\) has amplitude 3, period \(\pi\), midline \(y = -1\), range \([-4, 2]\).
Business Applications of Trig
Seasonal model: \(R(t) = R_0 + A\sin\left(\frac{2\pi}{T}(t - \phi)\right)\)
- \(R_0\): average value, \(A\): seasonal amplitude, \(T\): period (e.g., 12 months), \(\phi\): phase shift
- Use cosine when starting at maximum (e.g., tides at high point)
- Use negative cosine when starting at minimum
Rational Functions
Definition
\(f(x) = \frac{P(x)}{Q(x)}\) where \(P\) and \(Q\) are polynomials, \(Q(x) \neq 0\)
Domain: all real numbers except where \(Q(x) = 0\)
Always follow this order:
- Factor numerator and denominator completely
- Cancel common factors – these create holes (not asymptotes)
- Vertical asymptotes: remaining zeros of the denominator
- Horizontal asymptote: compare degrees of \(P\) and \(Q\):
- deg(\(P\)) \(<\) deg(\(Q\)): \(y = 0\)
- deg(\(P\)) \(=\) deg(\(Q\)): \(y = \frac{\text{leading coeff of } P}{\text{leading coeff of } Q}\)
- deg(\(P\)) \(>\) deg(\(Q\)): no horizontal asymptote (oblique if degree differs by 1)
- x-intercepts: zeros of simplified numerator
- y-intercept: evaluate \(f(0)\)
Holes vs Vertical Asymptotes
- Hole: factor cancels from both numerator and denominator. The function is undefined at that point, but the graph has a “missing point” (no blowup).
- Vertical asymptote: factor remains only in denominator after cancellation. Graph goes to \(\pm\infty\).
Example: \(f(x) = \frac{(x-2)(x+2)}{(x-2)(x+1)}\)
- Cancel \((x-2)\): hole at \(x = 2\), with \(y\)-value \(\frac{4}{3}\)
- Vertical asymptote at \(x = -1\)
- Horizontal asymptote at \(y = 1\)
Average Cost Functions
\[AC(x) = \frac{C(x)}{x} = \frac{FC + VC \cdot x}{x} = \frac{FC}{x} + VC\]
- Vertical asymptote at \(x = 0\) (cannot produce zero units)
- Horizontal asymptote at \(y = VC\) (minimum average cost approaches variable cost)
- Always decreasing for \(x > 0\) when cost is linear
Semi-Log and Log-Log Scales
Use these special scales to identify hidden patterns in data:
- Semi-log plot (y-axis logarithmic): exponential functions \(y = ae^{bx}\) appear as straight lines. The slope gives the growth rate.
- Log-log plot (both axes logarithmic): power functions \(y = ax^b\) appear as straight lines. The slope gives the exponent \(b\).
If your data looks curved on a normal plot, try a semi-log or log-log scale. A straight line on either reveals the underlying function type.
Inverse Trigonometric Functions
When you need to find the angle from a value:
- \(\arcsin(x)\) or \(\sin^{-1}(x)\): output range \([-\pi/2, \pi/2]\)
- \(\arccos(x)\) or \(\cos^{-1}(x)\): output range \([0, \pi]\)
- \(\arctan(x)\) or \(\tan^{-1}(x)\): output range \((-\pi/2, \pi/2)\)
Example: If \(\sin(\theta) = 0.5\), then \(\theta = \arcsin(0.5) = \pi/6\) (or \(30°\)).
Note: restricted output ranges ensure these are proper functions (one output per input).
Economies of Scale with Power Functions
Many production processes follow: \(C(x) = FC + kx^p\)
- If \(p < 1\) (e.g., \(x^{0.7}\)): cost grows slower than production (economies of scale)
- If \(p = 1\): linear cost (constant marginal cost)
- If \(p > 1\) (e.g., \(x^{1.5}\)): cost grows faster than production (diseconomies)
Average cost: \(AC(x) = \frac{C(x)}{x}\) often shows a U-shape:
- High at low production (fixed costs dominate)
- Minimum at optimal scale
- Rises again if diseconomies kick in
Function Transformations (Universal)
Standard Form: \(g(x) = a \cdot f(b(x - h)) + k\)
| Transformation | Formula | Effect |
|---|---|---|
| Shift up/down | \(f(x) \pm k\) | Vertical translation |
| Shift right/left | \(f(x \mp h)\) | Horizontal translation (counterintuitive!) |
| Vertical stretch/compress | \(a \cdot f(x)\) | \(|a| > 1\) stretches, \(|a| < 1\) compresses |
| Horizontal stretch/compress | \(f(bx)\) | \(|b| > 1\) compresses, \(|b| < 1\) stretches |
| Reflect over x-axis | \(-f(x)\) | Flip vertically |
| Reflect over y-axis | \(f(-x)\) | Flip horizontally |
Inside changes (to \(x\)) act opposite to intuition. Outside changes (to \(y\)) act as expected.
Key Vocabulary
| Term | Definition |
|---|---|
| Polynomial degree | Highest power of \(x\) in the polynomial |
| Leading coefficient | Coefficient of the highest-degree term |
| Multiplicity | Number of times a zero is repeated as a factor |
| Asymptote | Line that a graph approaches but may never reach |
| Hole | Removable discontinuity (canceled factor) |
| Amplitude | Half the distance between max and min of a wave |
| Period | Length of one complete cycle |
| Phase shift | Horizontal displacement of a periodic function |
| Carrying capacity | Maximum sustainable value in logistic growth |
| EAR | Effective Annual Rate for comparing investments |
Quick Reference
Growth Rate Comparison (for large \(x\))
\[\ln(x) \ll x^{0.5} \ll x \ll x^2 \ll x^3 \ll 2^x \ll e^x \ll 3^x\]
Logarithmic growth is slowest, polynomial is intermediate, exponential is fastest.
Function Type Identification
| Pattern in Data | Function Type | Model |
|---|---|---|
| Constant rate of change | Linear | \(y = mx + b\) |
| Accelerating/decelerating change | Polynomial | \(y = ax^n + \ldots\) |
| Constant percentage change | Exponential | \(y = a \cdot b^x\) |
| Repeating cycles | Trigonometric | \(y = A\sin(Bx) + D\) |
| Approaches a limit | Rational/Logistic | \(y = \frac{P(x)}{Q(x)}\) or \(\frac{L}{1+Ae^{-kt}}\) |
Problem-Solving Strategies
- Identify function type from context or equation structure
- Check domain restrictions (denominators, roots, logs)
- Find key features: intercepts, asymptotes, extrema, period
- Apply appropriate technique: factoring, vertex formula, log properties, etc.
- Verify results: substitute back, check domain, ensure business sense
- Polynomials: degree is the highest power, not the number of terms
- Even multiplicity: graph touches x-axis (bounces), does not cross
- Exponential rules: \((e^2)^3 = e^6\), not \(e^5\) – exponents multiply
- Logarithms: \(\log(a + b) \neq \log(a) + \log(b)\) – the product rule is \(\log(ab) = \log(a) + \log(b)\)
- Trig period: period of \(\sin(Bx)\) is \(2\pi/B\), not \(2\pi B\) – higher \(B\) means shorter period
- Rational functions: always factor and cancel before identifying asymptotes
- Growth rates: exponential always beats polynomial for large \(x\), regardless of the polynomial degree
- Degrees vs radians: check your calculator mode; formulas assume radians unless stated otherwise