Grade 11: Algebra 2 & Trigonometry

A polynomial function is a function of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer and the coefficients are real numbers. **Key Properties:** • The degree of a polynomial is the highest power of x with a non-zero coefficient • The leading coefficient is the coefficient of the term with the highest degree • End behavior describes what happens to f(x) as x approaches positive or negative infinity • A polynomial of degree n has at most n real zeros and at most n-1 turning points **End Behavior Rules:** • If degree is even and leading coefficient is positive: both ends go up • If degree is even and leading coefficient is negative: both ends go down • If degree is odd and leading coefficient is positive: left down, right up • If degree is odd and leading coefficient is negative: left up, right down **Finding Zeros:** • Factor the polynomial completely • Set each factor equal to zero • The multiplicity of a zero determines the behavior at that x-intercept: - Odd multiplicity: graph crosses the x-axis - Even multiplicity: graph touches the x-axis and turns around **Synthetic Division:** Used to divide a polynomial by (x - c). The remainder equals f(c) by the Remainder Theorem.

A rational function is a function of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0. **Domain:** All real numbers except values that make the denominator zero. **Vertical Asymptotes:** Occur at values of x that make the denominator zero but NOT the numerator (after simplification). The graph approaches ±∞ near vertical asymptotes. **Holes (Removable Discontinuities):** Occur when a factor cancels from both numerator and denominator. The graph has a "hole" at that x-value. **Horizontal Asymptotes:** Determined by comparing degrees of numerator (n) and denominator (m): • If n < m: horizontal asymptote at y = 0 • If n = m: horizontal asymptote at y = (leading coefficient of P)/(leading coefficient of Q) • If n > m: no horizontal asymptote (may have slant asymptote) **Slant (Oblique) Asymptotes:** Occur when degree of numerator is exactly 1 more than degree of denominator. Find by polynomial long division. **Graphing Strategy:** 1. Factor numerator and denominator 2. Find domain (exclude zeros of denominator) 3. Identify holes (common factors) 4. Find vertical asymptotes (remaining zeros of denominator) 5. Find horizontal or slant asymptote 6. Find intercepts 7. Plot points and sketch

A radical function contains a variable under a radical symbol. The most common are square root and cube root functions. **Radical Notation:** ⁿ√x where n is the index and x is the radicand • √x means ²√x (square root) • ³√x is cube root • ⁿ√x = x^(1/n) **Properties of Radicals:** • √(ab) = √a · √b • √(a/b) = √a / √b • (√a)² = a (for a ≥ 0) • ⁿ√(aⁿ) = |a| if n is even, = a if n is odd **Simplifying Radicals:** 1. Factor the radicand into perfect powers 2. Take roots of perfect powers 3. Simplify remaining factors **Rationalizing Denominators:** • For √a in denominator: multiply by √a/√a • For (a + √b) in denominator: multiply by conjugate (a - √b)/(a - √b) **Domain of Square Root Functions:** For f(x) = √g(x), domain is all x where g(x) ≥ 0 **Solving Radical Equations:** 1. Isolate the radical 2. Raise both sides to the power equal to the index 3. Solve the resulting equation 4. CHECK all solutions (extraneous solutions may occur!) **Rational Exponents:** • x^(m/n) = ⁿ√(x^m) = (ⁿ√x)^m • x^(-m/n) = 1/x^(m/n)

**Exponential Functions:** f(x) = a · b^x where a ≠ 0, b > 0, b ≠ 1 • If b > 1: exponential growth • If 0 < b < 1: exponential decay • Domain: all real numbers • Range: y > 0 (if a > 0) • Horizontal asymptote: y = 0 **Special Exponential Function:** f(x) = e^x where e ≈ 2.71828 (Euler's number) **Logarithmic Functions:** y = log_b(x) means b^y = x • log_b(x) is the inverse of b^x • Domain: x > 0 • Range: all real numbers • Vertical asymptote: x = 0 **Special Logarithms:** • Common log: log(x) = log₁₀(x) • Natural log: ln(x) = log_e(x) **Properties of Logarithms:** • log_b(xy) = log_b(x) + log_b(y) (Product Rule) • log_b(x/y) = log_b(x) - log_b(y) (Quotient Rule) • log_b(x^n) = n · log_b(x) (Power Rule) • log_b(b) = 1 • log_b(1) = 0 • b^(log_b(x)) = x • log_b(b^x) = x **Change of Base Formula:** log_b(x) = log(x)/log(b) = ln(x)/ln(b) **Solving Exponential Equations:** 1. If bases are the same: set exponents equal 2. If bases differ: take log of both sides **Applications:** • Compound Interest: A = P(1 + r/n)^(nt) • Continuous Growth: A = Pe^(rt) • Half-life: A = A₀(1/2)^(t/h)

**The Unit Circle:** A circle with radius 1 centered at the origin. Used to define trigonometric functions for all angles. **Radian Measure:** One radian is the angle formed when the arc length equals the radius. • Full circle: 2π radians = 360° • Half circle: π radians = 180° • Conversion: radians = degrees × (π/180) • Conversion: degrees = radians × (180/π) **Key Angle Conversions:** • 30° = π/6 rad • 45° = π/4 rad • 60° = π/3 rad • 90° = π/2 rad • 180° = π rad • 270° = 3π/2 rad • 360° = 2π rad **Unit Circle Coordinates:** For angle θ in standard position, the point on the unit circle is (cos θ, sin θ) **Special Angles (Exact Values):** • 0°: (1, 0) → cos = 1, sin = 0 • 30° (π/6): (√3/2, 1/2) • 45° (π/4): (√2/2, √2/2) • 60° (π/3): (1/2, √3/2) • 90° (π/2): (0, 1) → cos = 0, sin = 1 **Reference Angles:** The acute angle formed with the x-axis. Used to find trig values in all quadrants. **Quadrant Signs:** • QI: All positive • QII: Sin positive, Cos/Tan negative • QIII: Tan positive, Sin/Cos negative • QIV: Cos positive, Sin/Tan negative (Remember: "All Students Take Calculus") **Coterminal Angles:** Angles that share the same terminal side. Add or subtract 360° (or 2π radians).

**General Form of Sinusoidal Functions:** f(x) = A sin[B(x - C)] + D or f(x) = A cos[B(x - C)] + D Where: • |A| = amplitude (distance from midline to max/min) • Period = 2π/|B| • C = phase shift (horizontal shift) • D = vertical shift (midline) **Parent Functions:** • y = sin(x): starts at (0,0), period 2π, amplitude 1 • y = cos(x): starts at (0,1), period 2π, amplitude 1 • y = tan(x): period π, vertical asymptotes at x = π/2 + nπ **Key Features of y = sin(x):** • Domain: all real numbers • Range: [-1, 1] • Period: 2π • Zeros: x = nπ (where n is an integer) • Maximum: 1 at x = π/2 + 2nπ • Minimum: -1 at x = 3π/2 + 2nπ **Key Features of y = cos(x):** • Domain: all real numbers • Range: [-1, 1] • Period: 2π • Zeros: x = π/2 + nπ • Maximum: 1 at x = 2nπ • Minimum: -1 at x = π + 2nπ **Key Features of y = tan(x):** • Domain: all real numbers except x = π/2 + nπ • Range: all real numbers • Period: π • Zeros: x = nπ • Vertical asymptotes: x = π/2 + nπ **Transformations:** • A < 0: reflection over x-axis • B > 1: horizontal compression (shorter period) • 0 < B < 1: horizontal stretch (longer period) • C > 0: shift right • C < 0: shift left • D > 0: shift up • D < 0: shift down

**Fundamental Trigonometric Identities:** **Reciprocal Identities:** • csc(θ) = 1/sin(θ) • sec(θ) = 1/cos(θ) • cot(θ) = 1/tan(θ) **Quotient Identities:** • tan(θ) = sin(θ)/cos(θ) • cot(θ) = cos(θ)/sin(θ) **Pythagorean Identities:** • sin²(θ) + cos²(θ) = 1 • 1 + tan²(θ) = sec²(θ) • 1 + cot²(θ) = csc²(θ) **Cofunction Identities:** • sin(90° - θ) = cos(θ) • cos(90° - θ) = sin(θ) • tan(90° - θ) = cot(θ) **Even/Odd Identities:** • sin(-θ) = -sin(θ) (odd) • cos(-θ) = cos(θ) (even) • tan(-θ) = -tan(θ) (odd) **Strategies for Verifying Identities:** 1. Work with one side at a time (usually the more complex side) 2. Convert everything to sine and cosine 3. Use algebraic techniques (factor, combine fractions, multiply by conjugate) 4. Look for opportunities to use Pythagorean identities 5. Never move terms from one side to the other (not an equation to solve!) **Common Techniques:** • Factor when possible • Multiply by conjugate to eliminate radicals • Find common denominators • Substitute using fundamental identities • Split fractions