📚SAT (Digital) Unit 3 – SAT Math – Advanced Math

Key Concepts

• Polynomial functions involve variables with whole number exponents and can be added, subtracted, or multiplied
• Quadratic equations are polynomials of degree 2 and have the general form $ax^2 + bx + c = 0$
• Exponential functions have variables in the exponent and grow or decay at a constant rate (doubling time, half-life)
• Logarithms are the inverse of exponential functions and can be used to solve equations with variables in the exponent
• Rational functions are ratios of polynomial functions and have asymptotes where the denominator equals zero
  • Vertical asymptotes occur when the denominator equals zero for a specific x-value
  • Horizontal asymptotes describe the function's long-term behavior as x approaches positive or negative infinity
• Radical functions involve square roots, cube roots, or higher-order roots of variables
• Complex numbers consist of a real part and an imaginary part in the form $a + bi$ where $i = \sqrt{-1}$

Fundamental Equations

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves quadratic equations of the form $ax^2 + bx + c = 0$
• Exponential growth/decay: $A = A_0e^{kt}$ where $A_0$ is the initial amount, $k$ is the growth/decay rate, and $t$ is time
• Logarithmic properties: $\log_b(xy) = \log_b(x) + \log_b(y)$, $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$, $\log_b(x^n) = n\log_b(x)$
• Rational function asymptotes:
  • Vertical: Occur when the denominator equals zero
  • Horizontal: Determined by the ratio of the leading coefficients of the numerator and denominator polynomials
• Fundamental Theorem of Algebra: Every polynomial of degree $n$ has exactly $n$ complex roots (including repeated roots)
• Binomial Theorem: Expands $(x + y)^n$ into a sum of terms involving powers of $x$ and $y$ with binomial coefficients
• De Moivre's Theorem: $(cos\theta + i sin\theta)^n = cos(n\theta) + i sin(n\theta)$ relates complex numbers and trigonometry

Problem-Solving Strategies

• Factoring quadratic expressions to find roots and solve equations
• Completing the square to rewrite quadratic equations in vertex form and find the axis of symmetry
• Graphing functions to visualize their behavior, intercepts, and asymptotes
  • Identify the domain and range of the function
  • Determine the function's end behavior (as x approaches positive or negative infinity)
• Substitution to simplify expressions or solve systems of equations
• Logarithmic properties to simplify, expand, or condense logarithmic expressions
• Exponent rules to simplify expressions with variables in the exponent
  • Power rule: $x^a \cdot x^b = x^{a+b}$
  • Product rule: $(x^a)^b = x^{ab}$
• Synthetic division to efficiently divide polynomials by linear factors
• Rational root theorem to find potential rational roots of polynomial equations

Common Question Types

• Solving quadratic equations using factoring, completing the square, or the quadratic formula
• Graphing polynomial, exponential, logarithmic, and rational functions
  • Identifying key features such as intercepts, asymptotes, and end behavior
• Simplifying expressions using exponent rules, logarithmic properties, or complex number operations
• Analyzing the behavior of functions based on their equations or graphs (growth rates, asymptotes, periodicity)
• Solving systems of equations involving polynomial, exponential, or logarithmic functions
• Applying exponential and logarithmic functions to real-world problems (compound interest, population growth, radioactive decay)
• Manipulating rational expressions by factoring, simplifying, or performing arithmetic operations
• Solving equations with radicals or complex numbers

Advanced Techniques

• Partial fraction decomposition to break down complex rational expressions into simpler terms
  • Useful for integrating rational functions or solving certain types of differential equations
• L'Hôpital's rule to evaluate limits of indeterminate forms (0/0, $\infty/\infty$, 0 · $\infty$, $\infty - \infty$, $0^0$, $1^\infty$, $\infty^0$)
  • Differentiate the numerator and denominator separately and evaluate the new limit
• Euler's formula: $e^{ix} = cos(x) + i sin(x)$ connects exponential functions with complex numbers and trigonometry
• Polar form of complex numbers: $z = r(cos\theta + i sin\theta) = re^{i\theta}$ where $r$ is the modulus and $\theta$ is the argument
• Conic sections (circles, ellipses, hyperbolas, parabolas) in standard and general forms
  • Identify key features such as foci, directrices, and eccentricity
• Matrix operations to solve systems of linear equations or analyze transformations in the plane or space
• Sequences and series (arithmetic, geometric, Taylor series) to model patterns or approximate functions
  • Use formulas for the nth term, sum, or convergence properties

Practice Problems

• Solve the equation $2x^2 - 7x - 15 = 0$ using the quadratic formula
• Graph the rational function $f(x) = \frac{x^2 - 4}{x - 2}$ and identify its vertical and horizontal asymptotes
• Simplify the expression $\log_2(16) - \log_2(4) + \log_2(1/8)$
• Find the complex roots of the polynomial $x^4 + 1$ using De Moivre's Theorem
• Determine the end behavior and any holes in the graph of $g(x) = \frac{x^3 - 2x^2 + x - 2}{x^2 - 4}$
• Solve the system of equations $y = 2^x$ and $y = 3x - 1$ graphically or algebraically
• Use partial fraction decomposition to integrate $\int \frac{3x + 2}{(x - 1)(x + 2)} dx$
• Find the focus, directrix, and eccentricity of the ellipse $\frac{(x - 3)^2}{16} + \frac{(y + 1)^2}{9} = 1$

Tips and Tricks

• Memorize common quadratic factoring patterns (difference of squares, perfect square trinomials) to save time
• Use the rational root theorem to quickly identify potential rational roots of polynomial equations
• Recognize the graphs of basic functions (linear, quadratic, exponential, logarithmic) and their transformations
  • Shifts, reflections, stretches, and compressions can be identified from the function's equation
• Logarithms can be used to "undo" exponents and solve equations like $2^x = 10$ by applying $\log_2$ to both sides
• When graphing rational functions, first identify the asymptotes and then plot additional points to sketch the curve
• Euler's formula can simplify complex number calculations and prove trigonometric identities
• The discriminant $b^2 - 4ac$ of a quadratic equation determines the nature of its roots (real distinct, real repeated, or complex)
• In exponential and logarithmic equations, isolate the exponential or logarithmic term before solving for the variable

Potential Pitfalls

• Forgetting to use the correct order of operations (PEMDAS) when simplifying expressions
• Misidentifying the degree of a polynomial function, especially when terms are missing or out of order
• Incorrectly applying exponent rules, such as $(x^a)^b = x^{a+b}$ instead of $x^{ab}$
• Failing to consider the domain restrictions when working with rational, logarithmic, or radical functions
  • Rational functions: Denominator cannot equal zero
  • Logarithmic functions: Argument must be positive
  • Radical functions: Radicand must be non-negative for even roots
• Misinterpreting the end behavior of rational functions based on the degree of the numerator and denominator
• Confusing the properties of exponential and logarithmic functions, such as $e^{\ln(x)} = x$ and $\ln(e^x) = x$
• Oversimplifying or canceling terms in rational expressions without factoring first
• Neglecting to consider complex solutions when solving polynomial equations of degree 2 or higher