Advanced Math - Comprehensive Notes (12 Topics)

Linear & Nonlinear Equations in One Variable

• Solving linear equations and equations with:

  • Rational expressions (variables in denominators)

  • Radical expressions (variables under square roots)

  • Exponents (simple exponentials)

• Solving quadratic equations:

  • By factoring

  • Using the quadratic formula

  • Completing the square

• Recognizing the number of solutions based on discriminants

  • Discriminant: $\Delta = b^2 - 4ac$

  • If \Delta > 0 ⇒ two real solutions

  • If $\Delta = 0$ ⇒ one real solution (a repeated root)

  • If \Delta < 0 ⇒ no real solutions (two complex roots)

Linear & Nonlinear Inequalities

• Solving linear and quadratic inequalities

• Interpreting inequality graphs

• Solving inequalities involving:

  • Absolute value

  • Rational expressions

• Understanding inequality solution sets

  • Use sign analysis or interval notation

Functions and Their Properties

• Understanding function notation: $f(x),\, g(x)$

• Evaluating functions for specific values

• Determining domain and range

• Interpreting and analyzing:

  • Graphs of functions

  • Tables of values

• Recognizing whether a relation is a function

• Understanding compositions of functions: $f(g(x))$

  • Domain considerations for composite functions

Quadratic Functions & Graphs

• Graphing quadratics in:

  • Standard form: $f(x)=ax^2+bx+c$

  • Vertex form: $f(x)=a(x-h)^2+k$

  • Factored form: $f(x)=a(x-r<em>1)(x-r</em>2)$

• Finding:

  • Vertex: $V=(h,k)$ where $h= -\frac{b}{2a}$ and $k=f(h)$

  • Axis of symmetry: $x=h$

  • Intercepts: x-intercepts solve $ax^2+bx+c=0$; y-intercept at $x=0$ gives $f(0)=c$

  • Maximum or minimum values depending on sign of $a$

• Understanding how changing coefficients affects the graph

  • Changing a shifts width and direction (up/down) and affects symmetry

  • Changing b shifts the line of symmetry; changing c shifts the graph vertically

Polynomial Expressions

• Performing arithmetic operations:

  • Addition, subtraction, multiplication

  • Division by monomials and binomials

• Factoring expressions, including:

  • Difference of squares: $a^2-b^2=(a-b)(a+b)$

  • Perfect square trinomials: $a^2+2ab+b^2=(a+b)^2$

  • Sum/difference of cubes: $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$

• Using the Remainder Theorem

  • If a polynomial $P(x)$ is divided by $x-c$, the remainder is $P(c)$

• Using the Factor Theorem to find roots of polynomials

  • If $P(c)=0$ then $x-c$ is a factor of $P(x)$

Exponential Expressions & Equations

• Evaluating and simplifying expressions with exponents

• Solving exponential equations (when possible without logs)

• Understanding exponential growth and decay:

  • Model forms:

  • $A = A_0 \cdot r^t$

  • $A = A_0 (1 + r)^t$

• Interpreting exponential models in context (population, interest, etc.)

Rational Expressions

• Simplifying and performing operations on rational expressions

• Solving equations with rational expressions

• Identifying undefined values (values that make the denominator zero)

• Finding common denominators

Radical Expressions & Equations

• Simplifying expressions with square roots

• Rationalizing denominators

• Solving equations with radicals:

  • Example form: $x+1=5\sqrt{x+1}$

  • Squaring both sides may introduce extraneous solutions; always check

• Checking for extraneous solutions

Systems of Equations & Inequalities

• Solving systems of linear equations

• Solving systems with:

  • One linear + one quadratic equation

  • Nonlinear systems algebraically or graphically

• Applications: word problems with systems

Creating & Interpreting Expressions

• Constructing expressions that model real-world scenarios

• Interpreting:

  • Terms in an expression

  • How changes affect the overall output

• Rearranging formulas to isolate a variable:

  • Example: Solve for $r$ in $A = P(1 + r t)$

  • Solve: $A = P(1 + r t) \Rightarrow \frac{A}{P} = 1 + r t \Rightarrow r = \frac{\frac{A}{P}-1}{t}$

Graphing Nonlinear Functions

• Sketching and analyzing:

  • Quadratic graphs

  • Exponential growth/decay graphs

  • Radical and rational graphs (basic)

• Understanding symmetry, asymptotes, and intercepts

Word Problems & Modeling

• Translating word problems into equations

• Interpreting parameters in context:

  • For example, what does the constant term or slope represent in a linear model y = mx + b?

• Using equations and functions to model:

  • Motion

  • Economics

  • Biology, etc.

Additional Context and Connections

• Connections to foundational principles:

  • Domain, range, and function definitions

  • Properties of exponents and radicals

  • Patterns for factoring and solving polynomials

• Practical implications:

  • Modeling real-world situations accurately

  • Recognizing when extraneous solutions may arise (e.g., squaring both sides)

  • Choosing appropriate methods (factoring, formula, graphing, substitution) based on form of the problem