Algebra: Linear Equations and Quadratic Functions

Linear Equations

• Definition: An equation that makes a straight line when graphed; in the form y = mx + b.
• Slope (m): Represents the rate of change; calculated as (y2 - y1) / (x2 - x1).
• Y-intercept (b): The point where the line crosses the y-axis (x=0).
• Solution: The point(s) (x, y) that satisfy the equation.
• Standard form: Ax + By = C, where A, B, and C are constants.

Quadratic Functions

• Definition: A polynomial function of degree 2, typically in the form f(x) = ax² + bx + c.
• Graph: Parabola; opens upwards if a > 0 and downwards if a < 0.
• Vertex: The highest or lowest point of the parabola; found using x = -b/(2a).
• Roots/Zeros: Solutions to the equation ax² + bx + c = 0; can be found using:
  • Factoring
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
• Discriminant (D): b² - 4ac; determines the nature of the roots:
  • D > 0: 2 distinct real roots
  • D = 0: 1 real root (repeated)
  • D < 0: No real roots

Polynomials

• Definition: An expression consisting of variables, coefficients, and exponents; general form: a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0.
• Degree: The highest exponent of the variable in the polynomial.
• Leading Coefficient: The coefficient of the term with the highest degree.
• Types:
  • Monomial: One term (e.g., 3x²)
  • Binomial: Two terms (e.g., x² + 4)
  • Trinomial: Three terms (e.g., x² + 5x + 6)

Factoring

• Definition: Rewriting an expression as a product of its factors.
• Common methods:
  • Factoring out the greatest common factor (GCF).
  • Factoring by grouping.
  • Using special products (e.g., difference of squares, perfect square trinomials).
  • Quadratic trinomials: ax² + bx + c can often be factored into (px + q)(rx + s).

Inequalities

• Definition: Mathematical statements that relate expressions that are not equal; includes <, >, ≤, and ≥.
• Solution: Set of values that make the inequality true.
• Graphing:
  • Use a number line for one variable; open circles for < or >, closed circles for ≤ or ≥.
  • For two variables, graph the boundary line and shade the appropriate region.
• Types:
  • Linear inequalities: Can be expressed in the form ax + by < c.
  • Quadratic inequalities: Involves a quadratic expression; solved by determining the intervals of x where the inequality holds.

Use these notes as a quick reference to understand the foundational concepts of algebra.

Linear Equations

• Characterized by a linear graph; standard form is y = mx + b.
• Slope (m) shows how steep the line is, calculated from two points: (y2 - y1) / (x2 - x1).
• Y-intercept (b) indicates where the line intersects the y-axis, occurring at coordinates (0, b).
• Solutions are the specific points (x, y) that make the equation true.
• Standard form of a linear equation is Ax + By = C, involving constants A, B, and C.

Quadratic Functions

• Defined as a degree 2 polynomial, generally expressed as f(x) = ax² + bx + c.
• The graph resembles a parabola; opens upwards if the leading coefficient (a) is positive, downwards if negative.
• Vertex is the peak or trough of the parabola, calculated using x = -b / (2a).
• Roots/Zeros are x-values that solve the equation ax² + bx + c = 0, found through:
  • Factoring methods.
  • Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
• Discriminant (D) helps identify root characteristics:
  • D > 0: Two distinct real roots.
  • D = 0: One real root, which is repeated.
  • D < 0: No real roots exist.

Polynomials

• Comprised of variables, coefficients, and exponents; general form is a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0.
• Degree indicates the highest power of the variable present in the polynomial.
• Leading Coefficient is the coefficient associated with the term of the highest degree.
• Types are identified by the number of terms:
  • Monomial: Consists of one term (e.g., 3x²).
  • Binomial: Contains two terms (e.g., x² + 4).
  • Trinomial: Made up of three terms (e.g., x² + 5x + 6).

Factoring

• Involves rewriting a polynomial as a product of its factors.
• Common factoring techniques include:
  • Extracting the greatest common factor (GCF).
  • Factoring by grouping.
  • Utilizing special product identities, such as difference of squares or perfect square trinomials.
  • Quadratic trinomials, ax² + bx + c, are often factored as (px + q)(rx + s).

Inequalities

• Represent relationships where expressions are not equal, using symbols such as <, ≤, and ≥.
• Solutions denote the range of values that satisfy the inequality.
• Graphing strategies:
  • Use a number line for single-variable inequalities; apply open circles for < or > and closed circles for ≤ or ≥.
  • For two variables, draw the boundary line and shade the corresponding area to indicate solutions.
• Types include:
  • Linear inequalities fitting the form ax + by < c.
  • Quadratic inequalities involve quadratic expressions and require identifying valid x-intervals.