Algebra Equations and Functions

Algebra

• Equations and Inequalities
  • Linear equations: ax + by = c, where a, b, and c are constants
  • Quadratic equations: ax^2 + bx + c = 0, where a, b, and c are constants
  • Solving linear and quadratic equations using various methods (e.g. substitution, elimination, quadratic formula)
  • Graphing linear and quadratic equations on the coordinate plane
• Functions
  • Domain and range of a function
  • Composition of functions
  • Inverse functions
  • Graphing functions, including identifying maxima and minima

Geometry

• Points, Lines, and Planes
  • Midpoint and distance formulas
  • Slope of a line
  • Equation of a line (point-slope and slope-intercept forms)
  • Parallel and perpendicular lines
• Triangles and Quadrilaterals
  • Properties of congruent and similar triangles
  • Types of triangles (e.g. isosceles, right, equilateral)
  • Properties of quadrilaterals (e.g. rectangles, squares, trapezoids)

Trigonometry

• Angles and Triangles
  • Degree and radian measures of angles
  • Trigonometric ratios (sine, cosine, tangent)
  • Solving right triangles using trigonometric ratios
  • Graphing trigonometric functions
• Identities and Formulas
  • Pythagorean identity
  • Sum and difference formulas
  • Double and half-angle formulas

Calculus (Introduction)

• Limits and Continuity
  • Basic limit properties and rules
  • Continuity of functions
  • Intermediate Value Theorem
• Derivatives
  • Basic derivative rules (e.g. power rule, product rule)
  • Geometric interpretation of derivatives (e.g. tangent lines, slopes)

Algebra

• Equations and Inequalities
  • Linear equations have the form ax + by = c, where a, b, and c are constants
  • Quadratic equations have the form ax^2 + bx + c = 0, where a, b, and c are constants
  • Methods for solving linear and quadratic equations include substitution, elimination, and the quadratic formula
  • Graphing linear and quadratic equations involves plotting points on the coordinate plane
• Functions
  • The domain of a function is the set of input values, while the range is the set of output values
  • Composition of functions involves combining functions by applying one function to the output of another
  • Inverse functions "undo" each other, and can be used to solve equations
  • Graphing functions helps identify maxima and minima

Geometry

• Points, Lines, and Planes
  • The midpoint formula calculates the midpoint of a line segment: M = ((x1+x2)/2, (y1+y2)/2)
  • The distance formula calculates the distance between two points: √((x2-x1)^2 + (y2-y1)^2)
  • The slope of a line is a measure of its steepness, calculated as m = (y2-y1)/(x2-x1)
  • The equation of a line can be written in point-slope or slope-intercept form
  • Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals
• Triangles and Quadrilaterals
  • Congruent triangles have equal corresponding sides and angles, while similar triangles have proportional corresponding sides
  • There are different types of triangles, including isosceles, right, and equilateral
  • Quadrilaterals can be classified as rectangles, squares, trapezoids, and more, each with unique properties

Trigonometry

• Angles and Triangles
  • Angles can be measured in degrees or radians
  • Trigonometric ratios include sine, cosine, and tangent, which are used to solve right triangles
  • Graphing trigonometric functions involves plotting periodic curves
• Identities and Formulas
  • The Pythagorean identity states sin^2(θ) + cos^2(θ) = 1
  • The sum and difference formulas are used to find trigonometric ratios for sums and differences of angles
  • The double and half-angle formulas are used to find trigonometric ratios for double and half angles

Calculus (Introduction)

• Limits and Continuity
  • The basic limit properties include the sum and product rules, and the squeeze theorem
  • Continuity of functions is important for many calculus concepts
  • The Intermediate Value Theorem states that a continuous function on a closed interval takes on every value between its maximum and minimum
• Derivatives
  • The power rule is a basic derivative rule, stating that if f(x) = x^n, then f'(x) = nx^(n-1)
  • The product rule states that if f(x) = g(x) * h(x), then f'(x) = g'(x) * h(x) + g(x) * h'(x)
  • Geometrically, derivatives represent the slope of a tangent line to a curve