Math 2 Component Overview

Introduction

• Welcome to Math 2 of the online BAC program in Data Science and Programming.
• Review of concepts learned in Math 1, focusing on calculus and functions.

Key Topics from Math 1

• Functions: Including the definitions and examples.
• Straight Lines: Definition and properties.
• Quadratic Equations: Characteristics and forms.
• Polynomials: General understanding and examples.
• Exponential and Logarithmic Functions: Basic properties and behaviors.

Function of One Variable

• A function is a relation mapping inputs (domain) to exactly one output (codomain).
• Domain: Set of inputs.
• Codomain: Set of possible outputs.
• Range: Set of outputs actually produced by the function.
  • Example: If f(x) = m*x + c, then f(0) = c (y-intercept) and m is the slope of the line.

Linear Functions

• Form: f(x) = mx + c
  • m = slope, c = y-intercept.
• Graph is a straight line.
• Example: For f(x) = 5x + 2:
  • f(0) = 2
  • f(20) = 102.

Quadratic Functions

• Form: f(x) = a(x - b)² + c.
• Graph shape: Parabola.
  • Upwards if a > 0, downwards if a < 0.
• Vertex at (b, c).
• Example expansion: f(x) = ax² + bx + c.

Polynomial Functions

• General form: f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_0.
• Degree n polynomial
• Example: f(x) = x³ - 4x.

Exponential and Logarithmic Functions

• Exponential: f(x) = a^x.
  • Defined for all real numbers.
• Logarithmic: Inverse of exponential.
  • Function defined only for positive outputs of exponential functions.
• Relationship: log_a(a^x) = x.

Growth Comparisons of Functions

• Exponential functions grow faster than polynomials.
• Visual comparison of growth rates:
  • Polynomial functions vs. exponential functions.
  • Slow-growing functions: 0.8 power, root functions, and logarithms.

Tangent Lines

• Definition: A line that touches a curve at exactly one point.
• Examples: Tangents to y = x², y = x³, and y = 2^x.
• Importance: Relates to calculus for finding the equation of the tangent line.

Conclusion

• This overview reviews core ideas from Math 1.
• Students are encouraged to revisit Math 1 resources for deeper understanding.