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1. Introduction to Polynomial Functions and Number Systems

Polynomial functions build upon concepts from earlier grades (linear in Grade 9, quadratic in Grade 10, and general functions in Grade 11).

• Complex Number System: The real number system is a subset of the complex number system. Imaginary numbers involve $i = \sqrt{-1}$.

• Subsets of Real Numbers:

  • Rational Numbers: Can be expressed as fractions, containing integers, whole numbers (includes $0$), and natural numbers (counting numbers starting at $1$).

  • Irrational Numbers: Real numbers that cannot be written as simple fractions.

2. Characteristics of Power Functions

A power function is the simplest type of polynomial function, written in the form $y = a x^n$.

• Leading Coefficient ($a$): A real number coefficient in front of the term with the highest power.

• Degree ($n$): The highest exponent in the polynomial, which must be a whole number.

• Non-Polynomial Functions: Functions such as trigonometric functions (e.g., $P(x) = \cos(x)$) and exponential functions (e.g., $f(x) = 3^x$) are not polynomial functions.

3. Interval Notation and Graphing

Intervals describe domain, range, and key features of functions:

• Inequality Notation: $-5 < x \le 3$ indicates $x$ lies between $-5$ and $3$, excluding $-5$ and including $3$.

• Bracket Notation: Round brackets indicate non-inclusion (open circle), while square brackets indicate inclusion (closed circle), represented as $(-5, 3]$.

4. End Behavior of Power Functions

• Odd Degree Power Functions:

  • Positive leading coefficient ($a > 0$): Graph extends from Quadrant 3 to Quadrant 1.

  • Negative leading coefficient ($a < 0$): Graph extends from Quadrant 2 to Quadrant 4.

• Even Degree Power Functions:

  • Positive leading coefficient ($a > 0$): Graph extends from Quadrant 2 to Quadrant 1.

  • Negative leading coefficient ($a < 0$): Graph extends from Quadrant 3 to Quadrant 4.