Linear Systems, Functions & Polynomials

Functions and polynomials provide the language for modeling relationships in science, from population growth to reaction rates.

Linear Functions A linear function has the form y = mx + b, where:

m = slope (rate of change)

b = y-intercept (value where the line crosses the y-axis)

Slope Slope measures how steep a line is:

m = (change in y) / (change in x) = (y₂ - y₁) / (x₂ - x₁)

Positive slope means y increases as x increases; negative slope means y decreases.

Slope-Intercept Form y = mx + b is the most common form because it directly shows slope and intercept.

Example: y = 3x + 2 has slope 3 and y-intercept 2.

Rate of Change In real-world contexts, slope represents rate of change: mg/hour, mL/min, cells/hour, etc.

Function Notation Function notation f(x) shows that the output depends on the input x. For example, f(x) = 2x + 1 means the output is twice the input plus one.

Evaluating Functions Substitute the given input for x and simplify.

Example: If f(x) = 3x - 1, then f(4) = 3(4) - 1 = 11.

Polynomials A polynomial is an expression made of variables and coefficients using only addition, subtraction, multiplication, and non-negative integer exponents.

Examples of polynomials: x² + 3x - 5, 4x⁵, 7 Not polynomials: √x (radical), log(x) (logarithm), 1/x (rational with negative exponent)