Graphs Study Guide

Polynomial functions are expressions consisting of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Examples include linear functions (like y = 2x + 1), quadratic functions (like y = x^2 - 3x + 2), and cubic functions (like y = x^3 + 2x^2 - x + 5). The highest power of the variable determines the degree of the polynomial. For instance, x^2 - 3x + 2 is a polynomial of degree 2. These functions are smooth and continuous, meaning their graphs have no breaks, jumps, or sharp corners. Their behaviour is determined by their degree and leading coefficient. For example, a quadratic function generally forms a parabola, while a cubic function can have two turning points.

Reciprocal functions, on the other hand, are functions where the variable appears in the denominator, typically in the form y = k/x, where k is a constant. The most basic reciprocal function is y = 1/x. A key characteristic of reciprocal functions is that they are undefined when the denominator is zero. For y = 1/x, this means x cannot be 0. This leads to a vertical asymptote at x = 0, meaning the graph approaches the y-axis but never touches or crosses it. Similarly, as x gets very large (positive or negative), y approaches 0, leading to a horizontal asymptote at y = 0 (the x-axis). The graph of y = 1/x consists of two separate branches in opposite quadrants, illustrating its discontinuous nature. Understanding these fundamental properties - the smooth, continuous nature of polynomials versus the discontinuous nature with asymptotes for reciprocal functions - is crucial for analysing and sketching their graphs.

At foundation level, polynomials are algebraic expressions composed of variables and coefficients, involving only non-negative integer exponents of the variables, and operations of addition, subtraction, multiplication, and non-negative integer exponents. The simplest polynomial functions you will encounter are linear functions (e.g., y = 2x + 1) and quadratic functions (e.g., y = x² - 3x + 2). Polynomial functions have graphs that are smooth and continuous curves with no breaks or sharp corners. Linear graphs are straight lines. Quadratic graphs are parabolas, which are U-shaped curves.

Reciprocal functions, on the other hand, involve a variable in the denominator, usually in the form y = k/x, where k is a constant. The key characteristic of a reciprocal function is that it is undefined when the denominator is zero. For y = k/x, this means that x cannot be 0. As x gets very close to 0 (either from the positive or negative side), the value of y becomes very large (positive or negative). As the absolute value of x becomes very large (either positive or negative), the value of y gets very close to 0.

The graph of a reciprocal function like y = k/x consists of two separate curves, called hyperbolas, in opposite quadrants. These curves get closer and closer to the x and y axes but never actually touch them. The axes serve as asymptotes. For example, the graph of y = 1/x has branches in the first and third quadrants. For y = -1/x, the branches are in the second and fourth quadrants.

When working with these functions, you will often need to calculate values of y for given values of x, or vice versa, by substituting into the equation. You might also be asked to identify a type of graph from its shape or equation. Understanding the basic shapes and properties of these functions is crucial for solving related problems.