Graphs of Polynomials and Rational Functions with Linear Denominators

Graphs of Polynomials

Polynomials are expressions consisting of variables and coefficients.
Each term in the polynomial can only contain positive integer exponents.
Polynomial functions have graphs that are continuous curves, without any breaks or holes.
The degree of the polynomial function affects the shape of its graph. A polynomial of degree n may have at most n-1 turning points.
The highest exponent in a polynomial, known as the order, determines the end behaviour of the graph:
- Even order: both ends head in the same direction
- Odd order: ends head in opposite directions
The Factor Theorem can be used to find the roots of a polynomial graph, i.e., where it crosses the x-axis.
The roots of the polynomial correspond to its x-intercepts on the graph.

Graphs of Rational Functions with Linear Denominators

A rational function is a function that is the ratio of two polynomials.
For the purpose of this section, we’re focusing on rational functions where the denominator is linear.
The vertical asymptotes of a rational function occur at the zeros of the denominator.
There may be a horizontal asymptote depending on the degrees of the numerator and denominator polynomials.
When the degree of the polynomial in the numerator is less than in the denominator, the x-axis (y=0) is the horizontal asymptote.
When the degrees are the same, the ratio of the leading coefficients gives the equation of the horizontal asymptote.
Holes in the graph of a rational function occur when a factor in the denominator cancels with a factor in the numerator.
The position of the hole can be found by setting the cancelled factor equal to zero.
The y-coordinate of the hole is found by substituting the x-coordinate into the reduced (cancelled) function.

Plotting Graphs of Polynomials and Rational Functions

To plot these graphs, first locate the intercepts (where the graph crosses the axes).
Find the asymptotes if it is a rational function graph.
Plot the general shape according to the degree of the polynomial or the characteristics of the rational function.
Also factor in any local maximum and minimum points and their symmetry if any.

Remember to practise as much as possible. Understanding the characteristics and behaviours of these graph types will help you recognise them and sketch them accurately.