Polynomial and Rational Functions
In AP Precalculus, understanding polynomial and rational functions is essential for analyzing patterns, modeling real-world situations, and preparing for calculus. These types of functions extend what we learn about linear and quadratic equations and introduce more complex behaviors in graphs, roots, and asymptotes.
A polynomial function is a function made up of terms that involve variables raised to whole-number exponents. The general form of a polynomial is
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀,
where aₙ ≠ 0 and n is a non-negative integer. Each term includes a coefficient (aₖ) and a power of x. The highest power of x determines the degree of the polynomial, which influences the function’s end behavior and the possible number of turning points.
Polynomials can be classified by degree and number of terms. For example, a degree 2 polynomial is a quadratic, a degree 3 is a cubic, and a degree 4 is quartic. The end behavior of a polynomial graph depends on the degree and the leading coefficient. For instance, if the degree is even and the leading coefficient is positive, both ends of the graph rise; if the degree is odd, one end rises while the other falls.
Finding real zeros (or roots) of a polynomial is key to understanding its graph. These are the x-values where the function equals zero—where the graph crosses or touches the x-axis. Techniques such as factoring, synthetic division, and the Rational Root Theorem help identify these values. The multiplicity of a zero describes how many times it appears as a factor. A zero with even multiplicity causes the graph to touch and bounce off the x-axis, while odd multiplicity causes it to cross.
Rational functions, on the other hand, are the ratio of two polynomials. A general rational function has the form
f(x) = p(x)/q(x),
where p(x) and q(x) are polynomial functions and q(x) ≠ 0. These functions are more complex because they involve division, which introduces restrictions on the domain—values of x that make the denominator zero are undefined.
Rational functions often have asymptotes, which are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator equals zero (after simplifying), while horizontal or oblique asymptotes describe end behavior. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator. If the numerator’s degree is higher, the function has no horizontal asymptote, but it may have an oblique one.
Graphing rational functions involves identifying asymptotes, intercepts, holes (points of discontinuity caused by common factors in the numerator and denominator), and overall behavior. These graphs help visualize how rational functions behave near undefined values and at extreme x-values.
In summary, polynomial and rational functions form a core part of AP Precalculus. They represent a bridge between algebra and calculus, offering powerful tools for modeling and analysis. Whether examining the motion of a ball, the shape of a bridge, or the behavior of a market curve, these functions help describe real-world situations with precision and depth.