In Precalculus, one essential skill is understanding how quantities change with respect to each other. This concept is often represented through Polynomial and Rational Functions. Let's break down these two types of functions and explore how they help us describe the relationships between quantities.
Polynomial Functions
Imagine polynomial functions as a team of workers, with each worker responsible for a different task based on their position in the hierarchy. In a polynomial function, each "worker" is a term, and their "position" is represented by the exponent of the variable.
A polynomial function takes the form:
f(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_2 * x^2 + a_1 * x + a_0
where:
n is the degree of the polynomial (highest exponent)
a_n, a_(n-1), ..., a_2, a_1, and a_0 are coefficients
The degree of the polynomial determines the function's overall behavior and shape. For example, if the degree is even, the function has the same behavior at both ends (either both up or both down). If the degree is odd, the function has opposite behavior at its ends (one up, one down).
Rational Functions
Rational functions are like a delicate balance between two teams of workers, where one team is represented by a polynomial function in the numerator and the other team by a polynomial function in the denominator. A rational function takes the form:
f(x) = P(x) / Q(x)
where P(x) and Q(x) are both polynomial functions.
The behavior of rational functions depends on the degree of the polynomials in the numerator and denominator. For example, if the degree of the numerator is less than the degree of the denominator, the function approaches zero as x approaches infinity. If the degree of the numerator is equal to the degree of the denominator, the function approaches a constant value as x approaches infinity.
Describing Changes in Quantities
By analyzing polynomial and rational functions, we can describe how quantities change with respect to each other. We can determine relative maximums and minimums, points of inflection, and other key features of the functions that help us understand the relationships between the quantities involved.
To remember the relationship between polynomial and rational functions, think of this mnemonic: Polynomial Workers, Rational Balancers. It reminds us that polynomial functions are like workers with different tasks, while rational functions balance two teams of workers.