AP Precalculus
~ 1.6-1.7: End Behavior of Functions ~
~ 1.6-1.7: End Behavior of Functions ~
Describe end behaviors of polynomial functions.
As input values of a nonconstant polynomial function increase without bound, the output values will either increase or decrease without bound. Know the corresponding mathematical notation.
As input values of a nonconstant polynomial function decrease without bound, the output values will either increase or decrease without bound. Know the corresponding mathematical notation.
The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function, because as the input values increase or decrease without bound, the values of the leading term dominate the values of all lower-degree terms.
Describe end behaviors of rational functions.
A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function’s domain.
The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude.
For input values of large magnitude, a polynomial is dominated by its leading term.
The end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.
If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line.
If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function.
If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at y = 0.
When the graph of a rational function r has a horizontal asymptote at y = b, where b is a constant, the output values of the rational function get arbitrarily close to b and stay arbitrarily close to b as input values increase or decrease without bound. Know the corresponding mathematical notation.
The video for AP Precalculus, 1.6-1.7: End Behavior of Polynomials and Rational Functions will come out on Thursday, September 21, 2023.